HMH Go Math

Guided Practice – Page No. 268

Find the circumference of each circle.

Question 1.

________ in

Answer: 56.57 in

Explanation:
Circumference of the circle = 2πr = 2 x 22/7 x 9 = 56.57 in

Question 2.

________ cm

Answer: 44 cm

Explanation:
Circumference of the circle = 2πr = 2 x 22/7 x 7 = 44 cm

Find the circumference of each circle. Use 3.14 or \(\frac{22}{7}\) for π. Round to the nearest hundredth, if necessary.

Question 3.
______ m

Question 4.

______ yd

Answer: 30.15 yd

Explanation:
Circumference of the circle = 2πr = 2 x 3.14 x 4.8 = 30.144 yd

Question 5.

______ in

Answer: 7.5 in

Explanation:
Circumference of the circle = 2πr = 2 x 3.14 x 7.5 = 47.1 in

9.1 Circles and Circumference Answer Key Question 6.
A round swimming pool has a circumference of 66 feet. Carlos wants to buy a rope to put across the diameter of the pool. The rope costs $0.45 per foot, and Carlos needs 4 feet more than the diameter of the pool. How much will Carlos pay for the rope?
$ ______

Answer: $6.525

Explanation:
Circumference of the swimming pool = 66 feet
πd = 66
22/7 x d = 66
d = 66 x 7/ 22 = 10.5
The diameter of the pool = 10.5 feet
Carlos needs 4 feet more than the diameter of the pool.
Total rope needed = 10.5 + 4 = 14.5 feet
Cost of rope per foot = $0.45
Total cost of the rope = 14.5 x $0.45 = $6.525
Therefore the total cost of the rope = $6.525

Find each missing measurement to the nearest hundredth. Use 3.14 for π.

Question 7.
r =
d =
C = π yd
r = ________ yd
d = ________ yd

Answer:
r = 0.5 yd
d = 1 yd

Explanation:
Circumference = π yd
2πr = π yd
r = 1/2 yd = 0.5 yd
d = 2r = 2 [1/2] = 1 yd

Question 8.
r ≈
d ≈
C = 78.8 ft
r ≈ ________ ft
d ≈ ________ ft

Answer:
r = 495.31 ft
d = 990.62 ft

Explanation:
Circumference = 78.8 ft
2πr = 78.8 ft
r = 2 x 22/7 x 78.8 = 495.31 ft
d = 2 x 495.31 = 990.62 ft

Question 9.
r ≈
d ≈ 3.4 in
C =
r ≈ ________ in
C = ________ in

Answer:
r = 1.7 in
c = 10.68 in

Explanation:
Diameter = 3.4 in
Circumference = πd = 22/7 x 3.4 in = 10.68 in
r = d/2 = 1.7 in

Essential Question Check-In

Question 10.
Norah knows that the diameter of a circle is 13 meters. How would you tell her to find the circumference?
Type below:
____________

Answer: Circumference = 16.82 meters

Explanation:
Given,
Diameter = 13 meters
Circumference = πd = 22/7 x 13 = 16.82 meters

Independent Practice – Page No. 269

For 11–13, find the circumference of each circle. Use 3.14 or \(\frac{22}{7}\) for π. Round to the nearest hundredth, if necessary.

Question 11.

_______ ft

Answer:
Circumference = 18.526 ft = 19 ft (approx)

Explanation:
Given:
Diameter = 5.9 ft
Circumference = πd = 3.14 x 5.9 = 18.526 ft = 19 ft (approx)

9.1 Circles and Circumference Answer Key Question 12.

_______ cm

Answer:
Circumference =176 cm

Explanation:
Given:
Radius = 56 cm
Circumference = πd = 22/7 x 56 = 176 cm

Question 13.

_______ in

Answer:
Circumference = 110 in

Explanation:
Given:
Diameter = 35 in
Circumference = πd = 22/7 x 35 = 110 in

Question 14.
In Exercises 11–13, for which problems did you use \(\frac{22}{7}\) for π? Explain your choice.
Type below:
_____________

Answer:
The 11th question is 3.14 and the 12 and 13 questions as π

Explanation:
We can take 3.14 as π for 11 th question because the diameter is given in decimal points.
And in questions 12 and 13 we need to take π because the radius and diameter are given in whole number form.

Question 15.
A circular fountain has a radius of 9.4 feet. Find its diameter and circumference to the nearest tenth.
d = _________ ft
C = _________ ft

Answer:
d = 19 ft
C = 59 ft

Explanation:
Given:
Radius = 9.4 ft
Diameter = 2r = 2 x 9.4  = 18.8 ft = 19 ft (approx)
Circumference = πd = 22/7 x 18.8 = 59.08 = 59 ft (approx)

Question 16.
Find the radius and circumference of a CD with a diameter of 4.75 inches.
r = _________ in
C = _________ in

Answer:
r = 2.4 in
C = 15 in

Explanation:
Given:
Diameter = 4.75 in
Radius = r/2 = 4.75/2 = 2.37 in = 2.4 in (approx)
Circumference = πd = 22/7 x 4.75 = 14.92 in =15 in (approx)

Question 17.
A dartboard has a diameter of 18 inches. What are its radius and circumference?
r = _________ in
C = _________ in

Answer:
r = 9 in
C = 56.6 in

Explanation:
Given:
Diameter = 18 in
Radius = r/2 = 18/2 = 9 in
Circumference = πd = 22/7 x 18 = 56.57 in = 56.6 in (approx)

Question 18.
Multistep
Randy’s circular garden has a radius of 1.5 feet. He wants to enclose the garden with edging that costs $0.75 per foot. About how much will the edging cost? Explain.
$ _______

Answer:

Explanation:
Given:
The radius of the garden= 1.5 ft
Circumference of the garden = 2πr = 2 x 22/7 x 1.5 = 9.42 ft
The cost of enclosing the garden per foot = $0.75
Total cost of edging = 9.42 x $0.75 = $7.06 = $7 (approx)

Question 19.
Represent Real-World Problems
The Ferris wheel shown makes 12 revolutions per ride. How far would someone travel during one ride?

_______ ft

Answer: Total distance travelled in one ride is 4,752 ft

Explanation:
Given:
The diameter of the Ferris wheel= 63 ft
Circumference of the Ferris wheel = 2πr = 2 x 22/7 x 63 = 396 ft
Total number of revolutions = 12
Total distance travelled = 12 x 396 = 4,752 ft

Question 20.
The diameter of a bicycle wheel is 2 feet. About how many revolutions does the wheel make to travel 2 kilometers? Explain. Hint: 1 km ≈ 3,280 ft
_______ revolutions

Answer:
1044 revolutions

Explanation:
Given:
The diametre of the bicycle wheel = 2 feet
Total distance travelled = 2 kilometres
We know that,
1 km ≈ 3,280 ft
2 km = 2 x 3,280 = 6,560 ft
Circumference of the bicycle = Distance travelled in one revolution = πd = 22/7 x 2 = 6.28 ft = 6.3 ft
Total number of revolutions = Total distance traveled / distance traveled in one revolution
= 6560 / 6.28 = 1044  revolutions

9.1 Independent Practice Question 21.
Multistep
A map of a public park shows a circular pond. There is a bridge along the diameter of the pond that is 0.25 mi long. You walk across the bridge, while your friend walks halfway around the pond to meet you at the other side of the bridge. How much farther does your friend walk?
_______ mi

Answer:

Explanation:
Given,
The diameter of the pond = 0.25 mi
The length of the bridge = The diameter of the pond = 0.25 mi
Then the distance walked by the man = 0.25 mi
Distance travelled by the friend = Halfway around the pond to meet you at the other side of the bridge = πd/2
= 22/7 x 0.25/2  = 0.39 = 0.4 mi
The friend travelled more distance compared to the man
The more distance travelled by the friend = 0.39 – 0.25 = 0.14 mi

Page No. 270

Question 22.
Architecture
The Capitol Rotunda connects the House and the Senate sides of the U.S. Capitol. Complete the table. Round your answers to the nearest foot.

Type below:
_____________

Answer:
Radius = 48 ft
Diameter = 96 ft

Explanation:
Given
Height = 180 ft
Circumference = 301.5 ft
πd = 301.5
22/7 x d = 301.5
d = 95.93 = 96 ft
r = d/2 = 96/2 = 48 ft

H.O.T.

Focus on Higher Order Thinking

Question 23.
Multistep
A museum groundskeeper is creating a semicircular statuary garden with a diameter of 30 feet. There will be a fence around the garden. The fencing costs $9.25 per linear foot. About how much will the fencing cost altogether?
$ _______

Answer:
The total cost of fencing = $712

Explanation:
Given,
The diameter = 30 ft
Circumference of the garden in the shape of circle = 2πr
Circumference of the semicircle = πr = πd/2 =  22/7 x 30/2 = 47.14ft
Cost of fencing for each foot = $9.25
The total cost of fencing the semicircular garden = 47.14 x $9.25 + 30 x  $9.25  = $712 (approx)

Question 24.
Critical Thinking
Sam is placing rope lights around the edge of a circular patio with a diameter of 18 feet. The lights come in lengths of 54 inches. How many strands of lights does he need to surround the patio edge?
_______ strands

Answer: 12 and a half strands of light = 13 strands (approx)

Explanation:
Given,
The diameter of the circular patio = 18 ft = 216 inch
Circumference of the circular patio = πd = 22/7 x 216 = 678.85 inch
The lights will come in a length (in one strand)= 54 inches
Total number of strands of light required for the circular patio
= Circumference of the circular patio/ The lights will come in a length (in one strand) = 678.85/54 = 12.57 = 12 and a half strands of light

Question 25.
Represent Real-World Problems
A circular path 2 feet wide has an inner diameter of 150 feet. How much farther is it around the outer edge of the path than around the inner edge?
_______ feet

Answer: about 12.6 ft

Explanation:
Given,
Width of the circular path = 2 ft
The inner diameter of the circular path = 150 ft
The outer diameter of the circular path = 150 + 2(2) = 154 ft
Inner circumference = πd = 150 π
Outer circumference =  πd = 154π
Distance between the outer and inner edge = 154 π – 150 π = 4 π = 12.6 ft

Circumference Area and Volume Class 7 Question 26.
Critique Reasoning
The gear on a bicycle has the shape of a circle. One gear has a diameter of 4 inches, and a smaller one has a diameter of 2 inches. Justin says that the circumference of the larger gear is 2 inches more than the circumference of the smaller gear. Do you agree? Explain your answer.
_______

Answer:
Justin’s statement is incorrect.

Explanation:
The circumference of the larger gear = πd = 4π
The circumference of the smaller gear = πd = 2π
Since, 2 x 2π = 4π, the circumference of the larger gear is two times the circumference of the smaller gear.
Since = 4π – 2π = 2π = 6.28
Therefore, The larger circumference is not 2 inches more than the smaller circumference

Question 27.
Persevere in Problem-Solving
Consider two circular swimming pools. Pool A has a radius of 12 feet, and Pool B has a diameter of 7.5 meters. Which pool has a greater circumference? How much greater? Justify your answers.
_______

Answer:
Pool B about 0.9 meters

Explanation:
Given,
Pool A has a diameter = 24 ft
Pool B has a diameter = 7.5 m
We know that,
1 ft = 0.3 metres
24 ft = 7.2 metres
Pool B has a greater diameter so it has a greater circumference.
Circumference of the pool A = 7.2π
Circumference of the pool B = 7.5π
Difference between the circumferences = 7.5π – 7.2π = 0.9 meters.

Guided Practice – Page No. 274

Find the area of each circle. Round to the nearest tenth if necessary. Use 3.14 for π.

Question 1.

_______ m²

Answer: 153.9 m²

Explanation:
Given:
Diameter = 14 m
Radius = 14/2 = 7 m
Area of the circle = πr²
= 3.14 x 7 x 7 = 153.86 = 153.9 m²

Question 2.

_______ mm²

Answer: 452.2 mm²

Explanation:
Given:
Radius =12mm
Area of the circle = πr²
= 3.14 x 12 x 12 = 3.14(144) = 452.2mm²

Question 3.

_______ yd²

Answer: 314 yd²

Explanation:
Given:
Diameter = 20yd
Radius = 20/2 = 10yd
Area of the circle = πr²
= 3.14 x 10 x 10 = 3.14(100) = 314yd²

Solve. Use 3.14 for π.

Question 4.
A clock face has a radius of 8 inches. What is the area of the clock face? Round your answer to the nearest hundredth.
_______ in²

Answer: 200.96 in²

Explanation:
Given:
Radius = 8inches
Area of the clock face = πr²
= 3.14 x 8 x 8= 3.14(64) = 200.96 in²

Chapter 9 Circumference, Area and Volume Answer Key Question 5.
A DVD has a diameter of 12 centimeters. What is the area of the DVD? Round your answer to the nearest hundredth.
_______ cm²

Answer: 113.04 cm²

Explanation:
Given:
Diameter = 12 centimeters
Radius = 12/2 = 6 centimeters
Area of the DVD= πr²
= 3.14 x 6 x 6 = 3.14(36) = 113.04 cm²

Question 6.
A company makes steel lids that have a diameter of 13 inches. What is the area of each lid? Round your answer to the nearest hundredth.
_______ in²

Answer: 132.67 in²

Explanation:
Given:
Diameter = 13 inches
Radius = 13/2 = 6.5 inches
Area of each lid= πr²
= 3.14 x 6.5 x 6.5 = 3.14(42.25) = 132.67 in²

Find the area of each circle. Give your answers in terms of π.

Question 7.
C = 4π
A =
Type below:
______________

Answer: 4π

Explanation:
Given:
Circumcenter = 4π
2πr = 4π
Radius = 4/2 = 2 units
Area of the circle = πr²
= π x 2 x 2 = π(4) = 4π square units

Question 8.
C = 12π
A =
Type below:
______________

Answer: 36π

Explanation:
Given:
Circumcenter = 12π
2πr = 12π
Radius =6 units
Area of the circle = πr²
= π x 6 x 6 = π(36) = 36π square units

Question 9.
C = \(\frac{π}{2}\)
A =
Type below:
______________

Answer: π/16

Explanation:
Given:
Circumcenter = \(\frac{π}{2}\)
2πr = \(\frac{π}{2}\)
Radius = 1/4 units
Area of the circle = πr²
= π x 1/4 x 1/4 = π(1/16) = π/16 square units

Lesson 9.2 Area of Circles Practice and Problem-Solving a/b Answers Question 10.
A circular pen has an area of 64π square yards. What is the circumference of the pen? Give your answer in terms of π
Type below:
______________

Answer: 16π

Explanation:
Given:
Area of the circular pen = 64π square yards
πr² = 64π
r = 8 yards
Circumference of the circle = 2πr = 2 x 8 x π = 16π yards

Essential Question Check-In

Question 11.
What is the formula for the area A of a circle in terms of the radius r?
Type below:
______________

Answer: πr²

Explanation:
Area of a circle = πr²

Independent Practice – Page No. 275

Question 12.
The most popular pizza at Pavone’s Pizza is the 10-inch personal pizza with one topping. What is the area of a pizza with a diameter of 10 inches? Round your answer to the nearest hundredth.
_______ in²

Answer: 78.5 in²

Explanation:
Given:
Diameter = 10 inches
Radius = 10/2 = 5 inches
Area of a pizza = πr²
= 3.14 x 5 x 5 = 3.14(25) = 78.5 in²

Question 13.
A hubcap has a radius of 16 centimeters. What is the area of the hubcap? Round your answer to the nearest hundredth.

_______ cm²

Answer: 803.84 cm²

Explanation:
Given:
Radius = 16 cm
Area of the circle = πr²
= 3.14 x 16 x 16 = 3.14(256) = 803.84 cm²

Question 14.
A stained glass window is shaped like a semicircle. The bottom edge of the window is 36 inches long. What is the area of the stained glass window? Round your answer to the nearest hundredth.
_______ in²

Answer: 508.68 in²

Explanation:
Area of the semicircle = 1/2 πr² = 1/2(3.14)(18)(18) = 1/2 (3.14)(324) = 1.57(324) = 508.68 in ²

Question 15.
Analyze Relationships
The point (3,0) lies on a circle with the centre at the origin. What is the area of the circle to the nearest hundredth?
_______ units²

Answer: 28.26 units²

Explanation:
Radius = 3
Area of the circle = πr² = π(3)² = 3.14(9) = 28.26 units²

Question 16.
Multistep
A radio station broadcasts a signal over an area with a radius of 50 miles. The station can relay the signal and broadcast over an area with a radius of 75 miles. How much greater is the area of the broadcast region when the signal is relayed? Round your answer to the nearest square mile.
_______ mi²

Answer: 9813 mi²

Explanation:
Given:
The radius of a radio station broadcasting the signal (r) = 50 miles
The greatest radius to which the broadcast can be relayed (R) = 75 miles
The greatest area of the broadcast region when the signal is relayed = πR²-πr² = π(75) (75) – π (50) (50)
= 5625π – 2500π
= 3125π
= 3125(3.14) = 9813 mi²(approx)

Chapter 9 Practice Test Surface Area Question 17.
Multistep
The sides of a square field are 12 meters. A sprinkler in the center of the field sprays a circular area with a diameter that corresponds to a side of the field. How much of the field is not reached by the sprinkler? Round your answer to the nearest hundredth.
_______ m²

Answer:30.96 m²

Explanation:
Given:
The side of the square = 12 meters
The diameter circular area of the field in the centre = The side of the square = 12 meters
The radius of the field = 12/2 = 6 meters
Area of the field which is not reached by the sprinkler = Area of the square – Area of the circular area
= (side)²-πr² = (12)(12) – π (6) (6)
= 144 – 36 (3.14)
= 144 – 113.04
= 30.96 m²

Question 18.
Justify Reasoning
A small silver dollar pancake served at a restaurant has a circumference of 2π inches. A regular pancake has a circumference of 4π inches. Is the area of the regular pancake twice the area of the silver dollar pancake? Explain.
_______

Answer: No, the area of the regular pancake is 4 times the area of the silver dollar pancake

Explanation:
Silver Dollar pancake:
Circumference of the silver Dollar pancake = 2π inches
2πr = 2π
r = 1 inch
Area of the silver dollar pancake = πr² = π (1) (1) = π in²

Regular pancake:
Circumference of the regular pancake = 4π inches
2πr = 4π
r = 2 inch
Area of the silver dollar pancake = πr² = π (2) (2) = 4π in²

Therefore, the area of the regular pancake is 4 times the area of the silver dollar pancake

Question 19.
Analyze Relationships
A bakery offers a small circular cake with a diameter of 8 inches. It also offers a large circular cake with a diameter of 24 inches. Does the top of the large cake have three times the area of that of the small cake? If not, how much greater is its area? Explain.
_______

Answer: No, the area of the large cake is 9 times the area of the small cake

Explanation:
Small Cake:
The diameter of the small cake= 8 inches
The radius of the small cake = 8/2 = 4 inches
Area of the small cake  = πr² = π (4) (4) = 16 π in²

Large Cake:
The diameter of the large cake= 24 inches
The radius of the large cake = 24/2 = 12 inches
Area of the large cake  = πr² = π (12) (12) = 144 π in²

Since 144 π/ 16 π = 9
Therefore the

area of the large cake is 9 times the area of the small cake.

Page No. 276

Question 20.
Communicate Mathematical Ideas
You can use the formula A = \(\frac{C^{2}}{4π}\) to find the area of a circle given the circumference. Describe another way to find the area of a circle when given the circumference.
Type below:
____________

Answer: Area = C²/4π

Explanation:
Circumference of the circle = 2πr
C = 2πr
Divide both sides by 2π
then, r = C/2π
Area of the circle = πr²
Substitute C/2π for r:
Area = π(c/2π)² = C²/4π

Question 21.
Draw Conclusions
Mark wants to order a pizza. Which is the better deal? Explain.

_____

Answer: The pizza of 18 inches is a better deal

Explanation:
Given:
The diameter of the pizza = 12 inches
The radius of the pizza = 12/2= 6 inches
Area of the circle = πr²
= (3.14)(6)(6) = 113 (approx) in²
The total cost of the pizza = $10
Cost of the pizza per inch = $10/113 = $0.09 per square inch

The diameter of the pizza = 18 inches
The radius of the pizza = 18/2= 9 inches
Area of the circle = πr²
= (3.14)(9)(9) = 254 (approx) in²
The total cost of the pizza = $20
Cost of the pizza per inch = $20/254 = $0.08 per inch

Question 22.
Multistep
A bear was seen near a campground. Searchers were dispatched to the region to find the bear.
a. Assume the bear can walk in any direction at a rate of 2 miles per hour. Suppose the bear was last seen 4 hours ago. How large an area must the searchers cover? Use 3.14 for π. Round your answer to the nearest square mile.
_____ mi²

Answer: 201mi²

Explanation:
The bear can walk a distance = 2 x 4 = 8 miles
Since it is walking 2 miles per hour for 4 hours
The radius of the bear = 8 miles
Area of the circle = πr²
= (3.14)(8)(8) = 201 (approx) mi²

Question 22.
b. What If? How much additional area would the searchers have to cover if the bear were last seen 5 hours ago?
_____ mi²

Answer: 113mi²

Explanation:
If the bear for 5 hours then,
The bear can walk a distance = 2 x 5 = 10 miles
Since it is walking 2 miles per hour for 5 hours
The radius of the bear = 10 miles
Area of the circle = πr²
= (3.14)(10)(10) = 314 (approx) mi²

The additional area covered by the searches = 314 – 201 = 113 mi²

H.O.T.

Focus on Higher Order Thinking

Question 23.
Analyze Relationships
Two circles have the same radius. Is the combined area of the two circles the same as the area of a circle with twice the radius? Explain.
_____

Answer: No

Explanation:
If the radius of two circles is the same.
then, Let the radii of the circles be 1.
The area of each circle =  π square units
The combined area of 2 circles =π+π = 2π square units

If the radius is doubled.
then, Let the radii of the circles be 2
The area of each circle =  4π square units
The combined area of 2 circles =  4π+4π = 8π square units

Therefore the areas of both cases are not the same.

Question 24.
Look for a Pattern
How does the area of a circle change if the radius is multiplied by a factor of n, where n is a whole number?
Type below:
____________

Answer: The new area is then n² times the area of the original circle.

Explanation:
If the radius is multiplied by a factor “n”
then, the new radius = rn
The area of the circle (with radius rn) = π(rn)² = n² (πr²).
Therefore the new area is n² times the area of the original circle.

Question 25.
Represent Real World Problems
The bull’s-eye on a target has a diameter of 3 inches. The whole target has a diameter of 15 inches. What part of the whole target is the bull’s-eye? Explain.
Type below:
____________

Answer: 1/25 of the target

Explanation:
Bull’s eye:
Diameter of Bull’s eye = 3 inches
Radius of Bull’s eye = 3/2 = 1.5 inches
Area of the Bull’s eye = π(r)² = π(1.5)² = 2.25π
Target:
Diameter of the target = 15 inches
Radius of the target = 15/2 = 7.5 inches
Area of the target = π(r)² = π(7.5)² = 56.25π

The part of Bull’s eye in the whole target = 2.25π/ 56.25π = 1/25

Therefore the 1/25th part of the whole target is the Bull’s eye.

Guided Practice – Page No. 280

Question 1.
A tile installer plots an irregular shape on grid paper. Each square on the grid represents 1 square centimeter. What is the area of the irregular shape?

_____ cm²

Answer: Area of the irregular shape = 34 cm²

Explanation:
STEP1 First divide the irregular shapes into polygons.
STEP2 The irregular shape can be divided into a triangle, rectangle, parallelogram
STEP3 Areas of the polygons
Area of triangle = 1/2 (base x height) = 1/2 (4 x 2) = 4 cm²
Area of the rectangle = length x breadth = 5 x 3 = 15 cm²
Area of the parallelogram = base x height = 5 x 3 = 15 cm²
Area of the irregular shape = (15+15+5) cm²= 34cm²

Question 2.
Show two different ways to divide the composite figure. Find the area both ways. Show your work below.

_____ cm²

Answer: Area of the figure in both ways = 288 cm²

Explanation:
The first way to divide up the composite shape is to divide it into an 8 by 9 rectangle and a 12 by 18 rectangle.
The area of the first rectangle = Length x breadth = 9 x 8 = 72 cm²
The area of the second rectangle =  Length x breadth = 18 x 12 = 216 cm²
The total area of the figure = 72 + 216 = 288 cm²

Question 3.
Sal is tiling his entryway. The floor plan is drawn on a unit grid. Each unit length represents 1 foot. Tile costs $2.25 per square foot. How much will Sal pay to tile his entryway?

$ _____

Answer: Sal will pay $97.875

Explanation:
Separate this figure into trapezium and parallelogram.
Area of the trapezium = 1/2 (a+b)h = 1/2 (7+4) 5 = 1/2 (11) 5 = 27.5 ft²
Area of the parallelogram = base x height = 4 x 4 = 16 ft²

The total area of the figure = 27.5 + 16 = 43.5ft²
Cost of each square foot = $2.25
Amount paid by Sal = 43.5 x 2.25 = $97.875

Essential Question Check-In

Question 4.
What is the first step in finding the area of a composite figure?
Type below:
______________

Answer:
The first step in finding the area of a composite figure is to divide it up into smaller basic shapes.

Explanation:
The first step in finding the area of a composite figure is to divide it up into smaller basic shapes such as triangles, squares, rectangles, parallelograms, circles and trapezium.
Then calculate the area of each figure and add them to find the area of the figure.

Independent Practice – Page No. 281

Question 5.
A banner is made of a square and a semicircle. The square has side lengths of 26 inches. One side of the square is also the diameter of the semicircle. What is the total area of the banner? Use 3.14 for π.

_____ in²

Answer: 941.33 in²

Explanation:
Area of the square = side x side = 26 x 26 = 676 in²
Area of the semicircle =1/2 πr²= 1/2 (3.14) (13) (13) = 1/2 (3.14) (169) = 265.33 in²
Area of the figure = 676 + 265.33 = 941.33 in²

Question 6.
Multistep
Erin wants to carpet the floor of her closet. A floor plan of the closet is shown.

a. How much carpet does Erin need?
_____ ft²

Answer: 61 ft²

Explanation:
Area of the rectangle = length x breadth = 4 x 10 = 40 ft
Area of the triangle = 1/2 x base x height = 1/2 x 6 x 7 = 21 ft
The total area of the figure = 40+21 = 61 ft²

Question 6.
b. The carpet Erin has chosen costs $2.50 per square foot. How much will it cost her to carpet the floor?
$ _____

Answer: $152.50

Explanation:
Cost per square foot of the carpet = $2.50
The total cost of the carpet on the floor = 61 x $2.50 =$152.50

9.2 Practice A Geometry Answers Question 7.
Multiple Representations
Hexagon ABCDEF has vertices A(-2, 4), B(0, 4), C(2, 1), D(5, 1), E(5, -2), and F(-2, -2). Sketch the figure on a coordinate plane. What is the area of the hexagon?

_____ units²

Answer: The area of the figure is 30 square units

Explanation:
Separate the figure into a trapezium and a rectangle.
Area of a trapezium = 1/2 (a+b) h= 1/2 (2+4) x 3 = 1/2 (6) 3 = 9 square units
Area of a rectangle = length x breadth = 7 x 3 = 21 square units
The total area of the figure = 9+21 = 30 square units

Question 8.
A field is shaped like the figure shown. What is the area of the field? Use 3.14 for π.

_____ m²

Answer: 146.24 m²

Explanation:
Divide the figure into a square, triangle and a quarter of a circle.

Area of a square = side x side = 8 x 8 = 64 m²
Area of a quarter of a circle = 1/4 (πr²) = 1/4 (3.14 x 8²)
= 1/4 (200.96) = 50.24 m²
Area of the triangle = 1/2 x base x height = 1/2 x 8 x 8 = 32 m²
Total area of the figure = 64+32+50.24 = 146.24 m²

Question 9.
A bookmark is shaped like a rectangle with a semicircle attached at both ends. The rectangle is 12 cm long and 4 cm wide. The diameter of each semicircle is the width of the rectangle. What is the area of the bookmark? Use 3.14 for π.
_____ cm²

Answer: 60.56 cm²

Explanation:
The bookmark is divided into a rectangle, a semicircle.
Area of the rectangle = length x breadth = 12 x 4 = 48 cm²
The diameter of the semicircle = The width of the rectangle = 4 cm
The radius of the semicircle = 4/2 = 2 cm
The area of the semicircle = πr² = 3.14 x 2 x 2 = 12.56 cm²
The total area of the bookmark = 12.56 + 48 = 60.56 cm²

Question 10.
Multistep
Alex is making 12 pendants for the school fair. The pattern he is using to make the pennants is shown in the figure. The fabric for the pennants costs $1.25 per square foot. How much will it cost Alex to make 12 pennants?

$ _____

Answer: $52.50

Explanation:
Each pendant is made up of a rectangle and a triangle.
Area of the rectangle = length x breadth = 3 x 1 = 3 ft²
Area of the triangle = 1/2 x base x height = 1/2 x 1 x 1 = 0.5 ft²
The total area of the pendant = 3+0.5 = 3.5 ft²
Number of pendants = 12
Area of the pendants = 12 x 3.5 = 42 ft²
Cost of each square foot of the pendant = $1.25
Total cost for all the 12 pendants = 12 x $1.25  = $52.50

Question 11.
Reasoning
A composite figure is formed by combining a square and a triangle. Its total area is 32.5 ft². The area of the triangle is 7.5 ft². What is the length of each side of the square? Explain.
_____ ft

Answer: 5 ft

Explanation:
Given:
The area of the composite figure = 32.5 ft²
The area of the triangle = 7.5 ft²
The area of the square = 32.5 – 7.5 = 25
side x side = 25
side² = 25
side = root 25 = 5 ft

H.O.T. – Page No. 282

Focus on Higher Order Thinking

Question 12.
Represent Real-World Problems
Christina plotted the shape of her garden on graph paper. She estimates that she will get about 15 carrots from each square unit. She plans to use the entire garden for carrots. About how many carrots can she expect to grow? Explain.

______ carrots

Answer: 300 carrots

Explanation:
This shape is divided into two triangles and a square.
Area of figure = 2(1/2 x 2 x 2) + 4(4) = 4 + 16 = 20 square units
Number of carrots per square unit = 300
Total number of carrots = 20 x 15 = 300

Question 13.
Analyze Relationships
The figure shown is made up of a triangle and a square. The perimeter of the figure is 56 inches. What is the area of the figure? Explain.

_____ in²

Answer: 192 in²

Explanation:
Given:
The perimeter of the figure = 56 inches
The figure is divided into a square and a triangle.
10 + 10 + 3s = 56
3s = 36
s = 12
The area of a triangle = 1/2 x 12 x 8 = 48 in²
The area of a square = 12 x 12 = 144 in²
Total area of the figure = 144 + 48 = 192 in²

Question 14.
Critical Thinking
The pattern for a scarf is shown at right. What is the area of the scarf? Use 3.14 for π.

_____ in²

Answer: 243 in²

Explanation:
Area of the rectangle in the given figure = 28 x 15 = 420 in²
Area of two semicircles = 2 (1/2 πr² ) = 3.14 x 7.5 x 7.5 = 176.625 in²
Area of the shaded region = 420 – 176.625 = 243 in²(approx)

Question 15.
Persevere in Problem-Solving
The design for the palladium window shown includes a semicircular shape at the top. The bottom is formed by squares of equal size. A shade for the window will extend 4 inches beyond the perimeter of the window, as shown by the dashed line around the window. Each square in the window has an area of 100 in².

a. What is the area of the window? Use 3.14 for π.
_____ in²

Answer: a) 2228 in²

Explanation:
Area of the square = 100 in²
side x side = 100
Side = 10 in
Since the side of each square is 10 in and there are 4 squares.
The side length of the larger square (s) = 40 in
Area of the larger square = side x side = 40 x 40 = 1600 in²
Since the side of each square is 10 in and there are 2 squares.
The radius of the semi-circle = 20 in
Area of the semi-circle = 1/2(πr²) = 1/2(3.14 x 20²) = 628 in²
The area of the window = 1600 + 628 = 2228 in²

Question 15.
b. What is the area of the shade? Round your answer to the nearest whole number.
_____ in²

Answer: b) 3016 in²

Explanation:
The shade extends 4 inches beyond the shapes so the length of the bottom rectangle is 40+4+4 = 48 in
The length extends below the original square.
The height is now = 40+4 = 44 in
The radius of the semi-circle = 20+4 = 24 in
The new area of the figure = 48(44) + 1/2(3.14 x 24²) = 2112 + 904.32 = 3016.32 = 3016 in²

Guided Practice – Page No. 286

Find the surface area of each solid figure.

Question 1.

Total surface area: _____ ft²

Answer: 150 ft²

Explanation:
The base is a triangle with side lengths of 8 ft, 5 ft, 5 ft so the perimeter of the base = P = 8+5+5 = 18 ft
The height of the prism = 7 ft
The base is a triangle.
Area of the triangle = 1/2 (8) (3) = 12 ft²
The surface area formula for a prism is S = Ph + 2b
P = Perimeter = 18 h = height = 7 b = base = area of the triangle = 12
The surface area of the prism = 18(7) + 2(12) = 126 + 24 = 150 ft²

Practice and Homework Lesson 9.2 Answer Key Question 2.

Total surface area: _____ m²

Answer: 503 m²

Explanation:
Given:
Dimensions of the cuboid:
Length = 11 m
Breadth = 9 m
Height = 7 m
The surface area of the cuboid = 2(lb+bh+hl) = 2(11 x 9 + 9 x 7 + 7 x 11) = 478m²

The dimensions of the cube:
Length of the side = 2.5 m
The surface area of the cube = 6a² = 6 x 2.5 x 2.5 = 37.5 m²
The surface area of the rectangular prism = 2.5 x 2.5 = 6.25
The surface area of the figure = The overlapping area is the area of the base of the cube
= 37.5 + 478 – 2(6.25) = 503 m²

Essential Question Check-In

Question 3.
How can you find the surface area of a composite solid made up of prisms?
Type below:
_____________

Answer: The surface area of the prisms, add them up, and then subtract the overlapping areas twice.

Explanation:
The surface area of a composite solid is made up of prisms by finding the surface areas of the prisms, adding them up, and then up, and then subtracting the overlapping areas.

Independent Practice – Page No. 287

Question 4.
Carla is wrapping a present in the box shown. How much wrapping paper does she need, not including overlap?

_____ in²

Answer: 164 in²

Explanation:
The surface area of the cuboid excluding the top = 2h(l+b) + lb = 2 x 4 ( 13 ) + 10 x 3 =  164 in²
The length of the wrapping paper = The surface area of the cuboid excluding the top = 164 in²

Question 5.
Dmitri wants to cover the top and sides of the box shown with glass tiles that are 5 mm square. How many tiles does he need?

_____ tiles

Answer: 3720 tiles

Explanation:
The surface area of the cuboid excluding the bottom = 2h(l+b) + lb = 2 x 9 (35) + 20 x 15 = 930 cm²
5mm = 0.5 cm
Area of a tile = Area of the square = a² = 0.5cm x 0.5cm = 0.25 cm²
Total number of tiles = 930/0.25 = 3720 tiles

Question 6.
Shera is building a cabinet. She is making wooden braces for the corners of the cabinet. Find the surface area of each brace.

_____ in²

Answer: 45 in²

Explanation:
The perimeter of the figure = P = 3(3) + 2(1) = 11 in
Base = B = 3(2) = 6 in
Height = h = 3
The surface area of the figure = Ph + 2B = 11 x 3 +2(6) = 33 + 12 = 45 in²

Circumference and Area Answer Key Question 7.
The doghouse shown has a floor, but no windows. Find the total surface area of the doghouse, including the door.

_____ ft²

Answer:  66ft²

Explanation:
Perimeter of the pentagon base (P) = 2(2.5) + 2(2) + 3 = 5 + 4 + 3 = 12
Area of the pentagon base by adding the area of the triangle and the area of the rectangle (B) = 1/2(3)(2) + 2(3) = 9
Height (h) = 2 + 2 = 4
The surface area of the figure = Ph + 2B = 12(4) + 2(9) = 48 + 18 = 66ft²

Eddie built the ramp to train his puppy to do tricks. Use the figure for 8–9.

Question 8.
Analyze Relationships
Describe two ways to find the surface area of the ramp.
Type below:
____________

Answer: One way is to use the formula S = Ph + 2B. Another way is to find the area of each face of the prism and add them up to get the total surface area.

Explanation:
The very first way to use the formula S = Ph + 2B is where the trapeziums are the base. The second way is to find the area of each face of the prism and then add them up to get the total surface area.

Question 9.
What is the surface area of the ramp?
_____ in²

Answer: 3264 in²

Explanation:
P = Perimeter of the figure =  16(3) + 2 (20) + 16 = 104
B = Base of the figure = 1/2 (12) (16 + 3(16)) = 6 (16 + 48) = 6 (64) = 384
h = Height of the figure = 2
Surface area of the figure = Ph + 2B = 104(2) + 2(384) = 2496 + 768 = 3264 in²

Marco and Elaine are building a stand like the one shown to display trophies. Use the figure for 10–11.

Question 10.
What is the surface area of the stand?
_____ ft²

Answer:  58 ft²

Explanation:
Top:
Perimeter = P = 4(1) = 4
Base = B = 1(1) = 1
Height = h = 3
Top surface area = Ph + 2B = 4(3) + 2(1) = 14 ft²
Bottom :
Perimeter = P = 2(7) + 2(1) = 14 + 2 = 16
Base = B = 7(1) = 7
Height = h = 2
Top surface area = Ph + 2B = 16(2) + 2(7) = 46 ft²
Overlapping area = 1(1) = 1
The surface area of the figure = The surface area of the top + The surface area of the bottom – the overlapping area = 14 + 46 – 2 = 60 – 2 = 58 ft²

Question 11.
Critique Reasoning
Marco and Elaine want to paint the entire stand silver. A can of paint covers 25 square feet and costs $6.79. They set aside $15 for paint. Is that enough? Explain.
_____

Answer: No

Explanation:
Since the surface area is 58 ft², they will need 3 cans of paint. Since each can paints 25 ft² and we cannot buy a fraction of cans.
3 cans would then cost 6.79 x 3 = 20.37 so this is not enough.

Page No. 288

Question 12.
Henry wants to cover the box shown with paper without any overlap. How many square centimeters will be covered with paper?

_____ cm²

Answer: 2316 cm²

Explanation:
Given:
Length = 24cm  Breadth = 27cm Height = 10cm
P = Perimeter = 2(24) + 2(27) = 48 + 54 = 102
B = Base = 24(27) = 648
h = Height = 10
Surface area of the figure = Ph + 2B = 102(10) + 2(648) = 1020 + 1296 = 2316 cm²

Go Math Grade 7 Lesson 9.3 Answer Key Question 13.
What If?
Suppose the length and width of the box in Exercise 12 double. Does the surface area S double? Explain.
_____

Answer: No

Explanation:
Given :
Length = 24cm x 2 = 48 cm  Breadth = 27cm x 2 = 54 cm Height = 10cm
P = 2(48) + 2(54) = 96 + 108 = 204
B = 48(54) = 2592
New Surface area = Ph + 2B = 204(10) + 2(2592) = 2040 + 5184 = 7224 cm²
Double of surface area = 2 (2316) = 4632 cm²
So the new surface area is not double of the initial area.

H.O.T.

Focus on Higher Order Thinking

Question 14.
Persevere in Problem-Solving
Enya is building a storage cupboard in the shape of a rectangular prism. The rectangular prism has a square base with side lengths of 2.5 feet and a height of 3.5 feet. Compare the amount of paint she would use to paint all but the bottom surface of the prism to the amount she would use to paint the entire prism.
Type below:
______________

Answer: The difference would just be the area in the bottom surface. It would be 6.25 ft² less.

Explanation:
The difference in the amount of paint would just be the area of the bottom surface. The area of the bottom surface is (2.5)² = 6.25.
Therefore she would paint 6.25 ft² less if she painted all but the bottom surface compared to painting the entire prism.

Question 15.
Interpret the Answer
The oatmeal box shown is shaped like a cylinder. Use a net to find the surface area S of the oatmeal box to the nearest tenth. Then find the number of square feet of cardboard needed for 1,500 oatmeal boxes. Round your answer to the nearest whole number

_____ ft²

Answer: 138.28 in² , 1440 ft²

Explanation:
Given:
Dimensions of the cylinder:
Radius: 2 in
Height: 9 in
The total surface area of the cylinder = 2πr(r+h) = 2 x 22/7 x 2 (2 + 9) = 138.28 in²

The total number of square inches needed for 1,500 oatmeal boxes = 1,500 x 138.28 = 207,300 in²
1 ft = 12 in
(1 ft)² = (12 in)²
1 ft² = 144 in²
The total number of square feet needed for 1,500 oatmeal boxes (to the nearest whole number)
= 207,300/144 = 1440 ft²

Question 16.
Analyze Relationships
A prism is made of centimeter cubes. How can you find the surface area of the prism in Figure 1 without using a net or a formula? How does the surface area change in Figures 2, 3, and 4? Explain.

Type below:
______________

Answer: The surface area for the first 3 figures is the same. The surface area for Figure 4 is greater than the surface area of figures 1 – 3.

Explanation:
The surface area of the first 3 figures is the same. The 3 new faces in Figure 2 have the same areas as the 3 visible faces that were removed when the top corner cube was removed. The surface area is then the same as it is in Figure 1. Similarly, the areas of the new visible faces in Figure 3 are equal to the areas of the visible faces removed from removing the corner cubes so the surface areas are the same as in Figure 1. The surface area for Figure 4 is greater than the surface areas of figures 1 – 3. Removing the cube removed 2 of the visible faces (one from the top and one from the front side) but added 4 visible faces so the surface area increases.

Guided Practice – Solving Volume Problems – Page No. 292

Question 1.
Find the volume of the triangular prism.

_____ ft³

Answer: 84 ft³

Explanation:
Base area of the prism = 1/2 x 8 x 3 = 12 ft²
Height of the prism = 7 ft
Volume of the prism = (12 x 7) ft³

Question 2.
Find the volume of the trapezoidal prism.

_____ m³

Answer: 330 m³

Explanation:
Base area of the prism = 1/2 x (15 + 5) x 3 = 30 m²
Height of the prism = 11 m
Volume of the prism = (30 x 11) m³ = 330 m³

Lesson 9.5 Solving Volume Problems Answer Key Question 3.
Find the volume of the composite figure.

_____ ft²

Answer: Composite figure: 360 ft³

Explanation:
The volume of the triangular prism:
The base area of the prism = 1/2 x 4 x 6 = 12 ft²
Height = 6 ft
The volume of the triangular prism = 12 x 6 = 72 ft³

The volume of the rectangular prism:
The base area of the prism = 4 x 6 = 24 ft²
Height = 12 ft
The volume of the triangular prism = 12 x 24 = 288 ft³

Volume of the composite figure = (288 + 72)ft³ = 360 ft³

Find the volume of each figure.

Question 4.
The figure shows a barn that Mr. Fowler is building for his farm.

_____ ft³

Answer: 40,000 ft³

Explanation:
Triangular prism:
B = Base area = 1/2 x 10 (40) = 200 cm²
Height = 50 cm
The volume of the triangular prism = Bh = 200 x 50 = 10,000 cm³
Rectangular prism:
B = Base area =40 x 15 = 600 cm²
Height = 50 cm
The volume of the triangular prism = Bh = 600 x 50 = 30,000 cm³
Total volume of the prism = 10,000 + 30,000 = 40,000 cm³

Question 5.
The figure shows a container, in the shape of a trapezoidal prism, that Pete filled with sand.

_____ cm³

Answer: 385 cm³

Explanation:
B = Base area = 1/2 x 5 (10 + 12) = 55 cm²
Height = 7 cm
The volume of the container = Bh = 55 x 55 = 385 cm³

Essential Question Check-In

Question 6.
How do you find the volume of a composite solid formed by two or more prisms?
Type below:
______________

Answer: Finding the volume of each figure and adding them up to get the volume of the composite solid.

Explanation:
To find the volume of the composite figure that can be divided into 2 or more prisms, find the volume of each prism and add them up to get the volume of the composite solid.

Independent Practice – Page No. 293

Question 7.
A trap for insects is in the shape of a triangular prism. The area of the base is 3.5 in² and the height of the prism is 5 in. What is the volume of this trap?
_____ in³

Answer: 17.5 in³

Explanation:
The volume of the trap = Base area x height = 3.5 x 5 = 17.5 in³

Question 8.
Arletta built a cardboard ramp for her little brothers’ toy cars. Identify the shape of the ramp. Then find its volume.

Shape: _________
Area: _________ in³

Answer: 525 in³

Explanation:
Base area = 1/2 x 6 x 25 = 75 in²
Height  = 7 in
Volume of the figure = 75 x 7 = 525 in³

Question 9.
Alex made a sketch for a homemade soccer goal he plans to build. The goal will be in the shape of a triangular prism. The legs of the right triangles at the sides of his goal measure 4 ft and 8 ft, and the opening along the front is 24 ft. How much space is contained within this goal?

_____ ft³

Answer: 384 ft³

Explanation:
Base area = 1/2 x 4 x 8 = 16 ft²
Height  = 24 ft
Volume of the figure = 16 x 24 = 384 ft³

Question 10.
A gift box is in the shape of a trapezoidal prism with base lengths of 7 inches and 5 inches and a height of 4 inches. The height of the gift box is 8 inches. What is the volume of the gift box?
_____ in³

Answer: 192 in³

Explanation:
Base area = 1/2 x 4 x (7+5) = 24 in²
Height  = 8 in
Volume of the figure = 24 x 8 = 192 Base area = 1/2 x 6 x 25 = 75 in²
Height  = 7 in
Volume of the figure = 75 x 7 = 525 in³

Question 11.
Explain the Error
A student wrote this statement: “A triangular prism has a height of 15 inches and a base area of 20 square inches. The volume of the prism is 300 square inches.” Identify and correct the error.
Type below:
____________

Answer: The error is measurement unit.

Explanation:
The volume of the prism is:
base area x height = 20 x 15 = 300 in³

Find the volume of each figure. Round to the nearest hundredth if necessary.

Question 12.

_____ in³

Answer: 97.2 in³

Explanation:
The volume of the hexagonal prism = 23.4 x  3 = 70.2 in³

Base area of the rectangular prism = 3 x 3 = 9 in²
The volume of the rectangular prism = Bh = 9 x 3 = 27 in³

Total volume of the figure = 70.2 + 27 = 97.2 in³

Question 13.

_____ m³

Answer: 316.41 m³

Explanation:
The volume of the rectangular prism on the left = Bh = [7.5 x 3.75] (3.75) = 105.47 m³
The volume of the rectangular prism on the right = Bh = [7.5 x 3.75](7.5) = 210.94 m³
Total volume of the composite figure = 105.47 + 210.94 = 316.41 m³

Question 14.
Multi-Step
Josie has 260 cubic centimeters of candle wax. She wants to make a hexagonal prism candle with a base area of 21 square centimeters and a height of 8 centimeters. She also wants to make a triangular prism candle with a height of 14 centimeters. Can the base area of the triangular prism candle be 7 square centimeters? Explain.
_____

Answer: No

Explanation:
The volume of the hexagonal prism = 21 x 8 = 168
The total volume of wax, 260 is equal to the sum of the volumes of each prism.
B is the base area of the triangular prism.
168 + 14B = 260 cm³
14B = 260 – 168
B = 6.6 cm³

Page No. 294

Question 15.
A movie theater offers popcorn in two different containers for the same price. One container is a trapezoidal prism with a base area of 36 square inches and a height of 5 inches. The other container is a triangular prism with a base area of 32 square inches and a height of 6 inches. Which container is the better deal? Explain.

Type below:
___________

Answer: The triangular prism is a better deal since it has a larger volume

Explanation:
The base area of the trapezoidal prism = 36 in²
The volume of the trapezoidal prism = Bh = 36 x 5 = 175 in³

The base area of the triangular prism = 32 in²
The volume of the rectangular prism = Bh = 32 x 6 = 192 in³

The triangular prism is a better deal since it has a larger volume.

H.O.T.

Focus on Higher Order Thinking

Question 16.
Critical Thinking
The wading pool shown is a trapezoidal prism with a total volume of 286 cubic feet. What is the missing dimension?

______ ft.

Answer: 3.5 ft

Explanation:
Area of the trapezoidal prism = B = 1/2 x 13 (2+x)
Volume of the figure = 286 cubic feet
V = Bh
286 = 1/2 x 13 (2+x)(8)
5.5 = (2+x)
x = 3.5 ft

Question 17.
Persevere in Problem-Solving
Lynette has a metal doorstop with the dimensions shown. Each cubic centimeter of the metal in the doorstop has a mass of about 8.6 grams. Find the volume of the metal in the doorstop. Then find the mass of the doorstop.

______ grams

Answer: 75 cubic centimeter, 645 grams

Explanation:
V = Bh
B = Area of the triangle of base = 10 cm , height = 6 cm = 1/2 x 10 x 6 = 30 square centimeter
V = 30 x 2.5 = 75 cubic centimeter

1 cubic centimeter = 8.6 grams in mass
V = 75 cubic centimeter x 8.6 = 645 grams

Question 18.
Analyze Relationships
What effect would tripling all the dimensions of a triangular prism have on the volume of the prism? Explain your reasoning.
Type below:
____________

Answer: The volume is 27 times the original volume.

Explanation:
The area of the base = B = 1/2 (3b) (3h) = 9/2 (bh)
H is the height of the prism
The volume would be = 9/2 (bh) x (3H) = 27 [ 1/2 (bhH) ]

Therefore, The volume is 27 times the original volume.

Question 19.
Persevere in Problem Solving
Each of two trapezoidal prisms has a volume of 120 cubic centimetres. The prisms have no dimensions in common. Give possible dimensions for each prism.
Type below:
____________

Answer: A possible combination of dimension could be the height at 8 cm, base at 2 cm and 3 cm

Explanation:
The numbers that multiply to get 120 are 20 and 6 so let the first prism have a base area of 20 square centimetres and the height of 6 cm.
If the base area is 20, the height of the trapezoid and the length of the bases could be 8,2 and 3 respectively.

The other numbers that multiply to get 120 are 4 and 30 so let the second prism have a base area of 30 square centimetres and the height of 4 cm.
If the base area is 30, the height of the trapezoid and the length of the bases could be 10,1 and 5 respectively.

9.1, 9.2 Circumference and Area of Circles – Page No. 295

Find the circumference and area of each circle. Use 3.14 for π. Round to the nearest hundredth if necessary.

Question 1.

C = _________ m
A = _________ m²

Answer:
C = 43.96 m
A = 153.86 m²

Explanation:
C = 2 πr = 2 π(7) = 14 (3.14) = 43.96 m
A = πr² = 3.14 (7)² = 153.86 m²

Chapter 9 Circumference and Area of a Circle Answer Key Question 2.

C = _________ ft
A = _________ ft²

Answer:
C = 37.68 ft
A = 113.04 ft²

Explanation:
Diameter = 12 ft
Radius = d/2 = 12/2 = 6 ft
C = 2 πr = 2 π(6) = 6 (3.14) = 37.68 ft
A = πr² = 3.14 (6)² = 113.04 ft²

9.3 Area of Composite Figures

Find the area of each figure. Use 3.14 for π.

Question 3.

______ m²

Answer: 180.48 m²

Explanation:
Area of the triangle = 1/2 x 16 x 10 = 80 m²
Area of the semicircle = 1/2 πr² = 1/2 (3.14) (8)² = 100.48 m²
The total area of the figure = 80 + 100.48 = 180.48 m²

Question 4.

______ cm²

Answer: 200 cm²

Explanation:
Area of the parallelogram = 4.5(20) = 90 cm²
Area of the rectangle = 20(5.5) = 110 cm²
The total area of the figure = 90 + 110 = 200 cm²

9.4, 9.5 Solving Surface Area and Volume Problems

Find the surface area and volume of each figure.

Question 5.

S = _________ cm²
V = _________ cm³

Answer:
S = 132 cm²
V = 60 cm³

Explanation:
Perimeter = 3+4+5 = 12 cm
Base area = Area of the triangle = 1/2 x 3 x 4 = 6
S = Ph + 2B = 12(10) + 2(6) = 120 +12 = 132 cm²

V = Bh = 6 x 10 = 60 cm³

Question 6.

S = _________ yd²
V = _________ yd³

Answer:
S = 54.5 yd²
V = 27.5 yd³

Explanation:
Perimeter = 2(2.5) + 2(2) + 4 = 13 cm
Base area = Area of the triangle + Area of the rectangle = 1/2 x 1.5 x 4 + 4(2)= 11
S = Ph + 2B = 13(2.5) + 2(11) = 32.5 +22 = 54.5 yd²

V = Bh = 11 x 2.5 = 27.5 yd³

Essential Question

Area and Circumference of a Circle Iready Quiz Question 7.
How can you use geometric figures to solve real-world problems?
Type below:
______________

Answer: We can solve real-world problems by finding surface area and volume.
Example: We can find the amount of liquid in a tank by calculating its volume.

Explanation:
Real-world problems by finding surface area and volume.
Example 1: We can find the amount of liquid in a tank by calculating its volume.
Example 2: We can find the surface area of the house and find the amount of paint required to paint the house.

Page No. 296

Question 1.
What is the circumference of the circle?

a. 34.54 m
b. 69.08 m
c. 379.94 m
d. 1519.76 m

Answer: b. 69.08 m

Explanation:
Circumference = 2 πr = 2 π(11) = 22 (3.14) = 69.08 m

Question 2.
What is the area of the circle?

Options:
a. 23.55 m²
b. 47.1 m²
c. 176.625 m²
d. 706.5 m²

Answer: c. 176.625 m²

Explanation:
Diameter = 15 m
Radius = 7.5 m
Area of the circle = πr² = 3.14 (7.5)² = 176.625 m²

Question 3.
What is the area of the figure?

Options:
a. 28.26 m²
b. 36 m²
c. 64.26 m²
d. 92.52 m²

Answer: c. 64.26 m²

Explanation:
Area of the square = 6 x 6 = 36 m²
Radius = 6 m
Area of the quarter circle = 1/4 πr² = 1/4 x 3.14 (6)² = 28.26 m²
The total area of the figure = 36 + 28.26 = 64.26 m²

Question 4.
A one-year membership to a health club costs $480. This includes a $150 fee for new members that is paid when joining. Which equation represents the monthly cost x in dollars for a new member?
Options:
a. 12x + 150 = 480
b. \(\frac{x}{12}\) + 150 = 480
c. 12x + 480 = 150
d. \(\frac{x}{12}\) + 480 = 150

Answer: a. 12x + 150 = 480

Explanation:
If x is the monthly fee, then 12x is the total monthly fee.
The joining fee = $150
Total cost = $480
then,
12x + 150 = 480

Chapter 9 Practice Test Surface Area Answer Key Question 5.
What is the volume of the prism?

Options:
a. 192 ft³
b. 48 ft³
c. 69 ft³
d. 96 ft³

Answer: d. 96 ft³

Explanation:
B = Base area of the triangle = 1/2 x 8 x 2 = 8 ft²
Height = 12 ft
Volume of the triangular orism = Bh = 8(12) = 96 ft³

Question 6.
A school snack bar sells a mix of granola and raisins. The mix includes 2 pounds of granola for every 3 pounds of raisins. How many pounds of granola are needed for a mix that includes 24 pounds of raisins?
Options:
a. 16 pounds
b. 36 pounds
c. 48 pounds
d. 120 pounds
e. 120 pounds

Answer: a. 16 pounds

Explanation:
2/3 is equal to x/24 then 3 times 8 is equal to 24 and if 2 times 8 is equal to 16.

Question 7.
Find the percent change from $20 to $25.
Options:
a. 25% decrease
b. 25% increase
c. 20% decrease
d. 20% increase

Answer: b. 25% increase

Explanation:
25 – 20 = 5 divide by 20 = 1/4
When we find the percentage we get 25.
So we can say that there is an increase in 25%

Question 8.
Each dimension of the smaller prism is half the corresponding dimension of the larger prism.

a. What is the surface area of the figure?
_____ in²

Answer: 856 in²

Explanation:
Height of the top prism = 10/2 = 5
Length of the top prism = 16/2 = 8
Width of the top prism = 8/2 = 4
Perimeter = 2l + 2w = 2(8) + 2(4) = 16 + 8 = 24 in
B = lw = 8(4) = 32 in
Surface area of top prism= Ph + 2B = 24(5) + 2(32) = 184 in²

Height of the prism = 10
Length of the prism = 16
Width of the prism = 8
Perimeter = 2l + 2w = 2(16) + 2(8) = 32 + 16 = 48 in
B = lw = 16(8) = 128 in
Surface area of bottom prism= Ph + 2B = 48(10) + 2(128) = 736 in²

Area of overlapping region = 32 in²

The total surface area of the prism
= Surface area of top prism + Surface area of bottom prism – 2[Area of overlapping region ]
= 184 + 736 – 2(32) = 856 in²

Question 8.
b. What is the volume of the figure?
_____ in³

Answer: 1440 in³

Explanation:
Volume of top prism = Bh = 32(5) = 160 in³
Volume of bottom prism = Bh = 128(10) = 1280 in³
The total volume of the figure = 160 + 1280 = 1440 in³

EXERCISES – Page No. 298

Question 1.
In the scale drawing of a park, the scale is 1 cm: 10 m. Find the area of the actual park.

_____ m²

Answer: 450 m²

Explanation:
Multiply the dimensions of the scale drawing by 10 since 1 cm = 10 m
3cm by 1.5 cm = 30m by 15 m
Area = 30(15) = 450 m²

Question 2.
Find the value of y and the measure of ∠YPS.

y = __________ °
mYPS = __________ °

Answer: y = 8
mYPS = 40 °

Explanation:
140 + 5y = 180 [sum of angle on a line = 180°]
5y = 40
y = 8

mYPS = mRPZ = 5y [vertically opposite angles]
mYPS = 5(8) = 40°

Question 3.
Kanye wants to make a triangular flower bed using logs with the lengths shown below to form the border. Can Kanye form a triangle with the logs without cutting any of them? Explain.

_____

Answer: No

Explanation:
A side of a triangle must be greater than the difference of the other two sides and smaller than the sum of the other 2 sides.
The sum of the first 2 sides = 3+4 = 7 < 8
Therefore, he cannot form a triangle unless he cuts the logs.

Circumference Homework Help Question 4.
In shop class, Adriana makes a pyramid with a 4-inch square base and a height of 6 inches. She then cuts the pyramid vertically in half as shown. What is the area of each cut surface?

_____ in²

Answer: 12 in²

Explanation:
Base = 4 in
Height = 6 in
Area of the triangle = 1/2 x 6 x 4 = 12 in²

Page No. 300

Find the circumference and area of each circle. Round to the nearest hundredth.

Question 1.

C = __________ in
A = __________ in²

Answer:
C = 69.08 in
A = 379.94 in²

Explanation:
Diameter = 22 in
Radius = d/2 = 22/2 = 11 in
C = 2 πr = 2 π(11) = 22 (3.14) = 69.08 in
A = πr² = 3.14 (11)² = 379.94 in²

Question 2.

C = __________ m
A = __________ m²

Answer:
C = 28.26 m
A = 63.59m²

Explanation:
Radius = 4.5 m
C = 2 πr = 2 π(4.5) = 9 (3.14) = 28.26 m
A = πr² = 3.14 (4.5)² = 63.59 m²

Find the area of each composite figure. Round to the nearest hundredth if necessary.

Question 3.

______ in²

Answer: 99 in²

Explanation:
Area of the square = 9 x 9 = 81 in²
Base of the triangle = 13 – 9 = 4 in
Area of the triangle = 1/2 x 4 x 9 = 18 in²
The total area of the figure = 81 + 18 = 99 in²

Question 4.

______ cm²

Answer: 420.48 cm²

Explanation:
Area of the rectangle = 16 x 20 = 320 cm²
Diameter = 16 cm
Radius = 16/2 = 8 cm
Area of the semi circle = 1/2 πr² = 1/2 x 3.14 (8)² = 100.48 cm²
The total area of the figure = 320 + 100.48 = 420.48 cm²

Find the volume of each figure.

Question 5.

______ in³

Answer: 420 in³

Explanation:
B = 7(5) = 35 in²
V = Bh = 35 x 12 = 420 in³

Question 6.
The volume of a triangular prism is 264 cubic feet. The area of a base of the prism is 48 square feet. Find the height of the prism.
______ in

Answer: 5.5 ft

Explanation:
V = Bh
264 = 48h
h = 264/48 = 5.5ft

Page No. 301

A glass paperweight has a composite shape: a square pyramid fitting exactly on top of an 8 centimeter cube. The pyramid has a height of 3 cm. Each triangular face has a height of 5 centimeters.

Question 7.
What is the volume of the paperweight?
______ cm³

Answer: 576 cm³

Explanation:
Pyramid:
B = 8 x 8 = 64 cm²
V = 1/3 Bh = 1/3 x 64 x 3 = 64 cm³
Prism:
B = 8 x 8 = 64 cm²
V = Bh = 64 x 8 = 512 cm³

The total volume of the figure = 64 + 512 = 576 cm³

Question 8.
What is the total surface area of the paperweight?
______ cm²

Answer: 400 cm²

Explanation:
Pyramid:
P = 4(8) = 32 cm
S = 1/2 Pl + B = 80 + 64 = 144 cm²

Prism:
P = 4(8) = 32 cm
S = Ph + 2B = 32(8) + 2(64) = 384 cm²
The total surface area of the prism
= Area of the prism + Area of the pyramid – 2[Area of the overlapping region]
= 144 + 384 – 2(64) = 400

Unit 4 Performance Tasks

Question 9.
Product Design Engineer
Miranda is a product design engineer working for a sporting goods company. She designs a tent in the shape of a triangular prism. The dimensions of the tent are shown in the diagram.

a. How many square feet of material does Miranda need to make the tent (including the floor)? Show your work.
______ ft²

Answer: 261 3/4  ft²

Explanation:
P = 2 x 7 1/2 + 8 = 22 1/2
B = 4/2 (8) (6) = 24
S = Ph + 2B = 22 1/2 x 9 1/2 + 2(24) = 213 3/4 + 48 = 261 3/4 ft²

Question 9.
b. What is the volume of the tent? Show your work.
______ ft³

Answer: 228 ft³

Explanation:
V = Bh = 24 x 9 1/2 = 228 ft³

Question 9.
c. Suppose Miranda wants to increase the volume of the tent by 10%. The specifications for the height (6 feet) and the width (8 feet) must stay the same. How can Miranda meet this new requirement? Explain
Type below:
____________

Answer: Increase the height to 10.45 ft

Explanation:
New volume = 1.10 x 228 = 250.8
250.8 = 24h
h = 10.45 ft

Unit 4 Performance Tasks (cont’d) – Page No. 302

Question 10.
Li is making a stand to display a sculpture made in art class. The stand will be 45 centimeters wide, 25 centimeters long, and 1.2 meters high.
a. What is the volume of the stand? Write your answer in cubic centimeters.
______ cm³

Answer: 135,000 cm³

Explanation:
B = 45 x 25 = 1125 cm²
V = Bh = 1125 x 120 = 135,000 cm³

Question 10.
b. Li needs to fill the stand with sand so that it is heavy and stable. Each piece of wood is 1 centimeter thick. The boards are put together as shown in the figure, which is not drawn to scale. How many cubic centimeters of sand does she need to fill the stand? Explain how you found your answer.
______ cm³

Answer: 116,702 cm³

Explanation:
Width = 45 – 2(1) = 43 ft
Length = 25 – 2(1) =23ft
Height = 120-2(1) = 118ft
B = 43 x 23 = 989 ft²
V = Bh = 989 x 118 = 116,702 ft³

Selected Response – Page No. 303

Question 1.
A school flag is in the shape of a rectangle with a triangle removed as shown.

What is the measure of angle x?
Options:
a. 50°
b. 80°
c. 90°
d. 100°

Answer: d. 100°

Explanation:
x = 50 + 50 = 100° [ Sum of two angles created by the 2 lines]

Circumference Area and Volume Question 2.
On a map with a scale of 2 cm = 1 km, the distance from Beau’s house to the beach is 4.6 centimetres. What is the actual distance?
Options:
a. 2.3 km
b. 4.6 km
c. 6.5 km
d. 9.2 km

Answer: a. 2.3 km

Explanation:
2/1 = 4.6/x
x = 4.6/2 = 2.3 km

Question 3.
Lalasa and Yasmin are designing a triangular banner to hang in the school gymnasium. They first draw the design on paper. The triangle has a base of 5 inches and a height of 7 inches. If 1 inch on the drawing is equivalent to 1.5 feet on the actual banner, what will the area of the actual banner be?
Options:
a. 17.5 ft²
b. 52.5 ft²
c. 39.375 ft²
d. 78.75 ft²

Answer: c. 39.375 ft²

Explanation:
1in = 1.5ft
The base of the triangle = 5 in = 1.5(5) ft = 7.5 ft
Height = 7 in = 7(1.5) ft = 10.5 ft
Area of the triangle = 1/2 x 7.5 x 10.5 = 39.375 ft²

Question 4.
Sonya has four straws of different lengths: 2 cm, 8 cm, 14 cm, and 16 cm. How many triangles can she make using the straws?
Options:
a. no triangle
b. one triangle
c. two triangles
d. more than two triangles

Answer: b. one triangle

Explanation:
The third side of a triangle must be smaller than the sum of the other two sides to form a triangle.
2+8 = 10<14
2+8 = 10<16
8+14 = 22>14
8+14 = 22>16
2+14 = 16=16
2+16 = 18>16

Therefore, only one triangle can be formed using the sides 8, 14, 16.

Question 5.
A one-topping pizza costs $15.00. Each additional topping costs $1.25. Let x be the number of additional toppings. You have $20 to spend. Which equation can you solve to find the number of additional toppings you can get on your pizza?
Options:
a. 15x + 1.25 = 20
b. 1.25x + 15 = 20
c. 15x − 1.25 = 20
d. 1.25x − 15 = 20

Answer: b. 1.25x + 15 = 20

Explanation:
If x is the number of additional toppings, then 1.25 x is the cost of the additional toppings.
This gives the total cost is 1.25x + 15
then,
1.25x + 15 = 20

Question 6.
A bank offers a home improvement loan with simple interest at an annual rate of 12%. J.T. borrows $14,000 over a period of 3 years. How much will he pay back altogether?
Options:
a. $15680
b. $17360
c. $19040
d. $20720

Answer: c. $19040

Explanation:
Simple interest = 14,000 x 0.12 x 2 = $5,040
Amount = $14,000 + $5,040 = $19040

Question 7.
What is the volume of a triangular prism that is 75 centimeters long and that has a base with an area of 30 square centimeters?
Options:
a. 2.5 cm³
b. 750 cm³
c. 1125 cm³
d. 2250 cm³

Answer: d. 2250 cm³

Explanation:
V = Bh = 30(75) = 2250cm³

Question 8.
Consider the right circular cone shown.

If a vertical plane slices through the cone to create two identical half cones, what is the shape of the cross section?
Options:
a. a rectangle
b. a square
c. a triangle
d. a circle

Answer: c. a triangle

Explanation:
Slicing through the vertex to create 2 identical half cones would create a cross-section that  is a triangle.

Page No. 304

Question 9.
The radius of the circle is given in meters. What is the circumference of the circle? Use 3.14 for π.

a. 25.12 m
b. 50.24 m
c. 200.96 m
d. 803.84 m

Answer: b. 50.24 m

Explanation:
Circumference = 2 πr = 2 π(8) = 16 (3.14) = 50.24 m

Question 10.
The dimensions of the figure are given in millimeters. What is the area of the two-dimensional figure?

Options:
a. 39 mm²
b. 169 mm²
c. 208 mm²
d. 247 mm²

Answer: c. 208 mm²

Explanation:
Area of the square = 13 x 13 = 169 mm²
Area of the triangle = 1/2 x 13 x 6 = 39 mm²
The total area of the figure = 169 + 39 = 208 mm²

Question 11.
A forest ranger wants to determine the radius of the trunk of a tree. She measures the circumference to be 8.6 feet. What is the trunk’s radius to the nearest tenth of a foot?
Options:
a. 1.4 ft
b. 2.7 ft
c. 4.3 ft
d. 17.2 ft

Answer: a. 1.4 ft

Explanation:
Circumference = 2 πr = 8.6 ft
r = 8.6/2 π = 1.4 ft

Question 12.
What is the measure in degrees of an angle that is supplementary to a 74° angle?
Options:
a. 16°
b. 74°
c. 90°
d. 106°

Answer: d. 106°

Explanation:
Sum of supplementary angles = 180°
x + 74° = 180°
x = 106°

Question 13.
What is the volume in cubic centimeters of a rectangular prism that has a length of 6.2 centimeters, a width of 3.5 centimeters, and a height of 10 centimeters?
Options:
a. 19.7 cm³
b. 108.5 cm³
c. 217 cm³
d. 237.4 cm³

Answer: c. 217 cm³

Explanation:
V = Bh
B = 6.2 x 3.5 = 21.7 cm²
h = 10 cm
V = 21.7 x 10 = 217 cm³

Question 14.
A patio is the shape of a circle with diameter shown.

What is the area of the patio? Use 3.14 for π.
Options:
a. 9 m²
b. 28.26 m²
c. 254.34 m²
d. 1017.36 m²

Answer: c. 254.34 m²

Explanation:
Diameter = 18 m
Radius = 18/2 = 9 m
Area of the patio = πr² = 3.14 (9)² = 254.34 m²

Question 15.
Petra fills a small cardboard box with sand. The dimensions of the box are 3 inches by 4 inches by 2 inches.
a. What is the volume of the box?
______ in³

Answer: 24 in³

Explanation:
V = Bh
B = 3 x 4 = 12 in²
V = 12 x 2 = 24 in³

Question 15.
b. Petra decides to cover the box by gluing on wrapping paper. How much wrapping paper does she need to cover all six sides of the box?
______ in²

Answer: 76 in²

Explanation:
P = 2(3) + 2(4) = 6 + 8 = 14 in
S = Ph + 2B = 14 x 2 + 2 x 24 = 76 in²

Question 15.
c. Petra has a second, larger box that is 6 inches by 8 inches by 4 inches. How many times larger is the volume of this second box? The surface area?
Volume is _________ times greater.
Surface area is _________ times greater

Answer: Surface area is about 2.7 times larger

Explanation:
B = 6 x 8 = 48 in²
V = Bh = 48 x 4 = 192 in³
192/24 = 8
P = 2(6) + 2(8) = 12 + 16 = 28
S = Ph + 2B = 28(4) + 2(48) = 112 + 96 = 208 in²
208/76 = 2.7

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Go Math Grade 7 Answer Key Chapter 7 Writing and Solving One-Step Inequalities

Check out the topics of Chapter 7 Writing and Solving One-Step Inequalities before you start practicing. If you want to score the highest marks in the exams you have to practice the problems a number of times. Refer our Go Math Grade 7 Answer Key Chapter 7 Writing and Solving One-Step Inequalities while doing your homework. You can know how to draw the number line with the help of Go Math 7th Grade Chapter 7 Writing and Solving One-Step Inequalities.

Chapter 7 – Lesson 1:

• Guided Practice – Page No. 208
• Independent Practice Page No. 209
• Page No. 210

Chapter 7 – Lesson: 2

• Guided Practice – Page No. 214
• Independent Practice – Page No. 215
• Page No. 216

Chapter 7 – Lesson: 3

• Guided Practice – Page No. 220
• Independent Practice – Page No. 221
• Page No. 222

Chapter 7 – Writing and Solving One-Step Inequalities Lesson: 4

• 7.1 Writing and Solving One-Step Inequalities – Page No. 223
• Selected Response – Page No. 224

Chapter 7 – Lesson: 5

• EXERCISES – Page No. 226
• EXERCISES – Page No. 227

Chapter 7 – Performance Tasks

• Unit 3 Performance Tasks – Page No. 228
• Selected Response – Page No. 229
• Page No. 230

Guided Practice – Page No. 208

Write the resulting inequality.

Question 1.
−5 ≤ −2; Add 7 to both sides
Type below:
___________

Answer: 2 ≤ 5

Explanation:
Add 7 to both sides of the inequality.
-5 + 7 ≤ -2 + 7
2 ≤ 5

Question 2.
−6 < −3; Divide both sides by -3
Type below:
___________

Answer: 2 > 1

Explanation:
Divide both sides by -3. switch the inequality sign since you are dividing by a negative number.
-6/-3 > -3/-3
2 > 1

Equations and Inequalities Answer Key Question 3.
7 > −4; Subtract 7 from both sides
Type below:
___________

Answer: 0 > -11

Explanation:
7 – 7 > -4 – 7
Subtract 7 from both sides
0 > -11

Question 4.
−1 ≥ −8; Multiply both sides by -2
Type below:
___________

Answer: 2 ≤ 16

Explanation:
Multiply both sides by -2 switch the inequality sign since you are multiplying by a negative number.
-1(-2) ≤ -8(-2)
2 ≤ 16

Solve each inequality. Graph and check the solution.

Question 5.
n−5 ≥ −2
Type below:
___________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
Solve the inequality first:
n – 5 ≥ -2
n – 5 + 5 ≥ -2 + 5
n ≥ 3
The number opposite the variable is 3, we look for this in the number line. Since the inequality is ≥, we use a closed dot and shade the line going to the right. Its graph would look like the one below:

Question 6.
3 + x < 7
Type below:
___________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
Solve the inequality first:
3 + x < 7
3 – 3 + x < 7 – 3
x < 4
The number opposite the variable is 4, we look for this in the number line. Since the inequality is <, we use a hollow dot and shade the line going to the left. Its graph would look like the one below:

Question 7.
−7y ≤ 14
Type below:
___________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
Solve the inequality first:
−7y ≤ 14
-7y/-7 ≤ 14/-7
y ≥ -2
The number opposite the variable is -2, we look for this in the number line. Since the inequality is ≥, we use a closed dot and shade the line going to the right. Its graph would look like the one below:

Question 8.
\(\frac{b}{5}\) > −1
Type below:
___________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
Solve the inequality first:
\(\frac{b}{5}\) > −1
Multiply 5 on both sides.
(5)\(\frac{b}{5}\) > −1(5)
b > -5
The number opposite the variable is -5, we look for this in the number line. Since the inequality is >, we use a hollow dot and shade the line going to the right. Its graph would look like the one below:

Homework and Practice 7.1 Answer Key Question 9.
For a scientific experiment, a physicist must make sure that the temperature of a metal at 0 °C gets no colder than -80 °C. The physicist changes the metal’s temperature at a steady rate of -4 °C per hour. For how long can the physicist change the temperature?
a. Let t represent the temperature in degrees Celsius. Write an inequality. Use the fact that the rate of change in temperature times the number of hours equals the final temperature.
Type below:
___________

Answer:
We need to use the fact that the final temperature is equal to the rate of change in temperature times the number of hours.
We are given that the rate of change is -4°C per hour so the final temperature is -4 times the number of hours.
Let t represent the number of hours. The final temperature is then -4t degrees Celsius after t hours.
If the temperature must be no colder than -80°C, then the final temperature must be greater than or equal to -80.
The inequality is then -4t ≥ -80.

Question 9.
b. Solve the inequality in part a. How long can the physicist change the temperature of the metal?
Type below:
___________

Answer:
To solve the inequality for t, we need to divide both sides by -4. Remember to switch the inequality symbol since you are dividing by a negative number.
Dividing both sides by -4 then gives:
-4t/-4 ≤ -80/-4
t ≤ 20
The number of hours that the physicist can change the temperature of the metal is then at most 20 hours.

Question 9.
c. The physicist has to repeat the experiment if the metal gets cooler than -80 °C. How many hours would the physicist have to cool the metal for this to happen?
Type below:
___________

Answer:
From part (b), we know that the physicist can change the temperature for at most 20 hours to keep the temperature no colder than -80°C. This means the temperature will reach a temperature cooler than -80°C if he cools the metal for more than 20 hours.

Essential Question Check-In

Question 10.
Suppose you are solving an inequality. Under what circumstances do you reverse the inequality symbol?
Type below:
___________

Answer: You must reverse the inequality sign any time you multiply or divide both sides of the inequality by a negative number.

Page No. 209

In 11–16, solve each inequality. Graph and check the solution.

Question 11.
x − 35 > 15
Type below:
___________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
x − 35 > 15
Add 35 on both sides
x – 35 + 35 > 15 + 35
x > 50
The number opposite the variable is 50, we look for this in the number line. Since the inequality is >, we use a hollow dot and shade the line going to the right. Its graph would like the one below:

Question 12.
193 + y ≥ 201
Type below:
___________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
193 + y ≥ 201
193 + y – 193 ≥ 201 – 193
y ≥ 8
The number opposite the variable is 8, we look for this in the number line. Since the inequality is ≥, we use a closed dot and shade the line going to the right. Its graph would be like the one below:

Question 13.
−\(\frac{q}{7}\) ≥ −1
Type below:
___________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
−\(\frac{q}{7}\) ≥ −1
Multiply both sides by -7
(-7)−\(\frac{q}{7}\) ≥ −1(-7)
q ≤ 7
The number opposite the variable is 7, we look for this in the number line. Since the inequality is ≤, we use a closed dot and shade the line going to the left. Its graph would be like the one below:

Question 14.
−12x < 60
Type below:
___________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
−12x < 60
Divide both sides by -12
-12x/12 < 60/-12
x > -5
The number opposite the variable is -5, we look for this in the number line. Since the inequality is >, we use a hollow dot and shade the line going to the right. Its graph would be like the one below:

Question 15.
5 > z − 3
Type below:
___________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
5 > z − 3
Add both sides by 3
5 + 3 > z – 3 + 3
8 > z
z < 8
The number opposite the variable is 8, we look for this in the number line. Since the inequality is <, we use a hollow dot and shade the line going to the left. Its graph would like the one below:

Question 16.
0.5 ≤ \(\frac{y}{8}\)
Type below:
___________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
0.5 ≤ \(\frac{y}{8}\)
Multiply both sides by 8
(8)0.5 ≤ \(\frac{y}{8}\)(8)
4 ≤ y
y ≥ 4
The number opposite the variable is 4, we look for this in the number line. Since the inequality is ≥, we use a closed dot and shade the line going to the right. Its graph would like the one below:

Question 17.
The vet says that Lena’s puppy will grow to be at most 28 inches tall. Lena’s puppy is currently 1 foot tall. How many more inches will the puppy grow?
Type below:
___________

Answer: not more than 16 inches

Explanation:
Let x be the additional inches the puppy can grow remember 1 foot is 12 inches so the height of the puppy is 12 + x
12 + x ≤ 28
x ≤ 16

Practice and Homework Lesson 7.1 Answer Key Question 18.
In a litter of 7 kittens, each kitten weighs less than 3.5 ounces. Find all the possible values of the combined weights of the kittens.
Type below:
___________

Answer:
All of the kittens must weigh more than 0 ounces so the smallest combined weight is more than 0 ounces. Since there are 7 kittens, each kitten weighs less than 3.5 ounces, and 7 × 3.5 = 24.5, then the combined weights of the kittens must be less than 24.5 ounces.
This gives the inequality 0 < w < 24.5
where w is the combined weight of the kittens in ounces.

Question 19.
Geometry
The sides of the hexagon shown are equal in length. The perimeter of the hexagon is at most 42 inches. Find the possible side lengths of the hexagon.

Type below:
___________

Answer: 0 < s ≤ 7

Explanation:
Let s be the side lengths of the hexagon since its sides are all equal in length.
The side lengths of the hexagon must be greater than 0 since lengths can’t be negative or 0 so s > 0.
The perimeter of the figure is the sum of its side lengths so the perimeter of the hexagon must be 6s since it has 6 sides that are all s inches long.
The perimeter is at most 42 inches so 6s ≤ 42.
Dividing both sides by 6 then gives s ≤ 7.
Combining the inequalities s > 0 and s ≤ 7 then gives possible side lengths of 0 < s ≤ 7.

Question 20.
To get a free meal at his favorite restaurant, Tom needs to spend $50 or more at the restaurant. He has already spent $30.25. How much more does Tom need to spend to get his free meal?
Type below:
___________

Answer: at least $ 19.75

Explanation:
Let x be the additional amount he needs to spend. subtract 30.25 on both sides to solve for x.
x + 30.25 ≥ 50
x ≥ 19.75

Question 21.
To cover a rectangular region of her yard, Penny needs at least 170.5 square feet of sod. The length of the region is 15.5 feet. What are the possible widths of the region?
Type below:
___________

Answer: at least 11 feet

Explanation:
Area is the length times width so let w be the width.
Divide both sides by 15.5 to solve for w.
15.5w ≥ 170.5
w ≥ 11

Question 22.
Draw Conclusions
A submarine descends from sea level to the entrance of an underwater cave. The elevation of the entrance is -120 feet. The rate of change in the submarine’s elevation is no greater than -12 feet per second. Can the submarine reach the entrance to the cave in less than 10 seconds? Explain.
Type below:
___________

Answer:
No. Since the rate of descent is less than -12 feet per second and the submarine is descending for less than 10 seconds, the submarine elevation will still be greater than -120. The submarine would have to descend at a rate greater than -12 feet per second to reach the entrance in less than 10 seconds or descend for more than 10 seconds at a rate less than -12 feet per second to reach the entrance.

Page No. 210

The sign shows some prices at a produce stand.

Question 23.
Selena has $10. What is the greatest amount of spinach she can buy?
Type below:
___________

Answer: 3 \(\frac{1}{3}\) pounds

Explanation:
Let x be the number of pounds of spinach. divide both sides by 3 to solve for x.
3x ≤ 10
x ≤ \(\frac{10}{3}\)
x ≤ 3 \(\frac{1}{3}\) pounds

Question 24.
Gary has enough money to buy at most 5.5 pounds of potatoes. How much money does Gary have?
Type below:
___________

Answer: $2.75

Explanation:
Let x be the amount of money he has
Multiply the price per pound of potatoes by the number of pounds.
5.5(0.50) ≤ x
2.75 ≤ x

Question 25.
Florence wants to spend no more than $3 on onions. Will she be able to buy 2.5 pounds of onions? Explain.
Type below:
___________

Answer:
Since each pound of onions costs $1.25, then 2.5 pounds of onions cost $1.25 × 2.5 ≈ 3.13.
Since $3.13 is greater than $3, she will not have enough money if she wants to spend no more than $3.

H.O.T.

Focus on Higher Order Thinking

Question 26.
Counterexamples
John says that if one side of an inequality is 0, you don’t have to reverse the inequality symbol when you multiply or divide both sides by a negative number. Find an inequality that you can use to disprove John’s statement. Explain your thinking.
Type below:
___________

Answer:
A possible counterexample is -2x ≤ 0. Solving this correctly gives x ≥ 0 which means the inequality is true for all non-negative values. If you don’t switch the inequality sign you would get x ≤ 0 which means the inequality would be true for all non-positive numbers
x = -3 is a possible value for x ≤ 0 but -2x = -2(-3) = 6 which is not less than or equal to 0.

Question 27.
Look for a Pattern
Solve x + 1 > 10, x + 11 > 20, and x + 21 > 30. Describe a pattern. Then use the pattern to predict the solution of x + 9,991 > 10,000.
Type below:
___________

Answer:
x + 1 > 10
x > 9
Subtract both sides by 11
x + 11 > 20
x > 9
Subtract both sides by 21.
x + 21 > 30
x > 9
The pattern is that when the number on the left side of the inequality is 9 less than the number on the right side of the inequality, the answer is x > 9.
Since 9991 is 9 less than 10000, x + 9991 > 10,000 ha the solution x > 9

Question 28.
Persevere in Problem-Solving
The base of a rectangular prism has a length of 13 inches and a width of \(\frac{1}{2}\) inch. The volume of the prism is less than 65 cubic inches. Find all possible heights of the prism. Show your work.
Type below:
___________

Answer: 0 < h < 10

Explanation:
Let h be the height of the prism. It is given that the prism has a length of 13 inches and a width of 1/2 inches.
Using the formula v = lbh
13(1/2) h< 65
Multiply 2/13 on both sides
2/13 (13/2)h< 2/13 × 65
h < 2 × 5
h < 10
Since the height must be a positive number, then h > 0. Combining h > 0 and h < 10 then gives the final answer of 0 < h < 10.

Guided Practice – Page No. 214

Draw algebra tiles to model each two-step inequality.

Question 1.
4x − 5 < 7
Type below:
___________

Answer:
On the left side, draw 4 positive rectangles to model 4x and 5 negative squares to represent -5. On the right side, draw 7 positive squares to represent 7. then draw < in the middle.

Question 2.
−3x + 6 > 9
Type below:
___________

Answer:
On the left side, draw 3 negative rectangles to model -3x and 6 positive squares to represent 6. On the right side, draw 9 positive squares to represent 9. then draw > in the middle.

Question 3.
The booster club needs to raise at least $7,000 for new football uniforms. So far, they have raised $1,250. Write an inequality to find the average amounts each of the 92 members can raise to meet the club’s objective.
Type below:
___________

Answer: 1250 + 92a ≥ 7000

Explanation:
The amount to be raised is $7000. The amount already raised is $1250. The number of members is 92.
The inequality is then of the form: amount already raised + number of members × amount each member raises ≥ target amount.
The inequality is then:
1250 + 92a ≥ 7000

Two-Step Inequalities Worksheet with Answers Question 4.
Analyze what each part of 7x − 18 ≤ 32 means mathematically.
Type below:
___________

Answer:
x is the variable so it is the solution. 7x is the solution multiplied by 7. -18 means 7x is subtracted by 18. ≤ 32 means the result is no more than 32.

Question 5.
Write a real-world problem to represent 7x − 18 ≤ 32.
Type below:
___________

Answer:
A real-world problem could be: The temperature of a metal is currently at -18°C. A scientist will warm the metal at a rate of 7°C per hour until the temperature is 32°C. How many hours will it take to warm up the metal?

Essential Question Check-In

Question 6.
Describe the steps you would follow to write a two-step inequality you can use to solve a real-world problem.
Type below:
___________

Answer:
The first step is to translate the words into an algebraic expression.
The next step is to determine the target amount.
The third step is to determine what inequality sign to use by determining if you need to be greater than, greater than or equal to, less than, less than, or equal to the target amount to write the inequality. Then solve the inequality sign, and target amount to write the inequality.
Then solve the inequality for the unknown value. Finally, interpret the solution in the context of the problem.

Independent Practice – Page No. 215

Question 7.
Three friends earned more than $200 washing cars. They paid their parents $28 for supplies and divided the rest of the money equally. Write an inequality to find possible amounts each friend earned. Identify what your variable represents.
Type below:
___________

Answer: 3x + 28 > 200

Explanation:
Let x be the amount each friend received. Since there are 3 friends, then 3x is the amount of money they split evenly.
The amount of money they split evenly was the amount left over after paying their parents $28.
Therefore 3x + 28 is the total amount they earned.
3x + 28 > 200

Question 8.
Nick has $7.00. Bagels cost $0.75 each, and a small container of cream cheese costs $1.29. Write an inequality to find the number of bagels Nick can buy. Identify what your variable represents.
Type below:
___________

Answer: 0.75x + 1.29 ≤ 7

Explanation:
Let x represent the number of bagels he can buy.
Then 0.75x is the total cost of the bagels and 0.75x + 1.29 is the total cost of his purchase.

Question 9.
Chet needs to buy 4 work shirts, all costing the same amount. After he uses a $25 gift certificate, he can spend no more than $75. Write an inequality to find the possible costs for a shirt. Identify what your variable represents.
Type below:
___________

Answer: 4x – 25 ≤ 75

Explanation:
Let x represent the cost of each skirt the 4x is the total cost of the shirts. Since he has a $25 gift card, the total amount he is spending is 4x – 25.
4x – 25 ≤ 75

Question 10.
Due to fire laws, no more than 720 people may attend a performance at Metro Auditorium. The balcony holds 120 people. There are 32 rows on the ground floor, each with the same number of seats. Write an inequality to find the number of people that can sit in a ground-floor row if the balcony is full. Identify what your variable represents.
Type below:
___________

Answer: 32x + 120 ≤ 720

Explanation:
Let x represent the number of people that can sit in each ground floor row. then 32x is the total number of people sitting in the ground floor.
Since 120 people are sitting in the balcony, the total number of people is 32x + 120.

Question 11.
Liz earns a salary of $2,100 per month, plus a commission of 5% of her sales. She wants to earn at least $2,400 this month. Write an inequality to find amounts of sales that will meet her goal. Identify what your variable represents.
Type below:
___________

Answer: 2100 + 0.05x ≥ 2400

Explanation:
Let x represent the number of sales then 0.05x is the amount she earns in commission and 2100 + 0.05x is her total earnings.
2100 + 0.05x ≥ 2400

Question 12.
Lincoln Middle School plans to collect more than 2,000 cans of food in a food drive. So far, 668 cans have been collected. Write an inequality to find numbers of cans the school can collect on each of the final 7 days of the drive to meet this goal. Identify what your variable represents.
Type below:
___________

Answer: 7x + 668 > 2000

Explanation:
Let x represent the number of cans collected each day. Then 7x is the total number of cans collected on the final 7 days of the drive.
Since they have collected 668 cans already, the total number of cans collected is 7x + 668.
They want to collect more than 2,000 cans, so the inequality is:
7x + 668 > 2000

Practice and Homework Lesson 7.2 Answer Key Question 13.
Joanna joins a CD club. She pays $7 per month plus $10 for each CD that she orders. Write an inequality to find how many CDs she can purchase in a month if she spends no more than $100. Identify what your variable represents.
Type below:
___________

Answer: 7 + 10x ≤ 100

Explanation:
Let x represent the number of CDs then 10x is the total amount spent on CDs and 7 + 10x is the total purchase amount for the month.
7 + 10x ≤ 100

Question 14.
Lionel wants to buy a belt that costs $22. He also wants to buy some shirts that are on sale for $17 each. He has $80. What inequality can you write to find the number of shirts he can buy? Identify what your variable represents.
Type below:
___________

Answer: 22 + 17x ≤ 80

Explanation:
Let x represent the number of shirts he can buy then 17x is the total cost of the shirts and 22 + 17x is the total cost
22 + 17x ≤ 80

Page No. 216

Question 15.
Write a situation for 15x − 20 ≤ 130 and solve.
Type below:
___________

Answer:
You are given in the inequality 15x − 20 ≤ 130 and need to write a situation that is represented by this inequality. A possible situation could be:
You are going shopping to buy some shirts. The shirts cost $15 each. You have a $20 gift card and $130 in cash. How many shirts can you buy?
This solution is modeled by the inequality 15x − 20 ≤ 130 because if we let x be the number of shirts you buy, then 15x is the cost of the shirts before you use the gift card since each shirt is $15. The total cost after using the $20 gift card would then be 15x – 20 dollars. Since you have $130 in cash, you can spend at most $130 so the total cost of 15x – 20 must then be less than or equal to 130.
Therefore, 15x − 20 ≤ 130.
Solving this gives:
15x − 20 ≤ 130
Add 20 on both sides
15x ≤ 130
Divide both sides by 15.
x ≤ 10

Analyze Relationships

Write >, <, ≥, or ≤ in the blank to express the given relationship.

Question 16.
m is at least 25
______

Answer: m ≥ 25

Explanation:
m is at least 25 means m ≥ 25 since at least means it can equal or needs to be bigger.

Question 17.
k is no greater than 9
______

Answer: k ≤ 9

Explanation:
k is no greater than 9 means k ≤ 9 since no greater means it can equal or must be smaller.

Question 18.
p is less than 48
______

Answer: p < 48

Explanation:
p is less than 48 means p < 48 since the < symbol in words is “less than”.

Question 19.
b is no more than -5
______

Answer: b ≤ -5

Explanation:
b is no more than -5 means b ≤ -5 since no more means it can equal or must be smaller.

Question 20.
h is at most 56
______

Answer: h ≤ 56

Explanation:
h is at most 56 means h ≤ 56 since at most means it can equal or must be smaller.

Question 21.
w is no less than 0
______

Answer: w ≥ 0

Explanation:
w is no less than 0 means w ≥ 0 since no less than means it can equal or is bigger.

Solving Two-Step Inequalities Worksheet Answer Key Question 22.
Critical Thinking
Marie scored 95, 86, and 89 on three science tests. She wants her average score for 6 tests to be at least 90. What inequality can you write to find the average scores that she can get on her next three tests to meet this goal? Use s to represent the lowest average score.
Type below:
______

Answer: (95 + 86 + 89 + 3s)/6 ≥ 90

Explanation:
Let s be the average score on the remaining 3 tests. Then 3s is the sum of these 3 remaining tests. Since averages are found by adding up all the members, which gives 95 + 86 + 89 + 3s, and dividing by the total amount of numbers, which is 6, the inequality to the left can be used to find the lowest average she can get to have a minimum total average of 90.
(95 + 86 + 89 + 3s)/6 ≥ 90

H.O.T.

Focus on Higher Order Thinking

Question 23.
Communicate Mathematical Ideas
Write an inequality that expresses the reason the lengths 5 feet, 10 feet, and 20 feet could not be used to make a triangle. Explain how inequality demonstrates that fact.
Type below:
__________

Answer:
A side of a triangle must be greater than the difference of the other two sides and smaller than the sum of the other two sides.
Since 20 > 5 +10, the side with length 20 is not less than the sum of the other two sides.

Question 24.
Analyze Relationships
The number m satisfies the relationship m < 0. Write an inequality expressing the relationship between -m and 0. Explain your reasoning.
Type below:
__________

Answer: Multiplying both sides of m < 0 by -1 gives -m > 0 since you must switch the inequality sign when you multiply by a negative number.

Question 25.
Analyze Relationships The number n satisfies the relationship n > 0. Write three inequalities to express the relationship between n and \(\frac{1}{n}\).
Type below:
__________

Answer: 0 < n < 1: n < \(\frac{1}{n}\), n = 1: n = \(\frac{1}{n}\), n > 1: n > \(\frac{1}{n}\)

Explanation:
Since n is positive, there are three cases for the value of n to consider when comparing n and \(\frac{1}{n}\).
0 < n < 1: In case, n is a fraction smaller than 1 such as \(\frac{1}{4}\). Its reciprocal is the n bigger than 1, such as 1/\(\frac{1}{4}\) = 4.
Therefore, n < \(\frac{1}{n}\)
n = 1: If n = 1, then the reciprocal is also 1 so n = \(\frac{1}{n}\)
n > 1: If n is a value greater than 1, such as \(\frac{3}{2}\), then the reciprocal is smaller than 1, such as 1/\(\frac{3}{2}\) = \(\frac{2}{3}\).
Therefore n > \(\frac{1}{n}\)

Page No. 220

Question 1.
Describe how to solve the inequality 3x + 4 < 13 using algebra tiles.

Type below:
__________

Answer:
First, remove 4 positive squares from each side. This leaves 9 positive squares on the right side.
Then divide each side into 3 equal groups.
Each group would then have 3 positive squares on the left side. since the rectangles on the left are positive, the answer would be x < 3.

Solve each inequality. Graph and check the solution.

Question 2.
5d − 13 < 32
Type below:
__________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
5d − 13 < 32
Add 13 on both sides
5d – 13 + 13 < 32 + 13
5d < 45
Divide 5 on both sides
5d/5 < 45/5
d < 9
The number opposite to the variable is 9, we look for this in the number line. Since the inequality is <, we use a hollow dot and shade the line going to the left. Its graph would like the one below:

Question 3.
−4b + 9 ≤ −7
Type below:
__________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
−4b + 9 ≤ −7
Subtract 9 on both sides
-4b + 9 – 9 ≤ −7 – 9
-4b ≤ −16
Divide -4 by both sides
-4b/-4 ≤ −16/-4
b ≥ 4
The number opposite to the variable is 4, we look for this in the number line. Since the inequality is ≥, we use a closed dot and shade the line going to the right. Its graph would like the one below:

Circle any given values that make the inequality true.

Question 4.
2m + 18 > −4
m = −12; m = −11; m = −10
Type below:
__________

Answer: m = -10

Explanation:
m = -12
2m + 18 > −4
2(-12)  + 18 = -24 + 18 = -6 > -4 not true
m = -11
2m + 18 > −4
2(-11) + 18 > -4
-22 + 18 > -4
-4 > -4 not true
m = -10
2m + 18 > −4
2(-10) + 18 > -4
-20 + 18 > -4
-2 > -4 true

Unit 1 Equations and Inequalities Homework 3 Solving Equations Question 5.
−6y + 3 ≥ 0
y = 1; y = \(\frac{1}{2}\); y = 0
Type below:
__________

Answer: y = \(\frac{1}{2}\); y = 0

Explanation:
y = 1
−6y + 3 ≥ 0
-6(1) + 3 ≥ 0
-6 + 3 ≥ 0 not true
y = \(\frac{1}{2}\)
−6y + 3 ≥ 0
−6(\(\frac{1}{2}\)) + 3 ≥ 0
-3 + 3 ≥ 0
0 ≥ 0 true
y = 0
−6y + 3 ≥ 0
-6(0) + 3 ≥ 0
3 ≥ 0 true

Question 6.
Lizzy has 6.5 hours to tutor 4 students and spend 1.5 hours in a lab. She plans to tutor each student the same amount of time. The inequality 6.5 − 4t ≥ 1.5 can be used to find t, the amount of time in hours Lizzy could spend with each student. Solve the inequality. Graph and interpret the solution. Can Lizzy tutor each student for 1.5 hours? Explain.
Type below:
__________

Answer:
6.5 − 4t ≥ 1.5
Subtract both sides by 6.5
6.5 − 4t – 6.5 ≥ 1.5 – 6.5
− 4t ≥ 1.5
Divide both sides by -4 remember to switch the inequality sign since you are dividing by a negative number.
t ≤ 1.25
Since the inequality sign has an equal sign, draw a closed circle at 1.25. Since the inequality is less than, draw an arrow to the left the interpretation of this problem is that she can tutor each student no more than 1.25 hours.
Since 1.5 hours is more than 1.25 hours, she cannot tutor each student for 1.5 hours.

Essential Question Check-In

Question 7.
How do you solve a two-step inequality?
Type below:
__________

Answer:
To solve a two-step inequality you must use inverse operations.
Use subtraction to get rid of addition, and addition to get rid of subtraction for any terms that are being added or subtracted to the term that contains the variable. Use multiplication to get rid of any coefficient on the variable if it is a fraction and use division to get rid of any coefficient on the variable if its an integer. If the coefficient is negative, make sure to switch the inequality sign if you multiply or divide by a negative number.

Independent Practice – Page No. 221

Solve each inequality. Graph and check the solution.

Question 8.
2s + 5 ≥ 49
Type below:
__________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
2s + 5 ≥ 49
Subtract 5 on both sides
2s + 5 – 5 ≥ 49 – 5
2s ≥ 44
Divide both sides by 2.
s ≥ 22
The number opposite to the variable is 22, we look for this in the number line. Since the inequality is ≥, we use a closed dot and shade the line going to the right. Its graph would be like the one below:

Question 9.
−3t + 9 ≥ −21
Type below:
__________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
−3t + 9 ≥ −21
Subtract 9 on both sides
-3t + 9 – 9 ≥ −21 – 9
-3t ≥ −30
Divide by -3 on both sides
t ≤ 10
The number opposite to the variable is 10, we look for this in the number line. Since the inequality is ≤, we use a closed dot and shade the line going to the left. Its graph would be like the one below:

Lesson 2 Solve One-Step Inequalities Question 10.
55 > −7v + 6
Type below:
__________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
55 > −7v + 6
Subtract 6 on both sides
55 – 6 > −7v + 6 – 6
49 > -7v
Divide -7 on both sides
-7 < v
v > -7
The number opposite to the variable is -7, we look for this in the number line. Since the inequality is >, we use a hollow dot and shade the line going to the right. Its graph would be like the one below:

Question 11.
41 > 6m − 7
Type below:
__________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
41 > 6m − 7
Add 7 on both sides
41 + 7 > 6m – 7 + 7
48 > 6m
Divide by 6 on both sides
8 > m
m < 8
The number opposite to the variable is 8, we look for this in the number line. Since the inequality is <, we use a hollow dot and shade the line going to the left. Its graph would like the one below:

Question 12.
\(\frac{a}{-8}\) + 15 > 23
Type below:
__________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
\(\frac{a}{-8}\) + 15 > 23
Subtract 15 on both sides
\(\frac{a}{-8}\) + 15 – 15 > 23 – 15
(-8)\(\frac{a}{-8}\) > (8)(-8)
a < -64
The number opposite to the variable is -64, we look for this in the number line. Since the inequality is <, we use a hollow dot and shade the line going to the left. Its graph would be like the one below:

Question 13.
\(\frac{f}{2}\) − 22 < 48
Type below:
__________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
\(\frac{f}{2}\) − 22 < 48
Add 22 on both sides
\(\frac{f}{2}\) − 22 + 22 < 48 + 22
\(\frac{f}{2}\) < 70
Multiply 2 on both sides
(2)\(\frac{f}{2}\) < 70(2)
f < 140
The number opposite to the variable is 140, we look for this in the number line. Since the inequality is <, we use a hollow dot and shade the line going to the left. Its graph would like the one below:

Question 14.
−25 + \(\frac{t}{2}\) ≥ 50
Type below:
__________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
−25 + \(\frac{t}{2}\) ≥ 50
Add 25 on both sides
−25 + \(\frac{t}{2}\) + 25 ≥ 50 + 25
\(\frac{t}{2}\) ≥ 75
Multiply 2 on both sides
(2)\(\frac{t}{2}\) ≥ 75 (2)
t ≥ 150
The number opposite to the variable is 150, we look for this in the number line. Since the inequality is ≥, we use a closed dot and shade the line going to the right. Its graph would like the one below:

Question 15.
10 + \(\frac{g}{-9}\) > 12
Type below:
__________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
10 + \(\frac{g}{-9}\) > 12
Subtract 10 on both sides
10 + \(\frac{g}{-9}\) -10 > 12 – 10
\(\frac{g}{-9}\) > 2
Divide both sides by -9
g < -18
The number opposite to the variable is -18, we look for this in the number line. Since the inequality is <, we use a hollow dot and shade the line going to the left. Its graph would like the one below:

Question 16.
25.2 ≤ − 1.5y + 1.2
Type below:
__________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
25.2 ≤ − 1.5y + 1.2
Subtract 1.2 on both sides
25.2 – 1.2 ≤ − 1.5y + 1.2 – 1.2
24 ≤ -1.5y
Divide both sides by 1.5
-16 ≥ y
y ≤ -16
The number opposite to the variable is -16, we look for this in the number line. Since the inequality is ≤, we use a closed dot and shade the line going to the left. Its graph would like the one below:

Question 17.
−3.6 ≥ −0.3a + 1.2
Type below:
__________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
−3.6 ≥ −0.3a + 1.2
Subtract 1.2 on both sides
−3.6 – 1.2 ≥ −0.3a + 1.2 – 1.2
-4.8 ≥ −0.3a
Divide both sides by -0.3
16 ≤ a
a ≥ 16
The number opposite to the variable is 16, we look for this in the number line. Since the inequality is ≥, we use a closed dot and shade the line going to the right. Its graph would like the one below:

Question 18.
What If?
The perimeter of a rectangle is at most 80 inches. The length of the rectangle is 25 inches. The inequality 80 − 2w ≥ 50 can be used to find w, the width of the rectangle in inches. Solve the inequality and interpret the solution. How will the solution change if the width must be at least 10 inches and a whole number?
Type below:
__________

Answer:
The perimeter of a rectangle is at most 80 inches. The length of the rectangle is 25 inches. The inequality 80 − 2w ≥ 50 can be used to find w, the width of the rectangle in inches.
Subtract 80 on both sides
-2w ≥ -30
Divide -2 on both sides
w ≤ 15
The interpretation of this solution is that width must be a positive real number that is at most 15 inches.
If the width must be at least 10 inches and a whole number, then the solution would be the whole number lengths such that 10 ≤ w ≤ 15 which is 10, 11, 12, 13, 14 or 15 inches.

Page No. 222

Question 19.
Interpret the Answer
Grace earns $7 for each car she washes. She always saves $25 of her weekly earnings. This week, she wants to have at least $65 in spending money. How many cars must she wash? Write and solve an inequality to represent this situation. Interpret the solution in context.
Type below:
__________

Answer:
Let x represent the number of cars she must wash then 7x is the total amount she makes from washing cars. Since she is saving $25 of her earnings, 7x – 25 is the amount of earnings she will have to spend.
7x – 25 ≥ 65
Add 25 on both sides
7x – 25 + 25 ≥ 65 + 25
7x ≥ 90
Divide both sides by 7.
x ≥ 90/7 ≈ 13

H.O.T.

Focus on Higher Order Thinking

Question 20.
Critical Thinking
Is there any value of x with the property that x < x – 1? Explain your reasoning.
Type below:
__________

Answer:
Subtracting x on both sides of x < x – 1 gives x – x < x – 1 -x
0 < -1.
Since this is not a true statement, inequality is not true for any value of x.

Lesson 3 Homework Practice Solve One Step Inequalities Question 21.
Analyze Relationships A compound inequality consists of two simple equalities joined by the words “and” or “or.” Graph the solution sets of each of these compound inequalities. a. x > 2 and x < 7
Type below:
__________

Answer: Since the two inequalities are joined by “and”, we need to satisfy both inequalities. Therefore, we need all values greater than 2 but also we need values less than 7. Hence we can place a hollow dot on 2 and shade the line going to the right until another hollow dot on 7.

Question 21.
b. x < 2 or x > 7
Type below:
__________

Answer: Since the two inequalities are joined by “or”, we need to satisfy either of the inequalities. Therefore, we need all values less than 2 but also we need to graph the values greater than 7. Hence we can place a hollow dot on 2 and shade the line going to the left and another hollow dot on 7 and shade the line going to the right.

Question 21.
c. Describe the solution set of the compound inequality x < 2 and x > 7.
Type below:
__________

Answer: The solution set for x > 2 and x < 7 should be 2 < x < 7.

Question 21.
d. Describe the solution set of the compound inequality x > 2 or x < 7.
Type below:
__________

Answer: The solution set for x < 2 and x > 7 sould be (-∞, 2) ∪ (7, ∞).

Question 22.
Communicate Mathematical Ideas
Joseph used the problem-solving strategy Work Backward to solve the inequality 2n + 5 < 13. Shawnee solved the inequality using the algebraic method you used in this lesson. Compare the two methods.
Type below:
__________

Answer: Both involve using the same operations. The only difference is that working backward is done mostly mentally while algebraically is done on paper. It is easier to determine which direction the inequality is pointing when using the algebraic method.

7.1 Writing and Solving One-Step Inequalities – Page No. 223

Solve each inequality.

Question 1.
n + 7 < −3
Type below:
__________

Answer: n < -10

Explanation:
Subtract 7 on both sides
n + 7 – 7 < −3 – 7
n < -10

Question 2.
5p ≥ −30
Type below:
__________

Answer: p ≥ -6

Explanation:
5p ≥ −30
Divide by 5 on both sides
p ≥ -6

Solving One Step Inequalities by Adding/Subtracting Question 3.
14 < k + 11
Type below:
__________

Answer: 3 < k

Explanation:
14 < k + 11
Subtract 11 on both sides
14 – 11 < k + 11 – 11
3 < k

Question 4.
\(\frac{d}{-3}\) ≤ −6
Type below:
__________

Answer: d ≥ 18

Explanation:
\(\frac{d}{-3}\) ≤ −6
Multiply both sides by -3 remember to switch the inequality sign since you are multiplying both sides by a negative number.
d ≥ 18

Question 5.
c − 2.5 ≤ 2.5
Type below:
__________

Answer: c ≤ 5

Explanation:
c − 2.5 ≤ 2.5
Add 2.5 on both sides
c − 2.5 + 2.5 ≤ 2.5 + 2.5
c ≤ 5

Question 6.
12 ≥ −3b
Type below:
__________

Answer: -4 ≤ b

Explanation:
12 ≥ −3b
Divide by -3 on both sides
-4 ≤ b

Question 7.
Jose has scored 562 points on his math tests so far this semester. To get an A for the semester, he must score at least 650 points. Write and solve an inequality to find the minimum number of points he must score on the remaining tests in order to get an A.
Type below:
__________

Answer: x + 562 ≥ 650

Explanation:
Let x represent the score he must receive on the remaining tests. Then x + 562 is his total score for the semester.
x + 562 ≥ 650
Subtract 562 on both sides
x ≥ 88

7.2 Writing Two-Step Inequalities

Question 8.
During a scuba dive, Lainey descended to a point 20 feet below the ocean surface. She continued her descent at a rate of 20 feet per minute. Write an inequality you could solve to find the number of minutes she can continue to descend if she does not want to reach a point more than 100 feet below the ocean surface.
Type below:
__________

Answer:
Let x represent the number of minutes. Since she is descending 20 feet per minute, then -20x represents her altitude. It is negative since descending means her altitude is decreasing.
Since she started at 20 feet below the ocean surface, she started at -20 feet. Its negative since an altitude below the ocean surface must be represented by a negative number.
Her ending position is the sum of how far she has descended and her initial position so her ending position is -20 + (-20x) = -20 – 20x
She doesn’t want to travel more than 100 feet below the ocean surface so she needs to be higher than -100 feet.
The inequality is then -20 -20x ≥ -100.

7.3 Solving Two-Step Inequalities

Solve.

Question 9.
2s + 3 > 15
Type below:
__________

Answer: s > 6

Explanation:
2s + 3 > 15
Subtract 3 on both sides
2s + 3 – 3 > 15 – 3
2s > 12
Divide by 2 on both sides
s > 6

Question 10.
−\(\frac{d}{12}\) − 6 < 1
Type below:
__________

Answer: d > -84

Explanation:
−\(\frac{d}{12}\) − 6 < 1
Add 6 on both sides
−\(\frac{d}{12}\) − 6 + 6 < 1 + 6
d > -84

Question 11.
−6w − 18 ≥ 36
Type below:
__________

Answer: w ≤ -9

Explanation:
−6w − 18 ≥ 36
Add 18 on both sides
−6w − 18 + 18 ≥ 36 + 18
-6w ≥ 54
Divide by -6
w ≤ -9

Solving One-Step Inequalities Answer Key Question 12.
\(\frac{z}{4}\) + 22 ≤ 38
Type below:
__________

Answer: z ≤ 64

Explanation:
Subtract 22 on both sides
\(\frac{z}{4}\) + 22 – 22 ≤ 38 – 22
\(\frac{z}{4}\) ≤ 16
z ≤ 64

Question 13.
\(\frac{b}{9}\) − 34 < −36
Type below:
__________

Answer: b < -18

Explanation:
\(\frac{b}{9}\) − 34 < −36
Add 34 on both sides
\(\frac{b}{9}\) − 34 + 34 < −36 + 34
\(\frac{b}{9}\) < -2
b < -18

Question 14.
−2p + 12 > 8
Type below:
__________

Answer: p < 2

Explanation:
−2p + 12 > 8
Subtract 12 on both sides
-2p + 12 – 12 > 8 – 12
-2p > -4
p < 2

Essential Question

Question 15.
How can you recognize whether a real-world situation should be represented by an equation or an inequality?
Type below:
__________

Answer:
You use an equation when the situation involves finding an exact answer. You use an inequality when the solution can have more than one value. Problems that require the use of inequalities have phrases in them such as “at least”, “no more”, “at most” and “no less than”.

Selected Response – Page No. 224

Question 1.
Which graph models the solution of the inequality −6 ≤ −3x?
Options:
a.
b.
c.
d.

Answer:
Dividing both sides of −6 ≤ −3x by -3 gives 2 ≥ x.
Rewriting this so x is on the left side gives x ≤ 2. The graph must have a closed circle at 2 since the inequality has an equal sign and must be shaded to the left since its <.
Thus the correct answer is option C.

Question 2.
A taxi cab costs $1.75 for the first mile and $0.75 for each additional mile. You have $20 to spend on your ride. Which inequality could be solved to find how many miles you can travel if n is the number of additional miles?
Options:
a. 1.75n + 0.75 ≥ 20
b. 1.75n + 0.75 ≤ 20
c. 0.75n + 1.75 ≥ 20
d. 0.75n + 1.75 ≤ 20

Answer: 1.75n + 0.75 ≤ 20

Explanation:
Let n represent the number of additional miles. Then 0.75n is the cost of the additional miles which gives a total cost of 1.75 + 0.75n. You can spend a maximum of $20 so the inequality is ≤.
Thus the correct answer is option B.

Writing and Solving One-Step Inequalities Question 3.
The inequality \(\frac{9}{5}\)C + 32 < −40 can be used to find Celsius temperatures that are less than -40° Fahrenheit. What is the solution of the inequality?
Options:
a. C < 40
b. C < −\(\frac{40}{9}\)
c. C < −40
d. C < −\(\frac{72}{5}\)

Answer: C < −40

Explanation:
\(\frac{9}{5}\)C + 32 < −40
Subtract 32 on both sides.
\(\frac{9}{5}\)C + 32 – 32 < −40 – 32
\(\frac{9}{5}\)C < -72
c < -40
Thus the correct answer is option C.

Question 4.
The 30 members of a choir are trying to raise at least $1,500 to cover travel costs to a singing camp. They have already raised $600. Which inequality could you solve to find the average amounts each member can raise that will at least meet the goal?
Options:
a. 30x + 600 > 1,500
b. 30x + 600 ≥ 1,500
c. 30x + 600 < 1,500
d. 30x + 600 ≤ 1,500

Answer: 30x + 600 ≥ 1,500

Explanation:
Given,
The 30 members of a choir are trying to raise at least $1,500 to cover travel costs to a singing camp. They have already raised $600.
Let x represent the average amount each member raises.
There are 30 members so the members raise a combined amount of 30x.
Since they have already raised $600, the total amount raised is 30x + 600 they need to raise at least $1500 so the inequality is ≥.
Thus the correct answer is option B.

Question 5.
Which represents the solution for the inequality 3x − 7 > 5?
Options:
a. x < 4
b. x ≤ 4
c. x > 4
d. x ≥ 4

Answer: x > 4

Explanation:
Add 7 on both sides
3x − 7 > 5
3x − 7 + 7> 5 + 7
3x > 12
x > 4
Thus the correct answer is option C.

Question 6.
Which inequality has the following graphed solution?

Options:
a. 3x + 8 ≤ 2
b. 4x + 12 < 4
c. 2x + 5 ≤ 1
d. 3x + 6 < 3

Answer: 4x + 12 < 4

Explanation:
4x + 12 < 4
Subtract 12 on both sides
4x + 12 – 12 < 4 – 12
4x < -8
x < -2
Thus the correct answer is option B.

Question 7.
Divide: −36 ÷ 6.
Options:
a. 30
b. 6
c. -6
d. -30

Answer: -6

Explanation:
6 divides 36 six times
−36 ÷ 6 = -6
Thus the correct answer is option C.

Question 8.
Eleni bought 2 pounds of grapes at a cost of $3.49 per pound. She paid with a $10 bill. How much change did she get back?
Options:
a. $3.02
b. $4.51
c. $6.51
d. $6.98

Answer: $3.02

Explanation:
Given,
Eleni bought 2 pounds of grapes at a cost of $3.49 per pound. She paid with a $10 bill.
We have to find the total amount paid for the grapes
2 × 3.49 = 6.98
10 – 6.98 = 3.02
Thus the correct answer is option A.

Question 9.
In golf, the lower your score, the better. Negative scores are best of all. Teri scored +1 on each of the first three holes at a nine-hole miniature golf course. Her goal is a total score of -9 or better after she has completed the final six holes.
a. Let h represent the score Teri must average on each of the last six holes in order to meet her goal. Write a two-step inequality you can solve to find h.
Type below:
_____________

Answer: 3 +6h ≤ -9

Explanation:
If h is her average score for the last 6 holes, then 6h is her total score for the last 6 holes.
She currently has a score of 3 so her total score for all 9 holes is 3 + 6h.
She wants a score of -9 or better and since smaller scores are better, the inequality is ≤.

Question 9.
b. Solve the inequality.
Type below:
_____________

Answer: h ≤ -2

Explanation:
6h ≤ -12
Divide by 6 on both sides
h ≤ -2

EXERCISES – Page No. 226

Simplify each expression.

Question 1.
\(\left(2 x+3 \frac{2}{5}\right)+\left(5 x-\frac{4}{5}\right)\)
Type below:
_____________

Answer: 7x + 2 \(\frac{3}{5}\)

Explanation:
We are given  the expression,
\(\left(2 x+3 \frac{2}{5}\right)+\left(5 x-\frac{4}{5}\right)\)
Group the like terms
(2x + 5x) + (3\(\frac{2}{5}\) – \(\frac{4}{5}\))
7x + 2 \(\frac{3}{5}\)

Question 2.
(−0.5x − 4) − (1.5x + 2.3)
Type below:
_____________

Answer: -2x – 6.3

Explanation:
(−0.5x − 4) − (1.5x + 2.3)
(−0.5x − 4) − 1.5x – 2.3
Combine the like terms
-0.5x – 1.5x – 4 – 2.3
-2x – 6.3

Lesson 5 Inequalities Answer Key Question 3.
9(3t + 4b)
Type below:
_____________

Answer: 27t + 36b

Explanation:
9(3t + 4b)
9 × 3t + 9 × 4b
27t + 36b

Question 4.
0.7(5a − 13p)
Type below:
_____________

Answer: 3.5a – 9.1p

Explanation:
0.7(5a − 13p)
0.7 × 5a – 0.7 × 13p
3.5a – 9.1p

Factor each expression.

Question 5.
8x + 56
Type below:
_____________

Answer: 8(x + 7)

Explanation:
Since 56 ÷ 8 = 7 and 8 ÷ 8 = 1, factor out 8 from both terms
8x + 56 = 8(x + 7)

Question 6.
3x + 57
Type below:
_____________

Answer: 3(x + 19)

Explanation:
Since 3 ÷ 3 = 1 and 57 ÷ 3 = 19 factor out 3 from both terms.
3x + 57 = 3(x + 19)

Question 7.
1.6 + y = −7.3
_______

Answer: y = -8.9

Explanation:
subtract 1.6 on both sides
1.6 + y – 1.6 = -7.3 – 1.6
y = -8.9

Question 8.
−\(\frac{2}{3}\) n = 12
_______

Answer: n = -18

Explanation:
−\(\frac{2}{3}\) n = 12
n = −\(\frac{3}{2}\)(12)
n = -3 × 6
n = -18

Question 9.
The cost of a ticket to an amusement park is $42 per person. For groups of up to 8 people, the cost per ticket decreases by $3 for each person in the group. Marcos’s ticket cost $30. Write and solve an equation to find the number of people in Marcos’s group.
_______ people

Answer: 4

Explanation:
Given that,
The cost of a ticket to an amusement park is $42 per person.
For groups of up to 8 people, the cost per ticket decreases by $3 for each person in the group. Marcos’s ticket cost $30.
Let x represent the number of people in his group then the ticket price has changed by -3x dollars.
The original price is $42 so the discount price is 42 – 3x
42 – 3x = 30
-3x = 30 – 42
-3x = -12
Divide both sides by -3
x = 4
Thus there are 4 people in Marcos’s group.

Lesson 5 Homework Practice Solve One-Step Inequalities Question 10.
8x − 28 = 44
_______

Answer:
To plot a point, starting from 0, count the number of units going to the left or right.
Given equation is
8x − 28 = 44
Add 28 on both sides
8x – 28 + 28 = 44 + 28
8x = 72
Divide by 8 on both sides
x = 9
To plot 9 on a number line, from 0, we move 9 units to the right.

Question 11.
−5z + 4 = 34
_______

Answer:
To plot a point, starting from 0, count the number of units going to the left or right.
Given equation is
−5z + 4 = 34
Subtract 4 on both sides
-5z + 4 – 4 = 34 – 4
-5z = 30
Divide both sides by -5
z = -6
To plot -6 on the number line, from 0, we move 6 units to the left.

EXERCISES – Page No. 227

Question 1.
Prudie needs $90 or more to be able to take her family out to dinner. She has already saved $30 and wants to take her family out to eat in 4 days.
a. Suppose that Prudie earns the same each day. Write an inequality to find how much she needs to earn each day.
Type below:
___________

Answer:
Let x be the amount she makes each day then 4x is the amount she will make in the 4 days before she takes her family out to eat and 4x + 30 is the total amount she will have saved.
4x + 30 ≥ 90

Question 1.
b. Suppose that Prudie earns $18 each day. Will she have enough money to take her family to dinner in 4 days? Explain.
_______

Answer:
4(18) + 30 = 72 + 30 = 102
She will have saved $102 in total if she earns $18 each day so she will have enough money.

Solve each inequality. Graph and check the solution.

Question 2.
11 − 5y < −19
Type below:
___________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
11 − 5y < −19
Subtract 11 on both sides
11 – 5y – 11 < −19 – 11
-5y < -30
Divide by -5 on both sides
y > 6
The number opposite to the variable is 6, we look for this in the number line. Since the inequality is >, we use a hollow dot and shade the line going to the right. Its graph would like the one below:

Question 3.
7x − 2 ≤ 61
Type below:
___________

Answer:
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
7x − 2 ≤ 61
Add 2 on both sides
7x – 2 + 2 ≤ 61 + 2
7x ≤ 63
Divide by 7 into both sides
x ≤ 9
The number opposite to the variable is 9, we look for this in the number line. Since the inequality is ≤, we use a closed dot and shade the line going to the left. Its graph would like the one below:

Unit 3 Performance Tasks – Page No. 228

Question 1.
Mechanical Engineer
A mechanical engineer is testing the amount of force needed to make a spring stretch by a given amount. The force y is measured in units called Newtons, abbreviated N. The stretch x is measured in centimeters. Her results are shown in the graph.
a. Write an equation for the line. Explain, using the graph and then using the equation, why the relationship is proportional.

Type below:
___________

Answer: The graph is linear and passes through the origin so the relationship is proportional. find k by using the formula k = y/x where (x, y) is a point on the line then plug k into the equation of a line y = kx.

Question 1.
b. Identify the rate of change and the constant of proportionality.
Type below:
___________

Answer: k = 8

Explanation:
Observe part a the units N/cm since the units for y are N and the units for x are cm and the units for k must be the units for y divided by the units for x.
8 N/ cm
k = 8

Question 1.
c. What is the meaning of the constant of proportionality in the context of the problem?
Type below:
___________

Answer:
Since the rate of change is 8 N/ cm this means that for every 1 cm stretch in the spring, the force required in Newton increases by 8 N.

Question 1.
d. The engineer applies a force of 41.6 Newtons to the spring. Write and solve an equation to find the corresponding stretch in the spring.
______ cm

Answer:
y = 8x
41.6 = 8x
x = 41.6/8
x = 5.2 cm

Solving Two-Step Inequalities Worksheet Answer Key 7th Grade Question 2.
A math tutor charges $30 for a consultation, and then $25 per hour. An online tutoring service charges $30 per hour.
a. Does either service represent a proportional relationship? Explain.
Type below:
___________

Answer:
The math tutor charges $30 initially but has a constant rate of $25 per hour after. This means that it still is a proportional relationship. Online tutoring charges a constant rate of $30 per hour and thus is also a proportional relationship.

Question 2.
b. Write an equation for the cost c of h hours of tutoring for either service. Which service charges less for 4 hours of tutoring? Show your work.
Type below:
___________

Answer:
Using y as the total cost and x as the number of hours, we can represent each tutoring service. For the Math tutor, we can write this as y = 30 + 25x while for the online tutoring, we can write this as y = 30x.
Substituting x = 4, we can see that:
For Math tutor:
y = 30 + 25x
y = 30 + 25(4)
y = 30 + 100
y = 130
For Online tutoring:
y = 30x
y = 30(4)
y = 120
Therefore, the online tutoring service charges less at $120.

Selected Response – Page No. 229

Question 1.
Which expression is equivalent to (9x − 3 \(\frac{1}{8}\)) − (7x + 1 \(\frac{3}{8}\))?
Options:
a. 2x − 4 \(\frac{1}{2}\)
b. 16x−4 \(\frac{1}{2}\)
c. 2x − 1 \(\frac{3}{4}\)
d. 16x − 1 \(\frac{3}{4}\)

Answer: 2x − 4 \(\frac{1}{2}\)

Explanation:
(9x − 3 \(\frac{1}{8}\)) − (7x + 1 \(\frac{3}{8}\))
9x − 3 \(\frac{1}{8}\)− 7x – 1 \(\frac{3}{8}\)
Combine the like terms
2x – 4 \(\frac{1}{2}\)
Thus the correct answer is option A.

Question 2.
Timothy began the week with $35. He bought lunch at school, paying $2.25 for each meal. Let x be the number of meals he bought at school and y be the amount of money he had left at the end of the week. Which equation represents the relationship in the situation?
Options:
a. y = 2.25x + 35
b. y = 35 − 2.25x
c. x = 35 − 2.25y
d. y = 2.25x − 35

Answer: y = 35 − 2.25x

Explanation:
Let x be the number of meals he buys means 2.25x is the amount of money he has spent on meals.
The money he has left is then 35 – 2.25x
The expression is y = 35 – 2.25x
Thus the correct answer is option B.

Question 3.
Which expression factors to 8(x + 2)?
Options:
a. 8x + 2
b. 8x + 10
c. 16x
d. 8x + 16

Answer: 8x + 16

Explanation:
Given the expression
8(x + 2)
8x + 8(2)
8x + 16
Thus the correct answer is option D.

Question 4.
Ramón’s toll pass account has a value of $32. Each time he uses the toll road, $1.25 is deducted from the account. When the value drops below $10, he must add value to the toll pass. Which inequality represents how many times Ramón can use the toll road without having to add value to the toll pass?
Options:
a. 10 − 1.25t ≥ 0
b. −1.25t + 32 < 10
c. 32 − 1.25t ≥ 10
d. 32 − 10t ≥ 1.25

Answer: 32 − 1.25t ≥ 10

Explanation:
Let t represent the number of times he uses the toll road then 1.25t is the amount deducted from his account. the remaining balance is then 32 – 1.25t.
Since his balance must be at least $10 for him to not have to add value, the inequality sign is ≥
Thus the correct answer is option C.

Question 5.
A taxi costs $1.65 for the first mile and $0.85 for each additional mile. Which equation could be solved to find the number x of additional miles traveled in a taxi given that the total cost of the trip is $20?
Options:
a. 1.65x + 0.85 = 20
b. 0.85x + 1.65 = 20
c. 1.65x − 0.85 = 20
d. 0.85x − 1.65 = 20

Answer: 0.85x + 1.65 = 20

Explanation:
Let x be the number of additional miles means 0.85x is the cost of the additional miles The total cost is then 1.65 + 0.85x
1.65 + 0.85x = 20
Thus the correct answer is option B.

Question 6.
A sales tax of 6% is added to the price of an item. If Marisa buys an item, which expression indicates how much she will pay in all?
Options:
a. n + 0.06
b. 0.06n
c. n + 0.06n
d. 0.06 + 0.06n

Answer: n + 0.06n

Explanation:
The total cost she will pay is the cost of the item n plus the cost of tax 0.06n.
The expression is n + 0.06n
Thus the correct answer is option C.

Question 7.
Which equation has the solution x = 12?
Options:
a. 4x + 3 = 45
b. 3x + 6 = 42
c. 2x − 5 = 29
d. 5x −8 = 68

Answer: 3x + 6 = 42

Explanation:
a. 4x + 3 = 45
Substitute x = 12 in the above equation.
4(12) + 3 = 45
48 + 3 = 45
51 ≠ 45
b. 3x + 6 = 42
Substitute x = 12 in the above equation.
3(12) + 6 = 42
36 + 6 = 42
42 = 42
c. 2x − 5 = 29
Substitute x = 12 in the above equation.
2(12) – 5 = 29
24 – 5 = 29
19 ≠ 29
d. 5x −8 = 68
Substitute x = 12 in the above equation.
5(12) – 8 = 68
60 – 8 = 68
52 ≠ 68
Thus the correct answer is option B.

Question 8.
The 23 members of the school jazz band are trying to raise at least $1,800 to cover the cost of traveling to a competition. The members have already raised $750. Which inequality could you solve to find the amount that each member should raise to meet the goal?
Options:
a. 23x + 750 > 1,800
b. 23x + 750 ≥ 1,800
c. 23x + 750 < 1,800
d. 23x + 750 ≤ 1,800

Answer: 23x + 750 ≥ 1,800

Explanation:
Let x represent the amount each member raises means 23x is the amount the members raise individually.
The total amount raised is then 23x + 750 since they have already raised $750.
Since they are trying to raise at least $1800, the inequality is ≥
Thus the correct answer is option B.

Page No. 230

Question 9.
What is the solution of the inequality 2x − 9 < 7?
Options:
a. x < 8
b. x ≤ 8
c. x > 8
d. x ≥ 8

Answer: x < 8

Explanation:
Given the inequality 2x − 9 < 7
Add 9 on both sides 2x – 9 + 9 < 7 + 9
2x < 16
Divide by 2 on both sides
2x/2 < 16/2
x < 8
Thus the correct answer is option A.

Question 10.
Which inequality has the solution n < 5?
Options:
a. 4n + 11 > −9
b. 4n + 11 < −9
c. −4n + 11 < −9
d. −4n + 11 > −9

Answer: −4n + 11 > −9

Explanation:
Given the inequality n < 5
To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
a. 4n + 11 > −9
4n + 11 – 11 > -9 – 11
4n/4 > -20/4
n  -5
b. 4n + 11 < −9
4n + 11 – 11 < -9 – 11
4n < -20
4n/4 < -20/4
n < -5
c. −4n + 11 < −9
-4n + 11 – 11 < -9 – 11
-4n < -20
-4n/-4 < -20/-4
n > 5
d. −4n + 11 > −9
-4n + 11 – 11 > -9 – 11
-4n > -20
-4n/-4 > -20/-4
n < 5
Thus the correct answer is option D.

Question 11.
Which inequality has the solution shown?

Options:
a. 3x + 5 < 2
b. 4x + 12 < 4
c. 2x + 5 ≤ 1
d. 3x + 6 ≤ 3

Answer: 3x + 6 ≤ 3

Explanation:
The graph shows the inequality x ≤ -1 so the possible answers are C and D since A and B have < as the inequality signs. Solve C and D for x to see which one has x ≤ -1 as the solution.
c. 2x + 5 ≤ 1
2x + 5 – 5 ≤ 1 – 5
2x ≤ -4
x ≤ -2
d. 3x + 6 ≤ 3
3x + 6 – 6 ≤ 3 – 6
3x ≤ -3
x ≤ -1
Thus the correct answer is option D.

Question 12.
On a 4 \(\frac{1}{2}\) hour trip, Leslie drove \(\frac{2}{3}\) of the time. For how many hours did Leslie drive?
Options:
a. 3 hours
b. 3 \(\frac{1}{2}\) hours
c. 3 \(\frac{2}{3}\) hours
d. 3 \(\frac{5}{6}\) hours

Answer: 3 hours

Explanation:
Given that,
On a 4 \(\frac{1}{2}\) hour trip, Leslie drove \(\frac{2}{3}\) of the time.
Multiply the two fractions by first writing 4 \(\frac{1}{2}\) as an improper fraction then cancel the 2s and then simplifying the division.
4 \(\frac{1}{2}\)(\(\frac{2}{3}\)) = \(\frac{9}{2}\) × \(\frac{2}{3}\) = 3
Thus the correct answer is option A.

Question 13.
During a sale, the price of a sweater was changed from $20 to $16. What was the percent of decrease in the price of the sweater?
Options:
a. 4%
b. 20%
c. 25%
d. 40%

Answer:

Mini-Task

Question 14.
Max wants to buy some shorts that are priced at $8 each. He decided to buy a pair of sneakers for $39, but the total cost of the shorts and the sneakers must be less than $75.
a. Write an inequality to find out how many pairs of shorts Max can buy.
Type below:
____________

Answer: 39 + 8x < 75

Explanation:
Let x be the number of shorts he buys then 8x is the total cost of the shorts and 8x + 39 is the total cost of the shorts and sneakers his total must be less than $75 so the inequality is <.
39 + 8x < 75

Question 14.
b. Suppose that Max wants to buy 6 pairs of shorts. Will he have enough money? Explain.
______

Answer: No

Explanation:
Find the total amount he will spend buying 6 pairs of shorts this is more than the $75 he has so he will not have enough.
39 + 8(6) = 39 + 48 = 87

Question 14.
c. Solve the inequality to find the greatest number of pairs of shorts that Max can buy. Show your work.
______ pairs of shoes

Answer: 4 pairs of shoes

Explanation:
Use the above inequality,
39 + 8x < 75
Subtract 39 on both sides and then divide both sides by 8. Since you can’t buy a fraction of a pair of shorts the most pairs he can buy is 4.
39 + 8x – 39 < 75 – 39
8x < 36
x < 4.5

Conclusion:

I hope the answers displayed in this article is beneficial for all the students of 7th grade. Even parents can refer to this Go Math Answer Key of Grade 7 to help their children do the homework and to explain the real-time examples.

Download Go Math Grade 7 Answer Key Chapter 10 Random Samples and Populations for free of cost. This chapter contains random samples and populations which was explained in detail way that students can understand easily. Students can improve their knowledge and skills by practicing in Go Math Grade 7 Answer Key for Chapter 10 Random Samples and Populations.

Go Math Grade 7 Answer Key Chapter 10 Random Samples and Populations

With the help of Go Math Grade 7 Solution Key, Chapter 10 Random Samples and Populations students can easily understand the concepts without any difficulty. Students can check the below links for the solutions which are given with a detailed explanation.

Chapter 10 Random Samples and Populations – Lesson 1

• Guided Practice – Page No. 314
• Independent Practice – Page No. 315
• H.O.T. – Page No. 316

Chapter 10 Random Samples and Populations – Lesson: 2

• Guided Practice – Page No. 320
• Independent Practice – Page No. 321
• Page No. 322

Chapter 10 – Random Samples and Populations 

• Page No. 326
• Page No. 327
• Page No. 328
• 10.1 Populations and Samples – Page No. 329
• Selected Response – Page No. 330

Guided Practice – Page No. 314

Question 1.
Follow each method described below to collect data to estimate the average shoe size of seventh-grade boys.

Answer:
Method 1:
Select randomly 5 seventh-grade boys and record their shoe size in a table.

Answer:

The mean is \(\frac{10+8+7.5+9+10}{5}
= \frac{44.5}{5}\)
= 8.9

Method 2:
Find the 5 boys in the class who have the largest shoe size and record on a table.

Answer:

Question 2.
Method 1 produces results that are more/less representative of the entire student population because it is a random/biased sample.

Answer: Method 1 produces results that are more representative of the entire student population because it is a random sample.

Random Sampling Answer Key Question 3.
Method 2 produces results that are more/less representative of the entire student population because it is a random/biased sample.

Answer: Method 2 produces results that are less representative of the entire student population because it is a biased sample.

Question 4.
Heidi decides to use a random sample to determine her classmates’ favorite color. She asks, “Is green your favorite color?” Is Heidi’s question biased? If so, give an example of an unbiased question that would serve Heidi better.

Answer: Heidi’s question is biased as it suggests that people should say their favorite color is green. “What was your favorite color?” is an unbiased question, as it doesn’t suggest a certain answer.

Essential Question Check-In

Question 5.
How can you select a sample so that the information gained represents the entire population?

Answer: We should select a sample that is randomly chosen and is sufficiently large enough so that the result so that results are representative of the entire population.

Independent Practice – Page No. 315

Question 6.
Paul and his friends average their test grades and find that the average is 95. The teacher announces that the average grade of all of her classes is 83. Why are the averages so different?

Answer: As Paul and his friends are not a randomly chosen sample of the class population, so the averages are different.

Question 7.
Nancy hears a report that the average price of gasoline is $2.82. She averages the prices of stations near her home. She finds the average price of gas to be $3.03. Why are the averages different?

Answer: The gas stations around Nancy’s home are not a randomly chosen sample of all gas stations in the country, so the averages are so different.

For 8–10, determine whether each sample is a random sample or a biased sample. Explain.

Question 8.
Carol wants to find out the favorite foods of students at her middle school. She asks the boys’ basketball team about their favorite foods.

Answer: As Carol asks only boys and girls are not represented in the sample, so the sample is biased.

Question 9.
Dallas wants to know what elective subjects the students at his school like best. He surveys students who are leaving band class.

Answer: Dallas asked only students who are in band class and elective subject students are not represented, so the sample is biased.

Samples and Populations Worksheet Question 10.
To choose a sample for a survey of seventh graders, the student council puts pieces of paper with the names of all the seventh graders in a bag and selects 20 names.

Answer: As all students had an equal chance of being represented in the survey, the sample is random.

Question 11.
Members of a polling organization survey 700 of the 7,453 registered voters in a town by randomly choosing names from a list of all registered voters. Is their sample likely to be representative?

Answer: The sample is large enough and randomly chosen from all registered voters so that every voter gets a chance of being selected.
So the sample is likely to be representative.

For 12–13, determine whether each question may be biased. Explain.

Question 12.
Joey wants to find out what sport seventh-grade girls like most. He asks girls, “Is basketball your favorite sport?”

Answer: It mentions basketball and suggests that girls should give a certain answer. So the question is biased.

Question 13.
Jae wants to find out what type of art her fellow students enjoy most. She asks her classmates, “What is your favorite type of art?”

Answer: As it does not suggest students should give a certain answer, so it is not biased.

H.O.T. – Page No. 316

Focus on Higher Order Thinking

Question 14.
Draw Conclusions
Determine which sampling method will better represent the entire population. Justify your answer.

Answer: Collin’s survey is a better sampling method. Collin is randomly choosing names from the school directory, so each student has a chance of being chosen because they all appear in the school directory.
Karl’s survey is biased, as he is only choosing the students that were sitting near him during lunch which means the people he is asking are not representative of the entire population.

Question 15.
Multistep
Barbara surveyed students in her school by looking at an alphabetical list of 600 student names, dividing them into groups of 10, and randomly choosing one from each group.
a. How many students did she survey? What type of sample is this?
__________ people
This is a __________ sample

Answer: Barbara made 600÷10= 60 groups, so she chose one person in each group and surveyed 60 people. So this is a random sample because all the students are being represented and have an equal chance of being chosen.

Question 15.
b. Barbara found that 35 of the survey participants had pets. About what percent of the students she surveyed had pets? Is it safe to believe that about the same percentage of students in the school have pets? Explain your thinking.
__________ %

Answer: As there are 60 survey participants and in that 35/60= 0.58% ≈58%. Yes, it is safe to believe that about the same percent of students in the school have pets because the sample is large enough and all students are represented.

Random Sampling Worksheet Answer Key Question 16.
Communicating Mathematical Ideas
Carlo said a population can have more than one sample associated with it. Do you agree or disagree with his statement? Justify your answer.

Answer: Yes I agree. There are many different ways to randomly select a sample. Using the same way of choosing a sample multiple times could create a different sample. For example, picking a name out of a hat will not give you the same sample every time since the names will get mixed up every time you go to pick a name.

Guided Practice – Page No. 320

Patrons in the children’s section of a local branch library were randomly selected and asked for their ages. The librarian wants to use the data to infer the ages of all patrons of the children’s section so he can select age-appropriate activities. In 3–6, complete each inference.
7, 4, 7, 5, 4, 10, 11, 6, 7, 4

Question 1.
Make a dot plot of the sample population data.

Answer:

Question 2.
Make a box plot of the sample population data.

Answer: First we need to find the median, so we need to order the numbers from least to greatest: 4,4,4,5,6,7,7,7,10,11.
So median is (6+7)/2= 13/2= 6.5.
And the median for half of the data is 4,4,4,5,6= 4.
And the other half of the data is 7,7,7,10,11= 7.

Question 3.
The most common ages of children that use the library are _____ and _____.
_____ and _____

Answer: 4 and 7 are the numbers repeated the most in the data set, so the most common ages of the children that use the library are 4 and 7.

Question 4.
The range of ages of children that use the library is from _____ to _____.
_____ to _____

Answer: The lower that appears in the data set is 4 and the higher that appears in the data set is 11, so the range of ages of children that use the library is from 4 to 7.

Question 5.
The median age of children that use the library is _____.
_____

Answer: The median age of children who use the library is 6.5.

Question 6.
A manufacturer fills an order for 4,200 smartphones. The quality inspector selects a random sample of 60 phones and finds that 4 are defective. How many smartphones in the order are likely to be defective?
About _____ smartphones in the order are likely to be defective.
_____ smartphones

Answer: If we break down the whole order into samples of 60 phones we will get 4200÷60= 70 samples. So if we find 4 defective smartphones in every sample and we can expect about 4×70= 280 smartphones in the order are likely to be defective.

Lesson 2 Skills Practice Unbiased and Biased Samples Answer Key Question 7.
Part of the population of 4,500 elk at a wildlife preserve is infected with a parasite. A random sample of 50 elk shows that 8 of them are infected. How many elk are likely to be infected?
_____ elk

Answer: If we break down the whole elk population into samples of 50 elk, we get 4500÷50= 90 samples. So if we find 8 infected elk in every sample and we can expect about 8×90= 720 elk to be infected.

Essential Question Check-In

Question 8.
How can you use a random sample of a population to make predictions?

Answer: We can use a random sample of a population to make predictions by setting the ratio for the sample equal to the ratio for the population.

Independent Practice – Page No. 321

Question 9.
A manager samples the receipts of every fifth person who goes through the line. Out of 50 people, 4 had a mispriced item. If 600 people go to this store each day, how many people would you expect to have a mispriced item?
_____ people

Answer: 48 people.

Explanation:
Let X be the number of people with a mispriced item, so
4/50= X/600
50X= 2400
X= 48.
So there will be 48 people with a mispriced item.

Question 10.
Jerry randomly selects 20 boxes of crayons from the shelf and finds 2 boxes with at least one broken crayon. If the shelf holds 130 boxes, how many would you expect to have at least one broken crayon?
_____ boxes

Answer: 13 boxes.

Explanation:
Let X be the number of boxes with at least one broken crayon
2/20= X/130
20X= 260
X= 13.
So there will be 13 boxes with at least one broken crayon.

Random Sampling and Population Inferences Worksheet Answers Question 11.
A random sample of dogs at different animal shelters in a city shows that 12 of the 60 dogs are puppies. The city’s animal shelters collectively house 1,200 dogs each year. About how many dogs in all of the city’s animal shelters are puppies?
_____ dogs

Answer: 240 dogs.

Explanation:
Let X be the number of boxes with at least one broken crayon
12/60= X/1200
60X= 14400
X= 240.
So there will be 240 dogs in all of the city’s animal shelters are puppies.

Question 12.
Part of the population of 10,800 hawks at a national park is building a nest. A random sample of 72 hawks shows that 12 of them are building a nest. Estimate the number of hawks building a nest in the population.
_____ hawks

Answer: 1800 hawks.

Explanation:
Let X be the number of boxes with at least one broken crayon
12/72= X/10,800
72X= 10,800
X= 1800.
So there will be 1800 number of hawks building a nest in the population.

Question 13.
In a wildlife preserve, a random sample of the population of 150 raccoons was caught and weighed. The results, given in pounds, were 17, 19, 20, 21, 23, 27, 28, 28, 28 and 32. Jean made the qualitative statement, “The average weight of the raccoon population is 25 pounds.” Is her statement reasonable? Explain.
_____

Answer: Yes, Jean’s statement is reasonable.

Explanation: As the weights are not given for all 150 raccoons, so we don’t know how many raccoons at each of the weights given and we cannot calculate the average. So the best way to estimate the average is to find the median of the data set. So the median is
(23+27)/2= 25. As the median is 25 Jean’s statement is reasonable.

Question 14.
Greta collects the number of miles run each week from a random sample of female marathon runners. Her data are shown below. She made the qualitative statement, “25% of female marathoners run 13 or more miles a week.” Is her statement reasonable? Explain. Data: 13, 14, 18, 13, 12, 17, 15, 12, 13, 19, 11, 14, 14, 18, 22, 12.
_____

Answer: Greta’s statement is not reasonable.

Explanation: If we set the data from least to highest then 11,12,12,12,13,13,13,14,14,14,15,17,18,18,19,22. So there are 16 marathon runners, 12 of them run 13 miles or more each week. So
12/16= 0.75= 75%. So Greta’s statement is not reasonable.

Populations and Samples Practice and Problem-Solving a/b Answer Key Question 15.
A random sample of 20 of the 200 students at Garland Elementary is asked how many siblings each has. The data are ordered as shown. Make a dot plot of the data. Then make a qualitative statement about the population. Data: 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 6.

Answer: The mean is 2.

Explanation:
The mean is \(\frac{0+1+1+1+1+1+1+2+2+2+2+2+3+3+3+3+4+4+4+6}{20}
= \frac{46}{20}\)
= 2.3
So the mean is 2 as for the siblings the number must be a whole number.
Most of the students have at least 1 sibling and most of the students have fewer than 6 siblings, and the students have an average of two siblings.

Question 16.
Linda collects a random sample of 12 of the 98 Wilderness Club members’ ages. She makes an inference that most wilderness club members are between 20 and 40 years old. Describe what a box plot that would confirm Linda’s inference should look like.

Answer: Linda will conclude that most of the club members are between the ages of 20 and 40, so more than half of the 12 surveyed members must be between those ages. The box plot would have the lower quartile at 20 and the upper quartile at 40.

Page No. 322

Question 17.
What’s the Error?
Kudrey was making a box plot. He first plotted the least and greatest data values. He then divided the distance into half and then did this again for each half. What did Kudrey do wrong and what did his box plot look like?

Answer: By creating a box plot, the 3 middle values are not found by dividing the distance between the maximum and minimum values and then dividing the distances in half again. The 3 middle values are found by finding the median of the set values, the median of the first half of the values, and the median of the last half of the values.

H.O.T.

Focus on Higher Order Thinking

Question 18.
Communicating Mathematical Ideas
A dot plot includes all of the actual data values. Does a box plot include any of the actual data values?
______

Answer: Yes, a dot plot will include all of the actual data values. The only actual data values that a box plot must have the minimum and maximum values. The 3 median values may are may not be actual data values.

Question 19.
Make a Conjecture
Sammy counted the peanuts in several packages of roasted peanuts. He found that the bags had 102, 114, 97, 85, 106, 120, 107, and 111 peanuts. Should he make a box plot or dot plot to represent the data? Explain your reasoning.
______

Answer: Sammy should make a box plot to represent the data. As dot plots are helpful in finding the number of times each value occurs in a data set. As the values occur only once, so the box plot will better represent the data.

Question 20.
Represent Real-World Problems
The salaries for the eight employees at a small company are $20,000, $20,000, $22,000, $24,000, $24,000, $29,000, $34,000 and $79,000. Make a qualitative inference about a typical salary at this company. Would an advertisement that stated that the average salary earned at the company is $31,500 be misleading? Explain.
______

Answer: Yes, the statement is misleading.

Explanation: The median of the data set is \(\frac{$24,000+$24,000}{2}
= \frac48,000}{2}\)
= 24,000.
Yes, the statement is misleading because $31,500 is higher than 6 of the 8 salaries at the company.

Page No. 326

A manufacturer gets a shipment of 600 batteries of which 50 are defective. The store manager wants to be able to test random samples in future shipments. She tests a random sample of 20 batteries in this shipment to see whether a sample of that size produces a reasonable inference about the entire shipment.

Question 1.
The manager selects a random sample using the formula randInt( , ) to generate _____ random numbers.

Answer: Since 50 out of 600 batteries are defective and she is testing 20 batteries she can use randInt(1,600) to generate 20 random numbers.

Question 2.
She lets numbers from 1 to _____ represent defective batteries, and _____ to _____ represent working batteries. She generates this list: 120, 413, 472, 564, 38, 266, 344, 476, 486, 177, 26, 331, 358, 131, 352, 227, 31, 253, 31, 277.

Answer: She lets numbers from 1 to 50 represent defective batteries and 51 to 600 represent working batteries. She generates this list:
120, 413, 472, 564, 38, 266, 344, 476, 486, 177, 26, 331, 358, 131, 352, 227, 31, 253, 31, 277.

Go Math Middle School Grade 7 Solutions Question 3.
Does the sample produce a reasonable inference?
______

Answer: No, the sample does not produce a reasonable inference. In samples, 26, 31, 31, 38 numbers represent defective batteries, and in shipment 50 out of 600 of the batteries are defective.

Essential Question Check-In

Question 4.
What can happen if a sample is too small or is not random?

Answer: If the sample is too small or not random, it is likely to produce unrepresentative data values.

Page No. 327

Maureen owns three bagel shops. Each shop sells 500 bagels per day. Maureen asks her store managers to use a random sample to see how many whole-wheat bagels are sold at each store each day. The results are shown in the table. Use the table for 5–7.

Question 5.
If you assume the samples are representative, how many whole-wheat bagels might you infer are sold at each store?
Shop A: ___________
Shop B: ___________
Shop C: ___________

Answer:
Shop A: 100.
Shop B: 115.
Shop C: 140.

Explanation:
Shop A:
10/50×500
= 10×10
= 100.

Shop B:
23/100×500
= 23×5
= 115.

Shop C:
7/25×500
= 7×20
= 140.

Question 6.
Rank the samples for the shops in terms of how representative they are likely to be. Explain your rankings.

Answer: The samples can be ranked as C, A, B from least to most. Shop B’s is the most representative because it contained the most bagel. Shop C’s is the least representative because it contained the fewest bagels.

Question 7.
Which sample or samples should Maureen use to tell her managers how many whole-wheat bagels to make each day? Explain.

Answer: Maureen should use either Shop A or Shop B because the use a sufficient number of bagels to be considered accurate. Shop C’s sample would be the least representative because it contained the fewest bagels.

Question 8.
In a shipment of 1,000 T-shirts, 75 do not meet quality standards. The table below simulates a manager’s random sample of 20 T-shirts to inspect. For the simulation, the integers 1 to 75 represent the below-standard shirts.

Answer: In the sample, two values are from 1 to 75. So, 2 shirts are below the quality standards. So
= 2/20×1000
= 2×50
= 100.
The prediction would be that 100 shirts are below quality standards, which would be 25 more than the actual number.

Page No. 328

Question 9.
Multistep
A 64-acre coconut farm is arranged in an 8-by-8 array. Mika wants to know the average number of coconut palms on each acre. Each cell in the table represents an acre of land. The number in each cell tells how many coconut palms grow on that particular acre.

a. The numbers in green represent Mika’s random sample of 10 acres. What is the average number of coconut palms on the randomly selected acres?
______

Answer: The average is 49.8 coconut palms.

Explanation: The average is \(\frac{56+43+62+63+33+34+38+51+59+59}{10}
= \frac{498}{10}\)
= 49.8

Question 9.
b. Project the number of palms on the entire farm.
______

Answer: 3187 palms.

Explanation: As the average is 49.8 for each acre, so for 64 acres it is 64×49.8= 3187.2. So the number of palms on the entire farm is 3187.

H.O.T.

Focus on Higher Order Thinking

Question 10.
Draw Conclusions
A random sample of 15 of the 78 competitors at a middle school gymnastics competition are asked their height. The data set lists the heights in inches: 55, 57, 57, 58, 59, 59, 59, 59, 59, 61, 62, 62, 63, 64, 66. What is the mean height of the sample? Do you think this is a reasonable prediction of the mean height of all competitors? Explain.

Answer: Yes, this is a reasonable prediction.

Explanation: The mean height is \(\frac{55+57+57+58+59+59+59+59+59+61+62+62+63+64+66}{15}
= \frac{900}{15}\)
= 60 inches.
Yes, this is a reasonable prediction of the mean height of all competitors because it is a good sample generated randomly and contains sufficient values. So it should provide a good estimate of the mean height of all competitors.

Population and Samples Answer Key Question 11.
Critical Thinking
The six-by-six grid contains the ages of actors in a youth Shakespeare festival. Describe a method for randomly selecting 8 cells by using number cubes. Then calculate the average of the 8 values you found.

Answer: The average is 15.

Explanation: We can roll a number cube twice and record each value. The first value will be the row number and the second will be the column number we repeat the process 8 times in order to get 8 ages from the grid. 12,10,21,9,18,16,14,20.
The mean is \(\frac{12+10+21+9+18+16+14+20}{6}
= \frac{120}{8}\)
= 15

Question 12.
Communicating Mathematical Ideas
Describe how the size of a random sample affects how well it represents a population as a whole.

Answer: The bigger the size of the random sample, the more likely it so accurately represents the population.

10.1 Populations and Samples – Page No. 329

Question 1.
A company uses a computer to identify their 600 most loyal customers from its database and then surveys those customers to find out how they like their service. Identify the population and determine whether the sample is random or biased.
The sample is _______

Answer: The population is the customers in the company’s database. The sample is biased because instead of surveying all of their customers, the company only surveyed their most loyal customers.

10.2 Making Inferences from a Random Sample

Question 2.
A university has 30,330 students. In a random sample of 270 students, 18 speak three or more languages. Predict the number of students at the university who speak three or more languages.
_______ students

Answer: 2022 students.

Explanation: Let X be the number of students to speak three or more languages, so
18/270 = X/30,330
1/15 = X/30,330
X= 2022.

10.3 Generating Random Samples

A store receives a shipment of 5,000 MP3 players. In a previous shipment of 5,000 MP3 players, 300 were defective. A store clerk generates random numbers to simulate a random sample of this shipment. The clerk lets the numbers 1 through 300 represent defective MP3 players, and the numbers 301 through 5,000 represent working MP3 players. The results are given.
13 2,195 3,873 525 900 167 1,094 1,472 709 5,000

Question 3.
Based on the sample, how many of the MP3 players might the clerk predict would be defective?
_______ MP3’s

Answer: 1000 MP3’s.

Explanation: As the two random numbers are 13 and 167 as they are less than 300 and thus represent defective MP3 players. And the other 8 numbers are greater than 300 and represent working MP3 players. So the total number of randomly generated numbers is 10.
2/10 = X/5000
1/5 = X/5000
X = 1000.
So, about 1000 MP3 players are defective.

Populations and Samples 7th Grade Worksheets Pdf Answer Key Question 4.
Can the manufacturer assume the prediction is valid? Explain.
_______

Answer: No.

Explanation: The manufacturer cannot assume the prediction is valid. As the sample size of 10 is too small compared to the size of the shipment.

Essential Question

Question 5.
How can you use random samples to solve real-world problems?

Answer: We can use random samples to make a prediction about the population that is too large to survey.

Selected Response – Page No. 330

Question 1.
A farmer is using a random sample to predict the number of broken eggs in a shipment of 3,000 eggs. Using a calculator, the farmer generates the following random numbers. The numbers 1–250 represent broken eggs.
477 2,116 1,044 81 619 755 2,704 900 238 1,672 187 1,509
Options:
a. 250 broken eggs
b. 375 broken eggs
c. 750 broken eggs
d. 900 broken eggs

Answer: 750 broken eggs.

Explanation: Three random numbers are 81, 187, 238 which are less than 250 and represent broken eggs, so
3/12 = X/3000
1/4 = X/3000
4X = 3000
X= 750

Question 2.
A middle school has 490 students. Mae surveys a random sample of 60 students and finds that 24 of them have pet dogs. How many students are likely to have pet dogs?
Options:
a. 98
b. 196
c. 245
d. 294

Answer: 196.

Explanation: Let the number of students is likely to have pet dogs be X, so
24/60 = X/490
60X = 24×490
60X = 11,760
X = 196.

Question 3.
A pair of shoes that normally costs $75 is on sale for $55. What is the percent decrease in the price, to the nearest whole percent?
Options:
a. 20%
b. 27%
c. 36%
d. 73%

Answer: 27%

Explanation: The percent decrease in the price is \(\frac{75-55}{75}
= \frac{20}{75}\)
= 0.266= 27%

Samples and Populations 7th Grade Question 4.
Which of the following is a random sample?
Options:
a. A radio DJ asks the first 10 listeners who call in if they liked the last song.
b. 20 customers at a chicken restaurant are surveyed on their favorite food.
c. A polling organization numbers all registered voters, then generates 800 random integers. The polling organization interviews the 800 voters assigned those numbers.
d. Rebecca used an email poll to survey 100 students about how often they use the internet.

Answer:
A is biased because it is a voluntary survey.
B is biased because only 20 customers surveyed on their favorite food.
C is a sample because that is random.
D is biased students using email more likely to use the internet that students who don’t use email.

Question 5.
Each cell in the table represents the number of people who work in one 25-square-block section of the town of Middleton. The mayor uses a random sample to estimate the average number of workers per block.

a. The circled numbers represent the mayor’s random sample. What is the mean number of workers in this sample?
______

Answer: The mean is 54.

Explanation: The mean is \(\frac{56+60+50+43+62+53}{6}
= \frac{324}{6}\)
= 54

Question 5.
b. Predict the number of workers in the entire 25-block section of Middleton.
______

Answer: 1,350.

Explanation: As we know the mean is 54 per block, so for the entire 25-block section, the number is 54×25= 1,350.

Conclusion:

Now it is the time to redefine your true self using Go Math Answer Key. Learn the concepts and compare it with real-time examples. By learning the techniques you can perform well in the exams and also in time. For any queries, you can comment in the below comment box.

Go Math Grade 7 Answer Key Chapter 13 Theoretical Probability and Simulations: Students of 7th standard can Download the Answer Key of Go Math Chapter 13 from here. The Go Math Grade 7 Answer Key Chapter 13 Theoretical Probability and Simulations consists of the concepts like experimental and theoretical probability. We have provided a brief explanation for each and every problem to improve your math skills.

Go Math Grade 7 Answer Key Chapter 13 Theoretical Probability and Simulations

Theoretical Probability is the most interesting chapter in mathematics. Probability tells us how likely something is to happen in the long run. We can calculate the probability by looking at the outcomes of an experiment or by reasoning about the possible outcomes. If you understand the concepts of probability this is the scoring topic in the exams.

Tap on the below Go Math Grade 7 Answer Key Chapter 13 Theoretical Probability and Simulations link which is given topic-wise to get the solution with an explanation. Students can gradually improve their conceptual understanding of Math by following the HMH Go Math Grade 7 Chapter 13 Theoretical Probability and Simulations Answer Key pdf.

Chapter 13 Theoretical Probability and Simulations – Lesson: 1

• Guided Practice – Page No. 402
• Independent Practice – Page No. 403
• H.O.T. – Page No. 404

Chapter 13 Theoretical Probability and Simulati+ons – Lesson: 2

• Guided Practice – Page No. 408
• Independent Practice – Page No. 409
• Page No. 410

Chapter 13 Theoretical Probability and Simulations – Lesson: 3

• Guided Practice – Page No. 414
• Independent Practice – Page No. 415
• Page No. 416

Chapter 13 Theoretical Probability and Simulations – Lesson: 4

• Guided Practice – Page No. 420
• Independent Practice – Page No. 421
• Page No. 422

Chapter 13 Theoretical Probability and Simulations – Lesson: 5

• 13.1, 13.2 Theoretical Probability of Simple and Compound Events – Page No. 423
• Selected Response – Page No. 424

Chapter 13 Theoretical Probability and Simulations – Lesson: 6

• EXERCISES – Page No. 425
• Page No. 426

Chapter 13 Theoretical Probability and Simulations – Lesson: 7

• EXERCISES – Page No. 427
• Page No. 428

Chapter 13 Theoretical Probability and Simulations – Lesson: 8

• Selected Response – Page No. 429
• Page No. 430

Guided Practice – Page No. 402

At a school fair, you have a choice of randomly picking a ball from Basket A or Basket B. Basket A has 5 green balls, 3 red balls, and 8 yellow balls. Basket B has 7 green balls, 4 red balls, and 9 yellow balls. You can win a digital book reader if you pick a red ball.

Question 1.
Complete the chart. Write each answer in simplest form.
Type below:
______________

Answer:
We complete the table:

Question 2.
Which basket should you choose if you want the better chance of winning?
______

Answer: Basket B

Explanation:
In Exercise 1 we determined the probabilities Pa, Pb to pick a red ball from basket A, B
Pa = \(\frac{3}{16}\)
Pb = \(\frac{1}{5}\)
We compare the two probabilities
Pa = \(\frac{3}{16}\) . \(\frac{5}{5}\) = \(\frac{15}{80}\)
Pb = \(\frac{1}{5}\) . \(\frac{5}{5}\) = \(\frac{16}{80}\)
\(\frac{16}{80}\) > \(\frac{15}{80}\)
Pb > Pa
Since Pb > Pa, the better chance to win is in choosing Basket B.

A spinner has 11 equal-sized sections marked 1 through 11. Find each probability.

Question 3.
You spin once and land on an odd number.
\(\frac{□}{□}\)

Answer: \(\frac{6}{11}\)

Explanation:
We are given 11 equal-sized sections marked 1-11:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
We compute the probability that spinning once we would land on an odd number (1, 3, 5, 7, 9, 11):
P(odd) = number of odd sections/total number of sections = \(\frac{6}{11}\)

Making Predictions with Theoretical Probability Answer Key Question 4.
You spin once and land on an even number.
\(\frac{□}{□}\)

Answer: \(\frac{5}{11}\)

Explanation:
We are given 11 equal-sized sections marked 1-11:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
P(even) = number of even sections/total number of sections = \(\frac{5}{11}\)
We compute the probability that spinning once we would land on an even number (2, 4, 6, 8, 10)

You roll a number cube once.

Question 5.
What is the theoretical probability that you roll a 3 or 4?
\(\frac{□}{□}\)

Answer: \(\frac{1}{3}\)

Explanation:
P(3 or 4) = number of 3 or 4/total number of numbers on the number cube
\(\frac{2}{6}\) = \(\frac{1}{3}\)

Question 6.
Suppose you rolled the number cube 199 more times. Would you expect the experimental probability of rolling a 3 or 4 to be the same as your answer to Exercise 5?
Type below:
______________

Answer:
When rolling a number cube a large number of times, we expect the experimental probability not to be the same, but to get closer and closer to the theoretical probability.
Since 199 is not such a big number, we should not expect the experimental probability to be extremely close \(\frac{1}{3}\), but close enough.

Essential Question Check-In

Question 7.
How can you find the probability of a simple event if the total number of equally likely outcomes is 20?
Type below:
______________

Answer:
P(Simple event) = 1/total number of equally likely events
= \(\frac{1}{20}\)

Independent Practice – Page No. 403

Find the probability of each event. Write each answer as a fraction in simplest form, as a decimal to the nearest hundredth, and as a percent to the nearest whole number.

Question 8.
You spin the spinner shown. The spinner lands on yellow.

Type below:
______________

Answer: \(\frac{2}{6}\)

Explanation:
There are 2 yellow and 4 blue and we landed on yellow What is the probability of landing on yellow?
The probability is \(\frac{2}{6}\) because there are 2 yellow and the rest is blue.

Question 9.
You spin the spinner shown. The spinner lands on blue or green.

Type below:
______________

Answer: 67%

Explanation:
The yellow area, the blue area, and the green area have 3 sections each from the whole area.
We determine the probability that the spinner lands on a blue or green section:
P(spinner lands on blue or green) = (the number of blue sections + the number of green sections)/the total number f sections
= (4 + 4)/12 = \(\frac{8}{12}\) = \(\frac{2}{3}\)
\(\frac{2}{3}\) ≈ 0.67 = 67%

Question 10.
A jar contains 4 cherry cough drops and 10 honey cough drops. You choose one cough drop without looking. The cough drop is cherry.
Type below:
______________

Answer: 28%

Explanation:
We are given the data:
A jar contains 4 cherry cough drops and 10 honey cough drops.
P(to pick a cherry drop) = (the number of cherry drops)/the total number of drops
4/(4 + 10) = \(\frac{4}{14}\) = \(\frac{2}{7}\)
\(\frac{2}{3}\) ≈ 0.28 = 28%

Question 11.
You pick one card at random from a standard deck of 52 playing cards. You pick a black card.
Type below:
______________

Answer: 50%

Explanation:
We are given the data
You pick one card at random from a standard deck of 52 playing cards.
26 red cards
26 black cards
P(to pick a black card) = the number of black cards/the total number of cards

Question 12.
There are 12 pieces of fruit in a bowl. Five are lemons and the rest are limes. You choose a piece of fruit without looking. The piece of fruit is a lime.
Type below:
______________

Answer: 58%

Explanation:
There are 12 pieces of fruit in a bowl. Five are lemons and the rest are limes.
12 fruits:
5 lemons
7 limes
P(to pick a lime) =the number of lines/the total number of fruits
We determine the probability that we pick a lime:
\(\frac{7}{12}\) ≈ 0.58 = 58%

Question 13.
You choose a movie CD at random from a case containing 8 comedy CDs, 5 science fiction CDs, and 7 adventure CDs. The CD is not a comedy.
Type below:
______________

Answer: 60%

Explanation:
We are given the data:
8 comedy CDs
5 science fiction CDs
7 adventure CDs
P(to pick a CD which is not a comedy) = (the number of Sf CDs + the number of adventure CDs)/ the total number of CDs
= (5 + 7)/(8 + 5 + 7) = \(\frac{12}{20}\) = \(\frac{3}{5}\) = 0.60 = 60%

Question 14.
You roll a number cube. You roll a number that is greater than 2 and less than 5.
Type below:
______________

Answer: 33%

Explanation:
Rolling a number greater than 2 and less than 5 means to roll one of the numbers:
3, 4
P(to roll 3 or 4) = the number of 3 or 4 numbers/the total number of numbers
= (1 + 1)/6 = \(\frac{2}{6}\) = \(\frac{1}{3}\) = 0.33 = 33%

Making Predictions with Theoretical Probability Question 15.
Communicate Mathematical Ideas
The theoretical probability of a given event is \(\frac{9}{13}\). Explain what each number represents.
Type below:
______________

Answer:
The theoretical probability is the ratio between the number of favorable outcomes and the number of possible outcomes. The numerator 9 describes the number of desired events, while the denominator 13 describes the total number of events.
\(\frac{9}{13}\)

Question 16.
Leona has 4 nickels, 6 pennies, 4 dimes, and 2 quarters in a change purse. Leona lets her little sister Daisy pick a coin at random. If Daisy is equally likely to pick each type of coin, what is the probability that her coin is worth more than five cents? Explain.
\(\frac{□}{□}\)

Answer: \(\frac{3}{8}\)

Explanation:
Leona has 4 nickels, 6 pennies, 4 dimes, and 2 quarters in a change purse. Leona lets her little sister Daisy pick a coin at random.
1 penny = 1 cent
1 nickel = 5 cents
1 dime = 10 cents
1 quarter = 25 cents
We determine the probability that she picks a coin that is worth more than 5 cents is:
P(to pick a coin worth more than 5 cents) = the number of dimes+the number of quarters/the total number of coins
= (4 + 2)/(4 + 6 + 4 + 2) = \(\frac{6}{16}\) = \(\frac{3}{8}\) = 0.375 = 37.5%

H.O.T. – Page No. 404

Focus on Higher Order Thinking

Question 17.
Critique Reasoning
A bowl of flower seeds contains 5 petunia seeds and 15 begonia seeds. Riley calculated the probability that a randomly selected seed is a petunia seed as \(\frac{1}{3}\). Describe and correct Riley’s error.
Type below:
______________

Answer:
We are given the data
5 petunia seeds
15 begonia seeds
P(to pick a petunia seed) = the number of petunia seeds/the total number of seeds
We determine the probability that a randomly selected seed is the petunia seed
5/(5 + 15) = 5/20 = 1/4
Wrong:
Riley made the mistake of dividing the number of petunia seeds by the number of begonia seeds instead of dividing the number of petunia seeds by the total number of seeds:
P(to pick a petunia seed) = the number of petunia seeds/the total number of begonia seeds
= 5/15 = 1/3

Question 18.
There are 20 seventh graders and 15 eighth graders in a club. A club president will be chosen at random.
a. Analyze Relationships
Compare the probabilities of choosing a seventh grader or an eighth grader.
Type below:
______________

Answer:
We are given the data:
20 seventh graders
15 eighth graders
P(to pick a seventh-grader) = the number of seventh-graders/the total number of members
= 20/(20 + 15) = 20/35 = 4/7
We determine the probability of choosing a seventh-grader:
P(to pick an eighth-grader) = the number of eighth-graders/the total number of members
= 15/(20 + 15) = 15/35 = 3/7
Since 4/7 > 3/7, the probability of choosing a seventh-grader is higher than the probability of choosing an eighth-grader.

Question 18.
b. Critical Thinking
If a student from one grade is more likely to be chosen than a student from the other, is the method unfair? Explain.
Type below:
______________

Answer:
The method is not unfair because the number of seventh graders is greater than the number of eighth members (20 > 15), thus the seventh graders should be represented at a higher degree than the eighth graders.

A jar contains 8 red marbles, 10 blue ones, and 2 yellow ones. One marble is chosen at random. The color is recorded in the table, and then it is returned to the jar. This is repeated 40 times.

Question 19.
Communicate Mathematical Ideas
Use proportional reasoning to explain how you know that for each color, the theoretical and experimental probabilities are not the same.
Type below:
______________

Answer:
We are given the data
8 red marbles
10 blue marbles
2 yellow marbles
We determine the theoretical probability Pt of choosing each type of marble:
Pt(to pick a red marble) = the number of red marbles/the total number of marbles
= 8/(8 + 10 + 2) = 8/20 = 4/10
Pt(to pick a blue marble) = the number of blue marbles/the total number of marbles
= 10/(8 + 10 + 2) = 10/20 = 5/10
Pt(to pick a yellow marble) = the number of yellow marbles/the total number of marbles
= 2/(8 + 10 + 2) = 2/20 = 1/10
We determine the theoretical probability Pe of choosing each type of marble
Pe(to pick a red marble) = the number of red marbles/the total number of marbles
14/14+16+10 = 14/40 = 7/20
Pe(to pick a blue marble) = the number of blue marbles/the total number of marbles
16/14+16+10 = 16/40 = 8/20
Pe(to pick a yellow marble) = the number of yellow marbles/the total number of marbles
10/14+16+10 = 10/40 = 5/20
We notice that the number of red marbles is 4 times the number of yellow marbles, thus the theoretical probability to choose a red marble is 4 times greater than the one of choosing a yellow marble, while the experimental case shows that the probability of choosing a red marble is less than 1.5 times greater than the one of choosing a yellow one.
In the same way, we notice that the number of blue marbles is 5 times the number of yellow marbles, thus the theoretical probability to choose a blue marble is 5 times greater than the one of choosing a yellow marble, while the experimental case shows that the probability of choosing a blue marble is less than 2 times greater than the one of choosing a yellow one.
The exact probabilities are computed above.

Theoretical Probability Worksheet Grade 7 Question 20.
Persevere in Problem-Solving
For which color marble is the experimental probability closest to the theoretical probability? Explain.
______________

Answer:
We are given the data
8 red marbles
10 blue marbles
2 yellow marbles
Pt(to pick a red marble) = 8/20 = 4/10
Pt(to pick a blue marble) = 10/20 = 5/10
Pt(to pick a yellow marble) = 2/20 = 1/10
Pe(to pick a red marble) = 14/40 = 7/20
Pe(to pick a blue marble) = 16/40 = 8/20
Pe(to pick a yellow marble) = 10/40 = 5/20
|\(\frac{7}{20}\) – \(\frac{8}{20}\)| = \(\frac{1}{20}\)
|\(\frac{8}{20}\) – \(\frac{10}{20}\)| = \(\frac{2}{20}\)
|\(\frac{5}{20}\) – \(\frac{2}{20}\)| = \(\frac{3}{20}\)
\(\frac{1}{20}\) < \(\frac{2}{20}\) < \(\frac{3}{20}\)
Thus the answer is red.

Guided Practice – Page No. 408

Drake rolls two fair number cubes.

Question 1.
Complete the table to find the sample space for rolling a particular product on two number cubes.

Type below:
______________

Answer:
We complete the table to find the sample space for rolling a particular product on two number cubes:

Question 2.
What is the probability that the product of the two numbers Drake rolls is a multiple of 4?
\(\frac{□}{□}\)

Answer: \(\frac{5}{12}\)

Explanation:

We find the products which are multiples of 4:
4, 4, 8, 12, 12, 4, 8, 12, 14, 20, 24, 20, 12, 24, 36.
The number of multiples of 4 is 15.
The total number of products is
6 × 6 = 36
We determine the probability that the product is multiple of 4:
\(\frac{15}{36}\) = \(\frac{5}{12}\)

Topic 13 Lesson 13.2 Answer Key Question 3.
What is the probability that the product of the two numbers Drake rolls is less than 13?
\(\frac{□}{□}\)

Answer: \(\frac{23}{36}\)

Explanation:

We find products which are less than 13:
1, 2, 3, 4, 5, 6, 2, 4, 6, 8, 10, 12, 3, 6, 9, 12, 4, 8, 12, 5, 10, 6, 12
The number of products of less than 13 is 6 × 6 = 36.
The total number of products is
23/36

You flip three coins and want to explore probabilities of certain events.

Question 4.
Complete the tree diagram and make a list to find the sample space.

Type below:
______________

Answer:
We complete the given tree diagram placing one H and one T under each H and each T:

Question 5.
How many outcomes are in the sample space?
_______

Answer: 8 outcomes

Explanation:

Since each coin can land in two possible ways, the total possible number of outcomes is
2³ = 8
Thus there are 8 outcomes in the sample space.

Question 6.
List all the ways to get three tails.
Type below:
______________

Answer:
We are given the tree diagram we determined in Exercise 4:

The list of the 8 possible outcoes is
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
We list the outcomes containing 3 tails is TTT.

Question 7.
Complete the expression to find the probability of getting three tails.
\(\frac{□}{□}\)

Answer: \(\frac{1}{8}\)

Explanation:
P = number of outcomes with 3T/ total number of possible outcomes
The probability of getting three tails when three coins are flipped is \(\frac{1}{8}\)

Question 8.
What is the probability of getting exactly two heads?
\(\frac{□}{□}\)

Answer: \(\frac{3}{8}\)

Explanation:
The list of the 8 possible outcomes is:
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
We list the outcomes of 2H
HHT, HTH, THH
There are 3 ways to obtain exactly two heads is HHT, HTH, THH
P = number of outcomes with 3H/ total number of possible outcomes
P = \(\frac{3}{8}\)

Essential Question Check-In

Question 9.
There are 6 ways a given compound event can occur. What else do you need to know to find the theoretical probability of the event?
Type below:
______________

Answer:
We know that there are 6 ways in which a given compound event can occur and thus there are 6 favorable outcomes.
favorable outcomes = 6
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(event) = favorable outcomes/possible outcomes
Since we know the number of favorable outcomes, we also require the number of possible outcomes in order to determine the probability.

Independent Practice – Page No. 409

In Exercises 10–12, use the following information. Mattias gets dressed in the dark one morning and chooses his clothes at random. He chooses a shirt (green, red, or yellow), a pair of pants (black or blue), and a pair of shoes (checkered or red).

Question 10.
Use the space below to make a tree diagram to find the sample space.
Type below:
______________

Answer:
The sample space is:
Green Blue Red
Green Blue Checkered
Green Black Red
Green Black Checkered
Red Blue Red
Red Blue Checkered
Red Black Red
Red Black Checkered
Yellow Blue Red
Yellow Blue Checkered
yellow Black Red
Yellow Black Checkered

Question 11.
What is the probability that Mattias picks an outfit at random that includes red shoes?
\(\frac{□}{□}\)

Answer: \(\frac{1}{2}\)

Explanation:
Shirt Pants Shoes:
Green Blue Red
Green Blue Checkered
Green Black Red
Green Black Checkered
Red Blue Red
Red Blue Checkered
Red Black Red
Red Black Checkered
Yellow Blue Red
Yellow Blue Checkered
Yellow Black Red
Yellow Black Checkered
P = the number of outfits with red shoes/the total number of outfits
P = \(\frac{6}{12}\)
P = \(\frac{1}{2}\)

7th Grade Probability Quiz Answer Key Question 12.
What is the probability that no part of Mattias’s outfit is red?
\(\frac{□}{□}\)

Answer: \(\frac{1}{3}\)

Explanation:
Shirt Pants Shoes:
Green Blue Red
Green Blue Checkered
Green Black Red
Green Black Checkered
Red Blue Red
Red Blue Checkered
Red Black Red
Red Black Checkered
Yellow Blue Red
Yellow Blue Checkered
Yellow Black Red
Yellow Black Checkered
P = the number of outfits with no red shoes/the total number of outfits
P = \(\frac{4}{12}\)
P = \(\frac{1}{3}\)

Question 13.
Rhee and Pamela are two of the five members of the band. Every week, the band picks two members at random to play on their own for five minutes. What is the probability that Rhee and Pamela will be chosen this week?
\(\frac{□}{□}\)

Answer: \(\frac{1}{10}\)

Explanation:
Let’s note the 5 members of the band:
R = Rhee
P = Pamela
A, B, C = the other 3 members
The list of the possible outcomes is:
RP, RA, RB, RC, PR, PA, PB, AP, AR, AB, AC, BP, BR, BA, BC, CP, CR, CA, CB.
P = the number of outcomes containing P and R/the total number of outcomes
P = \(\frac{2}{20}\)
P = \(\frac{1}{10}\)

Question 14.
Ben rolls two number cubes. What is the probability that the sum of the numbers he rolls is less than 6?
\(\frac{□}{□}\)

Answer: \(\frac{5}{18}\)

The sums less than 6 are:
2, 3, 4, 5, 3, 4, 5, 4, 5, 5
P = the number of sums less than 6/the total number of sums
P = \(\frac{10}{36}\)
P = \(\frac{5}{18}\)

Question 15.
Nhan is getting dressed. He considers two different shirts, three pairs of pants, and three pairs of shoes. He chooses one of each of the articles at random. What is the probability that he will wear his jeans but not his sneakers?

\(\frac{□}{□}\)

Answer: \(\frac{2}{9}\)

Explanation:
We are given the data
Shirt: collared/T-shirt
Pants: Khakis/jeans/shorts
Shoes: sneakers/flip-flops/sandals
We determine the outcomes including jeans and not sneakers
P = the outcome including jeans and not sneakers/all possible outcomes
P = \(\frac{4}{18}\) = \(\frac{2}{9}\)

Question 16.
Communicate Mathematical Ideas
A ski resort has 3 chair lifts, each with access to 6 ski trails. Explain how you can find the number of possible outcomes when choosing a chair lift and a ski trail without making a list, a tree diagram, or a table.
Type below:
______________

Answer: 18

Explanation:
We are given the data:
Chair lifts: Chair lift 1/chair lift 2/chair lift 3
Ski trails: ski trail 1/ski trail 2/ski trail 3/ski trail 4/ski trail 5/ski trail 6
The sample space for choosing one of each is the product between the number of chair lifts and the number of ski lifts:
3 × 6 = 18

Question 17.
Explain the Error
For breakfast, Sarah can choose eggs, granola or oatmeal as a main course, and orange juice or milk for a drink. Sarah says that the sample space for choosing one of each contains 3² = 9 outcomes. What is her error? Explain.
Type below:
______________

Answer:
We are given the data:
Main course: eggs/granola/oatmeal
Drink: orange juice/milk
The sample space for choosing one of each is:
3 × 2 = 6
eggs-orange juice
eggs-milk
granola-orange juice
granola-milk
oatmeal-orange juice
oatmeal-milk
The error made by Sarah is that she considered only the number of main courses and forgot the number of drinks.

Page No. 410

Question 18.
Represent Real-World Problems
A new shoe comes in two colors, black or red, and in sizes from 5 to 12, including half sizes. If a pair of shoes is chosen at random for a store display, what is the probability it will be red and size 9 or larger?
\(\frac{□}{□}\)

Answer: \(\frac{7}{30}\)

Explanation:
We are given the data
Colors: black/red
Sizes: 5/5.5/6/6.5/7/7.5/8/8.5/9/9.5/10/10.5/11/11.5/12
The possible outcomes of red shoes with size greater or equal 9 are
red 9
red 9.5
red 10
red 10.5
red 11
red 11.5
red 12
P = the number of red shoes with size greater or equal to 9/the total number of outcomes
P = 7/(2 × 15) = \(\frac{7}{30}\)

H.O.T.

Focus on Higher Order Thinking

Question 19.
Analyze Relationships
At a diner, Sondra tells the server, “Give me one item from each column.” Gretchen says, “Give me one main dish and a vegetable.” Who has a greater probability of getting a meal that includes salmon? Explain.

______________

Answer:
We are given the data:
Main Dish: Pasta/salmon/beef/pork
Vegetables: carrots/peas/asparagus/sweet potato
Side: tomato soup/tossed salad
Psondra = (1 . 4 . 2)/(4 . 4 . 2) = \(\frac{8}{32}\) = \(\frac{1}{4}\)
Pgretchen = 4/16 = \(\frac{1}{4}\)

Theoretical Probability of Simple Events Worksheet Question 20.
The digits 1 through 5 are used for a set of locker codes.
a. Look for a Pattern
Suppose the digits cannot repeat. Find the number of possible two-digit codes and three-digit codes. Describe any pattern and use it to predict the number of possible five-digit codes.
Type below:
______________

Answer: 20, 60, 120

Explanation:
We are given the data
Digits: 1, 2, 3, 4, 5
We find the two-digit codes when digits do not repeat
12, 13, 14, 15
21, 23, 24, 25
31, 32, 34, 35
41, 42, 43, 45
51, 52, 53, 54
There are 5 × 4 = 20 possible codes.
We find the three digits codes when digits do not repeat:
123, 124, 125
132, 134, 135
142, 143, 145
152, 153, 154
213, 214, 215
231, 124, 135
…
….
512, 513, 514
521, 523, 524
531, 532, 534
541, 542, 543
There are 5 × 4 × 3 = 60 possible outcomes
If we use 5 digits and none can repeat, the first digit can be one of the numbers 1 2 3 4 5, the second digit can be one of the 4 remaining numbers, the third digit is one of the 3 remaining numbers, the fourth digit is one of the two remaining numbers, thus the number of possible outcomes is:
5 × 4 × 3 × 2 = 120

Question 20.
b. Look for a Pattern
Repeat part a, but allow digits to repeat.
Type below:
______________

Answer:
We find the two digits codes when digits can repeat:
11, 12, 13, 14, 15,
21, 22, 23, 24, 25
31, 32, 33, 34, 35
41, 42, 43, 44, 45
51, 52, 53, 54, 55
There are 5 . 5 = 25 possible codes.
There are 5 × 5 × 5 = 125 possible codes.
If we use 5 digits and they can repeat, the first digit can be one of the numbers 1 2 3 4 5, the second digit can be one of the same numbers 1 2 3 4 5, the third digit is one of the 5 numbers, the fourth digit is one of the 5 numbers, the fifth digit is one of the 5 numbers, thus the number of possible outcomes is
5 × 5 × 5 × 5 × 5 = 3125

Question 20.
c. Justify Reasoning
Suppose that a gym plans to issue numbered locker codes by choosing the digits at random. Should the gym use codes in which the digits can repeat or not? Justify your reasoning.
Type below:
______________

Answer:
The probability P1 to get a 2 digits code when digits do not repeat and the probability P2 to get a 2 digits code when digits can repeat:
P1 = 1/20
P2 = 1/25
The probability P1 to get a 3 digits code when digits do not repeat and the probability P2 to get a 3 digits code when digits can repeat:
P1 = 1/60
P2 = 1/125
The probability P1 to get a 5 digits code when digits do not repeat and the probability P2 to get a 5 digits code when digits can repeat:
P1 = 1/120
P2 = 1/3125
Thus the gym should use codes in which digits can repeat because the probability to be guessed is much smaller.

Guided Practice – Page No. 414

Question 1.
Bob works at a construction company. He has an equally likely chance to be assigned to work different crews every day. He can be assigned to work on crews building apartments, condominiums, or houses. If he works 18 days a month, about how many times should he expect to be assigned to the house crew?
_______ times

Answer:
Step 1:
Apartment: \(\frac{1}{3}\) Condo: \(\frac{1}{3}\) House: \(\frac{1}{3}\)
Probability of being assigned to the house crew: \(\frac{1}{3}\)
Step 2:
\(\frac{1}{3}\) = \(\frac{x}{18}\)
x = 6
6 times out of 18.

Question 2.
During a raffle drawing, half of the ticket holders will receive a prize. The winners are equally likely to win one of three prizes: a book, a gift certificate to a restaurant, or a movie ticket. If there are 300 ticket holders, predict the number of people who will win a movie ticket.
_______ people

Answer: 50 people

Explanation:
If 300 people buy tickets and half of them will receive a prize then 300 × 1/2 = 150 ticket holders will receive a prize. If they are equally likely to win one of the three prizes, then the probability of winning a movie ticket is 1/3. The number of people who will win a movie ticket is then 1/3 × 150 = 50 people.

Question 3.
In Mr. Jawarani’s first period math class, there are 9 students with hazel eyes, 10 students with brown eyes, 7 students with blue eyes, and 2 students with green eyes. Mr. Jawarani picks a student at random. Which color eyes is the student most likely to have? Explain.
______________

Answer: Brown

Explanation:
There are more students with brown eyes than any other colored eyes so if he picks a student at random, they will most likely have brown eyes.

Essential Question Check-In

Question 4.
How do you make predictions using theoretical probability?
Type below:
______________

Answer:
To make a prediction using theoretical probability, you can multiply the theoretical probability by the number of events to get a prediction. You can find the prediction by setting the theoretical probability equal to the ratio of x/number of events and then solving for x, where x is the prediction.

Independent Practice – Page No. 415

Question 5.
A bag contains 6 red marbles, 2 white marbles, and 1 gray marble. You randomly pick out a marble, record its color, and put it back in the bag. You repeat this process 45 times. How many white or gray marbles do you expect to get?
_______ marbles

Answer: 15

Explanation:
Given that there are 6 red marbles, 2 white marbles, and 1 gray marble, which are thus 6 + 2 + 1 = 9 marbles in total.
possible outcomes = 9
2 + 1 = 3 of the marbles are either white or gray and thus there are 3 favorable outcomes.
favorable outcomes = 3
The probability is the number of favorable outcomes divided by the number of possible outcomes.
P(white or gray) = favorable outcomes/possible outcomes = \(\frac{3}{9}\)
= \(\frac{1}{3}\)
The predicted number of white or gray marbles is then obtained by multiplying the number of repetitions by the probability.
Prediction = Number of repetitions × P (white or gray)
= 45 × \(\frac{1}{3}\)
= 15
Thus we predict that we obtain a white or gray marble about 15 times.

Probability Simulations 7th Grade Question 6.
Using the blank circle below, draw a spinner with 8 equal sections and 3 colors—red, green, and yellow. The spinner should be such that you are equally likely to land on green or yellow, but more likely to land on red than either on green or yellow.

Type below:
______________

Answer:
A possible spinner would be to have 4 red sections, 2 green sections, and 2 yellow sections. That way there is an equal chance of landing on yellow and green and there is a more likely chance of landing on yellow and green and there is a more likely chance of landing on red than landing on green or landing on yellow.
A second possible spinner could be to have 6 red sections, 1 yellow section, and 1 green section. This would still give an equal chance to land on green or yellow and a higher chance to land on red than to land on green or land on yellow.

Use the following for Exercises 7–9. In a standard 52-card deck, half of the cards are red and half are black. The 52 cards are divided evenly into 4 suits: spades, hearts, diamonds, and clubs. Each suit has three face cards (jack, queen, king), and an ace. Each suit also has 9 cards numbered from 2 to 10.

Question 7.
Dawn draws 1 card, replaces it, and draws another card. Is it more likely that she draws 2 red cards or 2 face cards?
______________

Answer: 2 red cards

Explanation:
There are 26 red cards in the deck and 12 face cards in the deck so it is more likely to draw two red cards than it is to draw two face cards.

Question 8.
Luis draws 1 card from a deck, 39 times. Predict how many times he draws an ace.
_______ times

Answer: About 3 times

Explanation:
A standard deck of cards contains 52 cards, of which 26 are red and 26 are black, 13 are of each suit (hearts, diamonds, spades, clubs), and of which 4 are of each denomination (A, 2 to 10, J, Q, K). The face cards are the Jacks J, Queens Q, and Kings K.
There are 52 cards in the deck of cards and thus there are 52 possible outcomes.
possible outcomes = 52
4 of the 52 cards in a standard deck of cards are aces and thus there are 4 favorable outcomes.
favorable outcomes = 4
The probability is the number of favorable outcomes divided by the number of possible outcomes.
P(white or gray) = favorable outcomes/possible outcomes = \(\frac{4}{52}\)
= \(\frac{1}{13}\)
The predicted number of aces is then obtained by multiplying the number of draws by the probability.
Prediction = Number of draws × P(Ace)
= 39 × \(\frac{1}{3}\)
Thus we predict that 3 of the drawn cards will be aces.

Question 9.
Suppose a solitaire player has played 1,000 games. Predict how many times the player turned over a red card as the first card.
_______ times

Answer: 500 times

Explanation:
A standard deck of cards contains 52 cards, of which 26 are red and 26 are black, 13 are of each suit (hearts, diamonds, spades, clubs), and of which 4 are of each denomination (A, 2 to 10, J, Q, K). The face cards are Jack’s J, queens Q, and King K.
There are 52 cards in the deck of cards and thus there are 52 possible outcomes.
possible outcomes = 52
26 of the 52 cards in a standard deck of cards are red. This then implies that there are 26 favorable outcomes.
favorable outcomes = 26
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(red) = favorable outcomes/possible outcomes = \(\frac{26}{52}\) = 1/2
The predicted number of aces is then obtained by multiplying the number of draws by the probability.
Prediction = Number of draws × P(Red)
= 1000 × \(\frac{1}{2}\)
= 500
Thus we predict that 500 of the drawn cards will be red.

Understand Experimental Probability Answer Key Question 10.
John and O’Neal are playing a board game in which they roll two number cubes. John needs to get a sum of 8 on the number of cubes to win. O’Neal needs a sum of 11. If they take turns rolling the number cube, who is more likely to win? Explain.
______________

Answer: John

Explanation:
To get a sum of 8, John can roll the following numbers:
2, 6
3, 5
4, 4
5, 3
6, 2
To get a sum of 11, O’Neal can roll the following numbers:
5, 6
6, 5
Since there are more ways to roll a sum of 8 than there are to roll a sum of 11, John is more likely to win.

Question 11.
Every day, Navya’s teacher randomly picks a number from 1 to 20 to be the number of the day. The number of the day can be repeated. There are 180 days in the school year. Predict how many days the number of the day will be greater than 15.
_______ days

Answer: 45 days

Explanation:
There are 20 numbers from 1 to 20 and thus there are 20 possible outcomes.
possible outcomes = 20
5 of the 20 numbers from 1 to 20 are greater than 15 (16, 17, 18, 19, 20) and thus there are 5 favorable outcomes.
favorable outcomes = 5
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(greater than 15) = favorable outcomes/possible outcomes = \(\frac{5}{20}\) = \(\frac{1}{4}\)
The predicted is the number of favorable outcomes divided by the number of possible outcomes/
Prediction = Number of days × P(Greater than 15)
180 × \(\frac{1}{4}\)
= 45
Thus we predict that 45 of the days have a number greater than 15.

Question 12.
Eben rolls two standard number cubes 36 times. Predict how many times he will roll a sum of 4.
_______ times

Answer: 3 times

Explanation:
A number cube has 6 possible outcomes: 1, 2, 3, 4, 5, 6.
There are then 6 × 6 = 36 possible outcomes when rolling 2 dice.
possible outcomes = 6 . 6 = 36
3 of the outcomes in the image below result in a sum of 4 ((1, 3), (2, 2), (3, 1)) and thus there are 3 favorable outcomes.
favorable outcomes = 3
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(sum is 4) = favorable outcomes/possible outcomes = \(\frac{3}{36}\) = 1/12
The predicted number of rolls that result in a sum of 4 is then obtained by multiplying the number of rolls by the probability.
Prediction = Number of rolls × P(sum is 4)
= 36 × 1/12
= 3
Thus we predict that 3 of the rolls result in a sum of 4.

Probability Test Answer Key 7th Grade Question 13.
Communicate Mathematical Ideas
Can you always show that a prediction based on theoretical probability is true by performing the event often enough? If so, explain why. If not, describe a situation that justifies your response.
Type below:
______________

Answer:
You cannot show that a prediction based on theoretical probability is true by performing the event often enough. The prediction value will get closer to the actual value as more events are performed but will not always equal the actual value.

Page No. 416

Question 14.
Represent Real-World Problems
Give a real-world example of an experiment in which all of the outcomes are not equally likely. Can you make a prediction for this experiment, using theoretical probability?
Type below:
______________

Answer:
A real work example of an experiment in which all of the outcomes are not equally likely could be spinning a spinner that has 1 red section, 2 orange sections, and 3 blue sections, and the sections are of the same size. Since there are not the same number of sections for each other, the outcomes of red, orange, and blue do not have the same probabilities. A prediction can still be made because the theoretical probabilities of landing on each color can be found. If you wanted to predict the number of times you would land on blue in 100 spins, you would first need to find the theoretical probability of landing on blue. Since there are 3 blue sections and a total of 6 sections, the theoretical probability is \(\frac{3}{6}\) = \(\frac{1}{2}\). The prediction would then be \(\frac{1}{2}\) × 100 = 50 times.

H.O.T.

Focus on Higher Order Thinking

Question 15.
Critical Thinking
Pierre asks Sherry a question involving the theoretical probability of a compound event in which you flip a coin and draw a marble from a bag of marbles. The bag of marbles contains 3 white marbles, 8 green marbles, and 9 black marbles. Sherry’s answer, which is correct, is \(\frac{12}{40}\). What was Pierre’s question?
Type below:
______________

Answer: What is the probability of drawing a white or black marble and flipping heads?

Question 16.
Make a Prediction
Horace is going to roll a standard number cube and flip a coin. He wonders if it is more likely that he rolls a 5 and the coin lands on heads, or that he rolls a 5 or the coin lands on heads. Which event do you think is more likely to happen? Find the probability of both events to justify or reject your initial prediction.
Type below:
______________

Answer:
It is more likely that he rolls a 5 or flips heads than it is to roll a 5 and flip heads. This is because the probability of two events occurring at the same time is always less than the probability of one or another event occurring. The probability of rolling a 5 is 1/6 and the probability of flipping heads is 1/2 so the probability of both occurring is 1/6 × 1/2 = 1/12.
There are 12 possible outcomes to rolling a number cube and flipping a coin since there are 6 outcomes for the cube and 2 outcomes for the coin and 6 × 2 = 12.
Of those 12 outcomes, 7 of them are rolling a 5 or flipping heads (1H, 2H, 3H, 4H, 5H, 6H, 5T). The probability of rolling a 5 or flipping heads is then 7/12 which is greater than 1/12.

Question 17.
Communicate Mathematical Ideas
Cecil solved a theoretical prediction problem and got this answer: “The spinner will land on the red section 4.5 times.” Is it possible to have a prediction that is not a whole number? If so, give an example.
Type below:
______________

Answer: Yes
It is possible if what is being predicted does not have to be a whole number, like time. A possible example could be, the theoretical probability that there will be 50 people in a line at a store during a one-hour interval is 1/12. What is the predicted number of hours that there will be 50 people in line if the store is open for 9 hours? The prediction would then be 1/12 × 9 = 0.75 hours.

Guided Practice – Page No. 420

There is a 30% chance that T’Shana’s county will have a drought during any given year. She performs a simulation to find the experimental probability of a drought in at least 1 of the next 4 years.

Question 1.
T’Shana’s model involves whole numbers from 1 to 10. Complete the description of her model.
Type below:
______________

Answer:
Since the chance of drought is 30%, let the numbers 1 to 3 represent a drought year and the numbers 4 to 10 represent a year without a drought. Since you are concerned with the number of droughts in the next 4 years, generate 4 random numbers in each trial.

Question 2.
Suppose T’Shana used the model described in Exercise 1 and got the results shown in the table. Complete the table.

Type below:
______________

Answer:
The number of drought years is the number of times 1 to 3 were generated in each trial so count the number of times in each trial that the number 1 to 3 occurred:

7th Grade Theoretical Probability Question 3.
According to the simulation, what is the experimental probability that there will be a drought in the county in at least 1 of the next 4 years?
\(\frac{□}{□}\)

Answer: \(\frac{4}{5}\)

Explanation:
There are 10 trials and thus there are 10 possible outcomes.
possible outcomes = 10
In the previous exercise, we obtained at least 1 drought year in 8 of the 10 trials and thus there are 8 favorable outcomes.
favorable outcomes = 8
The probability is the number of favorable outcomes divided by the number of possible outcomes.
P(At least 1 drought year) = favorable outcomes/possible outcomes = \(\frac{8}{10}\)
= \(\frac{4}{5}\) = 0.8 = 80%

Essential Question Check-In

Question 4.
You want to generate random numbers to simulate an event with a 75% chance of occurring. Describe a model you could use.
Type below:
______________

Answer:
75% in fraction form is \(\frac{3}{4}\) so you can randomly generate numbers from 1 to 4. The numbers 1 to 3 would mean success and 4 would mean unsuccessful.

Independent Practice – Page No. 421

Every contestant on a game show has a 40% chance of winning. In the simulation below, the numbers 1–4 represent a winner, and the numbers 5–10 represent a nonwinner. Numbers were generated until one that represented a winner was produced.

Question 5.
In how many of the trials did it take exactly 4 contestants to get a winner?
_____ trial(s)

Answer: 1

Explanation:
Only trial 6 took 4 contestants to get a winner so 1 trial.

Question 6.
Based on the simulation, what is the experimental probability that it will take exactly 4 contestants to get a winner?
\(\frac{□}{□}\)

Answer: \(\frac{1}{10}\)

Explanation:
We have data about 10 trials and thus there are 10 possible outcomes.
possible outcomes = 10
1 of the 10 trials required exactly 4 numbers to get a winner and thus there is 1 favorable outcome.
favorable outcomes = 1
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(exactly 4 contestants) = favorable outcomes/possible outcomes = \(\frac{1}{10}\) = 0.1 = 10%

Over a 100-year period, the probability that a hurricane struck Rob’s city in any given year was 20%. Rob performed a simulation to find an experimental probability that a hurricane would strike the city in at least 4 of the next 10 years. In Rob’s simulation, 1 represents a year with a hurricane.

Question 7.
According to Rob’s simulation, what was the experimental probability that a hurricane would strike the city in at least 4 of the next 10 years?
\(\frac{□}{□}\)

Answer: \(\frac{1}{5}\)

Explanation:
We have been given data about 10 trials and thus there are 10 possible outcomes.
possible outcomes = 10
A 1 represents a hurricane. We then note that trial 2 and trial 7 both have at least 4 ones and thus there are 4 trials that result in at least 4 hurricanes.
Thus there are 2 favorable outcomes.
favorable outcomes = 2
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(At least 4) = favorable outcomes/possible outcomes = \(\frac{2}{10}\) = \(\frac{1}{5}\)
= 0.2 = 20%

Probability Questions and Answers Pdf Grade 7 Question 8.
Analyze Relationships
Suppose that over the 10 years following Rob’s simulation, there was actually 1 year in which a hurricane struck. How did this compare to the results of Rob’s simulation?
Type below:
______________

Answer:
If a hurricane struck in 1 year the next 10 years following the simulation, it would match the results of his simulation. In 3 of his trials, exactly 1 year had a hurricane which means the experimental probability that there will be 1 hurricane in 10 years is \(\frac{3}{10}\).
In all of the trials, there was at least 1 year with a hurricane which means the experimental probability is 100% that a hurricane will occur the next 10 years.

Page No. 422

Question 9.
Communicate Mathematical Ideas
You generate three random whole numbers from 1 to 10. Do you think that it is unlikely or even impossible that all of the numbers could be 10? Explain?
Type below:
______________

Answer:
It is unlikely that all three numbers would be 10. The theoretical probability that a random whole number from 1 to 10 is 10 is 1/10.
The theoretical probability that three random whole numbers from 1 to 10 are all 10s is then \(\frac{1}{10}\) × \(\frac{1}{10}\) × \(\frac{1}{10}\) = \(\frac{1}{1000}\).
This is a very small probability so it is unlikely.

Question 10.
Erika collects baseball cards, and 60% of the packs contain a player from her favorite team. Use a simulation to find an experimental probability that she has to buy exactly 2 packs before she gets a player from her favorite team
Type below:
______________

Answer:
Generate random numbers from 1 to 10 using 10 trials. Since 60% of the packs contain a player from her favorite team, let the numbers 1 to 6 represent a pack with a player from her favorite team and the numbers 7 to 10 represent packs without a player from her favorite team.
Out of 10 trials she had to buy exactly 10 packs before getting a player from her favorite team only in 2 trials so the experimental probability is \(\frac{2}{10}\) = \(\frac{1}{5}\).

H.O.T.

Focus on Higher Order Thinking

Question 11.
Represent Real-World Problems
When Kate plays basketball, she usually makes 37.5% of her shots. Design and conduct a simulation to find the experimental probability that she makes at least 3 of her next 10 shots. Justify the model for your simulation.
Type below:
______________

Answer: \(\frac{7}{10}\)

Explanation:
Since 37.5% = 3/8 perform simulation by randomly generating 10 members from 1 to 8 where the numbers 1 to 3 are when she makes the shot and 4 to 8 are when she doesn’t make the shot. Perform 10 trials.
She made at least 3  shots in 7 of the 10 trials so the experimental probability is \(\frac{7}{10}\)

Question 12.
Justify Reasoning
George and Susannah used a simulation to simulate the flipping of 8 coins 50 times. In all of the trials, at least 5 heads came up. What can you say about their simulation? Explain.
Type below:
______________

Answer:
If at least 5 heads came up in every trial, then the simulation they used does not accurately model flipping a coin 8 times. Since each coin has a theoretical probability of 1/2 and \(\frac{1}{2}\) × 8 = 4, there should be around 4 heads in each trial. Getting at least 5 heads in every trial means that the coin is more likely to land on heads than to land on tails.

13.1, 13.2 Theoretical Probability of Simple and Compound Events – Page No. 423

Find the probability of each event. Write your answer as a fraction, as a decimal, and as a percent.

Question 1.
You choose a marble at random from a bag containing 12 red, 12 blue, 15 green, 9 yellow, and 12 black marbles. The marble is red.
Type below:
______________

Answer:
The bag contains 12 red, 12 blue, 15 green, 9 yellow, and 12 black marbles, which are thus 12 + 12 + 15 + 9 + 12 = 60 marbles in total and thus there are 60 possible outcomes.
possible outcomes = 60
12 of the 60 marbles are red and thus there are 12 favorable outcomes.
favorable outcomes = 12
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(red) = favorable outcomes/possible outcomes = \(\frac{12}{60}\) = 1/5 = 0.2 = 20%

Question 2.
You draw a card at random from a shuffled deck of 52 cards. The deck has four 13-card suits (diamonds, hearts, clubs, spades). The card is a diamond or a spade.
Type below:
______________

Answer:
A standard deck of cards contains 52 cards, of which 26 are red and 26 are black, 13 are of each suit (hearts, diamonds, spades, clubs), and of which 4 are of each denomination (A, 2 to 10, J, Q, K). The face cards are Jacks J, Queens Q, and Kings K.
There are 52 cards in the deck of cards and thus there 52 possible outcomes.
possible outcomes = 52
13 of the cards are diamonds and 13 of the cards are spades, thus there are 13 + 13 = 26 cards that are diamonds or spades. This then implies that there are 26 favorable outcomes.
favorable outcomes = 26
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(red) = favorable outcomes/possible outcomes = \(\frac{26}{52}\) = 1/2 = 50%

13.3 Making Predictions with Theoretical Probability

Question 3.
A bag contains 23 red marbles, 25 green marbles, and 18 blue marbles. You choose a marble at random from the bag. What color marble will you most likely choose?
______________

Answer: Green

Explanation:
There are more green marbles than any other color so you are more likely to choose a green marble.

13.4 Using Technology to Conduct a Simulation

Question 4.
Bay City has a 25% chance of having a flood in any given decade. The table shows the results of a simulation using random numbers to find the experimental probability that there will be a flood in Bay City in at least 1 of the next 5 decades. In the table, the number 1 represents a decade with a flood. The numbers 2 through 5 represent a decade without a flood.

According to the simulation, what is the experimental probability of a flood in Bay City in at least 1 of the next 5 decades?
\(\frac{□}{□}\)

Answer: \(\frac{2}{5}\)

Explanation:
We have been given information about 10 trials and thus there are 10 possible outcomes.
possible outcomes = 10
The number 1 represents a decade with a flood. We then note that 4 of the 10 trials contained at least one 1 and thus there are 4 favorable outcomes.
favorable outcomes = 4
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(Flood) = favorable outcomes/possible outcomes = 4/10 = \(\frac{2}{5}\)

Essential Question

Question 5.
How can you use theoretical probability to make predictions in real-world situations?
Type below:
______________

Answer:
You can use theoretical probabilities to make predictions by multiplying the theoretical probability times the number of events.
An example would be flipping a coin 50 times and wanting to predict the number of heads.
Since the theoretical probability of landing on heads is 1/2, a prediction is
50 × 1/2 = 25 heads.

Selected Response – Page No. 424

Question 1.
What is the probability of flipping two fair coins and having both show tails?
Options:
a. \(\frac{1}{8}\)
b. \(\frac{1}{4}\)
c. \(\frac{1}{3}\)
d. \(\frac{1}{2}\)

Answer: \(\frac{1}{4}\)

Explanation:
Each coin has 2 possible outcomes: Heads H and tails T
We then note that there are 4 possible outcomes for the 2 coins: HH, HT, TH, TT
Possible outcomes = 4
1 of the 4 possible outcomes results in two tails TT and thus there is 1 favorable outcome.
favorable outcomes = 1
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(two tails) = favorable outcomes/possible outcomes = \(\frac{1}{4}\)
Thus the correct answer is option B.

Question 2.
A bag contains 8 white marbles and 2 black marbles. You pick out a marble, record its color, and put the marble back in the bag. If you repeat this process 45 times, how many times would you expect to remove a white marble from the bag?
Options:
a. 9
b. 32
c. 36
d. 40

Answer: 36

Explanation:
The bag contains 8 white marbles and 2 black marbles, which are thus 8 + 2 = 10 marbles in total and thus there are 10 possible outcomes.
possible outcomes = 10
We note that 8 of the marbles in the bag are white and thus there are 8 favorable outcomes.
favorable outcomes = 8
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(white) = favorable outcomes/possible outcomes = \(\frac{8}{10}\) = \(\frac{4}{5}\)
The predicted number of times we select a white marble is then obtained by multiplying the number of repetitions by the probability.
Prediction = Number of repetitions × \(\frac{4}{5}\)
= 45 × \(\frac{4}{5}\)
= 9 × 4 = 36
Thus we predict that we will get a white marble about 36 times.
Thus the correct answer is option C.

Question 3.
Philip rolls a standard number cube 24 times. Which is the best prediction for the number of times he will roll a number that is even and less than 4?
Options:
a. 2
b. 3
c. 4
d. 6

Answer: 4

Explanation:
A number cube has 6 possible outcomes: 1, 2, 3, 4, 5, 6.
possible outcomes = 6
1 of the 6 possible outcomes results in an even number of less than 4, that is outcome 2.
favorable outcome = 1
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(even and less than 4) = favorable outcomes/possible outcomes = 1/6
The predicted number of times we select an even number less than 4 is then obtained by multiplying the number of rolls by the probability.
Prediction = Number of rolls × P(even and less than 4)
= 24 × 1/6
= 4
Thus we predict that we roll an even number less than 4 about 4 times.
Thus the correct answer is option C.

Question 4.
A set of cards includes 24 yellow cards, 18 green cards, and 18 blue cards. What is the probability that a card chosen at random is not green?
Options:
a. \(\frac{3}{10}\)
b. \(\frac{4}{10}\)
c. \(\frac{3}{5}\)
d. \(\frac{7}{10}\)

Answer: \(\frac{7}{10}\)

Explanation:
The set of cards includes 24 yellow, 18 green, and 18 blue cards, which are thus 24 + 18 + 18 = 60 cards in total and thus there are 60 possible outcomes.
possible outcomes = 60
18 of the 60 cards are green and thus 60 – 18 = 42 of the 60 cards are not green. This then implies that there are 42 favorable outcomes.
favorable outcomes = 42
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(not green) = favorable outcomes/possible outcomes = 42/60 = \(\frac{7}{10}\)
Thus the correct answer is option D.

Question 5.
A rectangle made of square tiles measures 10 tiles long and 8 tiles wide. What is the width of a similar rectangle whose length is 15 tiles?
Options:
a. 3 tiles
b. 12 tiles
c. 13 tiles
d. 18.75 tiles

Answer: 12 tiles

Explanation:
Write the proportion relating to the lengths and widths of each rectangle.
length/width = 10/8 = 15/w
10w = 120
w = 12
Thus the correct answer is option B.

Question 6.
The Fernandez family drove 273 miles in 5.25 hours. How far would they have driven at that rate in 4 hours?
Options:
a. 208 miles
b. 220 miles
c. 280 miles
d. 358 miles

Answer: 208 miles

Explanation:
Write the proportion relating the number of miles and hours.
miles/hours = 273/5.25 = m/4
5.25m = 1092
m = 208 miles
Thus the correct answer is option A.

Question 7.
There are 20 tennis balls in a bag. Five are orange, 7 are white, 2 are yellow, and 6 are green. You choose one at random. Which color ball are you least likely to choose?
Options:
a. green
b. orange
c. white
d. yellow

Answer: Yellow

Explanation:
The color with the fewest number of balls is yellow so you are least likely to choose yellow.
Thus the correct answer is option D.

Mini-Task

Question 8.
Center County has had a 1 in 6 (or about 16.7%) chance of a tornado in any given decade. In a simulation to consider the probability of tornadoes in the next 5 decades, Ava rolled a number cube. She let a 1 represent a decade with a tornado, and 2–6 represent decades without tornadoes. What experimental probability did Ava find for each event?

a. That Center County has a tornado in at least one of the next five decades.
\(\frac{□}{□}\)

Answer: \(\frac{3}{5}\)

Explanation:
We have been given the data about 10 trials and thus there are 10 possible outcomes.
possible outcomes = 10
The number 1 represents a tornado. We then note that 6 of the 10 trials contain at least one 1 and thus 6 of the 10 trials result in at least one tornado. This then implies that there are 6 favorable outcomes.
favorable outcomes = 6
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(at least one tornado) = favorable outcomes/possible outcomes = 6/10 = \(\frac{3}{5}\)

Question 8.
b. That Center County has a tornado in exactly one of the next five decades
\(\frac{□}{□}\)

Answer: \(\frac{3}{10}\)

Explanation:
We have been given data about 10 trials and thus there are 10 possible outcomes.
possible outcomes = 10
The number 1 represents a tornado. We then note that 3 of the 10 trials contain at least one 1 and thus 3 of the 10 trials result in exactly one tornado. This then implies that there are 3 favorable outcomes.
favorable outcomes = 3
The probability is the number of favorable outcomes divided by the number of possible outcomes
P(exactly one tornado) = favorable outcomes/possible outcomes = 3/10 = 0.3

EXERCISES – Page No. 425

Find the probability of each event.

Question 1.
Rolling a 5 on a fair number cube.
\(\frac{□}{□}\)

Answer: \(\frac{1}{6}\)

Explanation:
A number cube has 6 possible outcomes: 1, 2, 3, 4, 5, 6
possible outcomes = 6
We note that 1 of the 6 possible outcomes results in a 5 and thus there is a favorable outcome.
favorable outcomes = 1
The probability is the number of favorable outcomes divided by the number of possible outcomes.
P(5) = favorable outcomes/possible outcomes
\(\frac{1}{6}\) ≈ 0.1667 = 16.67%

Question 2.
Picking a 7 from a standard deck of 52 cards. A standard deck includes 4 cards of each number from 2 to 10.
\(\frac{□}{□}\)

Answer: \(\frac{1}{13}\)

Explanation:
There are 52 cards in the standard deck of cards and thus there are 52 possible outcomes.
possible outcomes = 52
We note that 4 of the 52 cards are 7’s and thus there are 4 favorable outcomes.
favorable outcomes = 4
The probability is the number of favorable outcomes divided by the number of possible outcomes.
P(7) = favorable outcomes/possible outcomes
4/52 = \(\frac{1}{13}\) ≈ 0.0769 = 7.69%

13.6 Skills Practice Theoretical and Experimental Probability Answer Key Question 3.
Picking a blue marble from a bag of 4 red marbles, 6 blue marbles, and 1 white marble.
\(\frac{□}{□}\)

Answer: \(\frac{6}{11}\)

Explanation:
The bag contains 4 red, 6 blue, and 1 white marble, thus the bag contains 4 + 6 + 1 = 11 marbles in total and thus there are 11 possible outcomes.
possible outcomes = 11
We note that 6 of the 11 marbles in the bag are blue and thus there are 6 favorable outcomes.
favorable outcomes = 6
The probability is the number of favorable outcomes divided by the number of possible outcomes.
P(blue) = favorable outcomes/possible outcomes = \(\frac{6}{11}\) ≈ 0.5455 = 54.55%

Question 4.
Rolling a number greater than 7 on a 12-sided number cube.
\(\frac{□}{□}\)

Answer: \(\frac{5}{12}\)

Explanation:
A 12 side number cube has 12 possible outcomes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
possible outcomes = 12
We note that 5 of the 12 possible outcomes result in a number greater than 7 (that is 8, 9, 10, 11, 12) and thus there are 5 favorable outcomes.
favorable outcomes = 5
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(Greater than 7) = favorable outcomes/possible outcomes = \(\frac{5}{12}\) ≈ 0.4167 = 41.67%

Page No. 426

Question 5.
Christopher picked coins randomly from his piggy bank and got the numbers of coins shown in the table. Find each experimental probability.

a. The next coin that Christopher picks is a quarter.
\(\frac{□}{□}\)

Answer: \(\frac{6}{23}\)

Explanation:
The table contains 7 pennies, 2 nickels, 8 dimes and 6 quarters, which are 7 + 2 + 8 + 6 = 23 coins in total and thus there are 23 possible outcomes.
possible outcomes = 23
We note that 6 of the 23 coins are quarters and thus there are 6 favorable outcomes.
favorable outcomes = 6
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(Quarter) = favorable outcomes/possible outcomes = \(\frac{6}{23}\) ≈ 0.2609 = 26.09%

Question 5.
b. The next coin that Christopher picks is not a quarter.
\(\frac{□}{□}\)

Answer: \(\frac{17}{23}\)

Explanation:
The sum of the probabilities of an event and its complement is always equal to 1.
P(not a Quarter) + P(Quarter) = 1
Let us then determine the probability of picking the marble that is not marked with the number 5.
P(not a Quarter) + P(Quarter) = 1
P(not a Quarter) = 1 – P(Quarter)
1 – \(\frac{6}{23}\)
= \(\frac{17}{23}\) ≈ 0.7391 = 73..91%

Question 5.
c. The next coin that Christopher picks is a penny or a nickel.
\(\frac{□}{□}\)

Answer: \(\frac{9}{23}\)

Explanation:
The table contains 7 pennies, 2 nickels, 8 dimes, and 6 quarters, which are 7 + 2 + 8 + 6 = 23 coins in total and thus there are 23 possible outcomes.
possible outcomes = 23
There are 7 pennies and 2 nickels, thus 7 + 2 = 9 of the coins are pennies or nickels and thus there are 9 favorable outcomes.
favorable outcomes = 9
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(Penny or nickel) = favorable outcomes/possible outcomes = \(\frac{9}{23}\) ≈ 0.3913 = 39.13%

Practice 13.6 Theoretical and Experimental Probability Answers Question 6.
A grocery store manager found that 54% of customers usually bring their own bags. In one afternoon, 82 out of 124 customers brought their own grocery bags. Did a greater or lesser number of people than usual bring their own bags?
_____________

Answer: Greater

Explanation:
54% of 124 is 0.54 × 124 ≈ 67 so more customers than usual brought their own bag.

EXERCISES – Page No. 427

Find the probability of each event.

Question 1.
Graciela picks a white mouse at random from a bin of 8 white mice, 2 gray mice, and 2 brown mice.
\(\frac{□}{□}\)

Answer: \(\frac{2}{3}\)

Explanation:
There are 8 white, 2 gray, and 2 brown mice, thus there are 8 + 2 + 2 = 12 mice in total and thus there are 12 possible outcomes.
possible outcomes = 12
8 of the mice are white and thus there are 8 favorable outcomes
favorable outcomes = 8
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(white) = favorable outcomes/possible outcomes = \(\frac{8}{12}\) ≈ 0.6667 = 66.67%

Question 2.
Theo spins a spinner that has 12 equal sections marked 1 through 12. It does not land on 1.
\(\frac{□}{□}\)

Answer: \(\frac{11}{12}\)

Explanation:
There are 12 numbers from 1 to 12 and thus there are 12 possible outcomes.
possible outcomes = 12
11 of the 12 numbers from 1 to 12 are not 1 and thus there are 11 favorable outcomes
favorable outcomes = 11
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(not 1) = favorable outcomes/possible outcomes = \(\frac{11}{12}\) ≈ 0.9167 = 91.67%

Question 3.
Tania flips a coin three times. The coin lands on heads twice and on tails once, not necessarily in that order.
\(\frac{□}{□}\)

Answer: \(\frac{3}{8}\)

Explanation:
A fair coin has 2 possible outcomes: Heads and Tails T.
There are then 8 possible outcomes when tossing 3 coins: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.
possible outcomes = 8
We note that 3 of the possible outcomes result in two heads and one tail HHT, HTH, TTH and thus there are 3 favorable outcomes
favorable outcomes = 3
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(two heads and one tail) = favorable outcomes/possible outcomes = \(\frac{3}{8}\)

Question 4.
Students are randomly assigned two-digit codes. Each digit is either 1, 2, 3, or 4. Guy is given the number 11.
\(\frac{□}{□}\)

Answer: \(\frac{1}{16}\)

Explanation:
Each digit has 4 possible outcomes so there are 4 × 4 = 16 possible two-digit numbers with digits of 1, 2, 3 or 4. There is only one way to 11 as the two-digit number so the probability is \(\frac{1}{16}\)

Question 5.
Patty tosses a coin and rolls a number cube.
a. Find the probability that the coin lands on heads and the cube lands on an even number.
\(\frac{□}{□}\)

Answer:
A coin has 2 possible outcomes: heads H and tails T.
A number cube has 6 possible outcomes: 1, 2, 3, 4, 5, 6
We then note that there are 2 . 6 = 12 possible outcomes for the coin and the number cube: H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6
possible outcomes = 12
We then note that 3 of the 12 possible outcomes result in heads and an even number: H2, H4, H6.
favorable outcomes = 3
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(heads and even) = favorable outcomes/possible outcomes = \(\frac{3}{12}\) = \(\frac{1}{4}\)

Question 5.
b. Patty tosses the coin and rolls the number cube 60 times. Predict how many times the coin will land on heads and the cube will land on an even number.
______ times

Answer:
The predicted number of repetitions resulting in heads and an even number is then obtained by multiplying the number of repetitions by the probability.
Prediction = Number of repetitions × P
= 60 × \(\frac{1}{4}\)
= 15
Thus we predict taht we obtain heads with an even number about 15 times.

Question 6.
Rajan’s school is having a raffle. The school sold raffle tickets with 3-digit numbers. Each digit is either 1, 2, or 3. The school also sold 2 tickets with the number 000. Which number is more likely to be picked, 123 or 000?
____________

Answer: 000

Explanation:
There is only 1 ticket that has the number 123 and 2 tickets that have 000 so it is more likely that 000 will be picked.

Page No. 428

Question 7.
Suppose you know that over the last 10 years, the probability that your town would have at least one major storm was 40%. Describe a simulation that you could use to find the experimental probability that your town will have at least one major storm in at least 3 of the next 5 years.
Type below:
____________

Answer:
Since the probability is 40% = 4/10 = 2/5, randomly generate numbers from 1 to 5 where 1 and 2 is a year with a major storm and 3 to 5 is a year without a major storm.

Unit 6 Performance Tasks

Question 8.
Meteorologist
A meteorologist predicts a 20% chance of rain for the next two nights and a 75% chance of rain on the third night.
a. On which night is it most likely to rain? On that night, is it likely to rain or unlikely to rain?
Type below:
____________

Answer: 3rd night

Explanation:
The third night it is most likely to rain since the probability of rain is higher that night. Since the probability of 75% is greater than 50%, it is likely that it will rain.

Question 8.
b. Tara would like to go camping for the next 3 nights, but will not go if it is likely to rain on all 3 nights. Should she go? Use probability to justify your answer.
Type below:
____________

Answer:
The probability that it will rain all three nights is 0.2 × 0.2 × 0.75 = 0.03 = 3%. It is unlikely that it will rain all 3 nights since the probability is 3% so she should go.

Lesson 7 Homework Practice Theoretical and Experimental Probability Question 9.
Sinead tossed 4 coins at the same time. She did this 50 times, and 6 of those times, all 4 coins showed the same result (heads or tails).
a. Find the experimental probability that all 4 coins show the same result when tossed.
\(\frac{□}{□}\)

Answer: \(\frac{3}{25}\)

Explanation:
The 4 coins were tossed 50 times and thus there are 50 possible outcomes.
possible outcomes = 50
The result showed that all 4 coins have the same result on 6 of the 50 tosses.
favorable outcomes = 6
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(Same result) = favorable outcomes/possible outcomes = \(\frac{6}{50}\) = \(\frac{3}{25}\)

Question 9.
b. Can you determine the experimental probability that no coin shows heads? Explain.
Type below:
____________

Answer:
The 4 coins were tossed 50 times and thus there are 50 possible outcomes.
possible outcomes = 50
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(No heads) = favorable outcomes/possible outcomes
Since we know the number of possible outcomes, we require the knowledge of the number of favorable outcomes to determine the probability.
That is, we require the knowledge of how many of the tosses resulted in no heads. Since this has not been given, we cannot determine the experimental probability that no coin show heads.

Question 9.
c. Suppose Sinead tosses the coins 125 more times. Use experimental probability to predict the number of times that all 4 coins will show heads or tails. Show your work.
_______ times

Answer:
The predicted number of times that all 4 coins will show heads or tails is then obtained by multiplying the number of times by the probability.
Since the coins were tossed 50 times initially and now were tossed 125 more times, the coins were tossed 50 + 125 = 175 times in total.
Prediction = Number of times × P
= 175 × \(\frac{3}{25}\)
= 7 × 3 = 21
Thus we predict that we obtain that all 4 coins will show heads or tails about 21 times.

Selected Response – Page No. 429

Question 1.
A pizza parlor offers thin, thick, and traditional-style pizza crusts. You can get pepperoni, beef, mushrooms, olives, or peppers for toppings. You order a one-topping pizza. How many outcomes are in the sample space?
Options:
a. 3
b. 5
c. 8
d. 15

Answer: 15

Explanation:
The Fundamental Counting Principle (also called the counting rule) is a way to figure out the number of outcomes in a probability problem. Basically, you multiply the events together to get the total number of outcomes.
Crust: 3 ways (thin, thick, traditional)
Topping: 5 ways (pepperoni, beef, mushrooms, olives, peppers)
Use the Fundamental Counting Principle:
3 × 5 = 15
Thus there are 15 possible outcomes in the sample space.
Thus the correct answer is option D.

Question 2.
A bag contains 9 purple marbles, 2 blue marbles, and 4 pink marbles. The probability of randomly drawing a blue marble is \(\frac{2}{15}\). What is the probability of not drawing a blue marble?
Options:
a. \(\frac{2}{15}\)
b. \(\frac{4}{15}\)
c. \(\frac{11}{15}\)
d. \(\frac{13}{15}\)

Answer: \(\frac{13}{15}\)

Explanation:
Given,
P(blue) = \(\frac{2}{15}\)
The sum of the probabilities of an event and its complement is always equal to 1.
P(not blue) + P(blue) = 1
Let us determine the probability of picking the marble that is not marked with the number 5.
P(not blue) = 1 – P(blue)
= 1 – \(\frac{2}{15}\)
= \(\frac{15}{15}\) – \(\frac{2}{15}\)
= \(\frac{13}{15}\)
Thus the correct answer is option D.

Question 3.
During the month of April, Dora kept track of the bugs she saw in her garden. She saw a ladybug on 23 days of the month. What is the experimental probability that she will see a ladybug on May 1?
Options:
a. \(\frac{1}{23}\)
b. \(\frac{7}{30}\)
c. \(\frac{1}{2}\)
d. \(\frac{23}{30}\)

Answer: \(\frac{23}{30}\)

Explanation:
There are 30 days in the month of April and thus there are 30 possible outcomes.
possible outcomes = 30
A ladybug was seen on 23 of the 30 days and thus there are 23 favorable outcomes.
favorable outcomes = 23
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(see ladybug) = favorable outcomes/possible outcomes = \(\frac{23}{30}\)
Thus the correct answer is option D.

Question 4.
Ryan flips a coin 8 times and gets tails all 8 times. What is the experimental probability that Ryan will get heads the next time he flips the coin?
Options:
a. 1
b. \(\frac{1}{2}\)
c. \(\frac{1}{8}\)
d. 0

Answer: 0

Explanation:
The coin was flipped 8 times and thus there are 8 possible outcomes.
possible outcomes = 5
All 8 flips resulted in tails and thus heads occurred on 0 of the flips, which implies that there are 0 favorable outcomes.
favorable outcomes = 0
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(heads) = favorable outcomes/possible outcomes = \(\frac{0}{8}\) = 0
Thus the correct answer is option D.

Question 5.
A used guitar is on sale for $280. Derek offers the seller \(\frac{3}{4}\) of the advertised price. How much does Derek offer for the guitar?
Options:
a. $180
b. $210
c. $240
d. $270

Answer: $210

Explanation:
Since 280(3/4) = 210, he offered $210 for the guitar.
Thus the correct answer is option B.

Question 6.
Jay tossed two coins several times and then recorded the results in the table below

What is the experimental probability that the coins will land on different sides on his next toss?
Options:
a. \(\frac{1}{5}\)
b. \(\frac{2}{5}\)
c. \(\frac{3}{5}\)
d. \(\frac{4}{5}\)

Answer: \(\frac{3}{5}\)

Explanation:
We have the coin toss results of 5 tosses and thus there are 5 possible outcomes.
possible outcomes = 5
Wwe note that 3 of the 5 tosses resulted in two different sides (H, T or T, H) and thus there are 3 favorable outcomes.
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(different sides) = favorable outcomes/possible outcomes = \(\frac{3}{5}\)
Thus the correct answer is option C.

Question 7.
What is the probability of tossing two fair coins and having exactly one land tails side up?
Options:
a. \(\frac{1}{8}\)
b. \(\frac{1}{4}\)
c. \(\frac{1}{3}\)
d. \(\frac{1}{2}\)

Answer: \(\frac{1}{2}\)

Explanation:
Each coin has 2 possible outcomes: Heads H and Tails T.
When tossing 2 fair coins, there are 4 possible outcomes: HH, HT, TH, TT.
possible outcomes = 4
We note that 2 of the 4 possible outcomes result in exactly one tail (TH or HT) and thus there are 2 favorable outcomes.
favorable outcomes = 2
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(exactly one tail) = favorable outcomes/possible outcomes = \(\frac{2}{4}\) = \(\frac{1}{2}\)
Thus the correct answer is option D.

Question 8.
Find the percent change from 60 to 96.
Options:
a. 37.5% decrease
b. 37.5% increase
c. 60% decrease
d. 60% increase

Answer: 60% increase

Explanation:
Percent change = (amount of change)/(original amount).
The amount of change is 96 – 60 = 36 and the original amount is 60.
The percent change is then 36/60 = 0.6 = 60%.
Since the amounts got larger, it is an increase.
Thus the correct answer is option D.

Question 9.
A bag contains 6 white beads and 4 black beads. You pick out a bead, record its color, and put the bead back in the bag. You repeat this process 35 times. Which is the best prediction of how many times you would expect to remove a white bead from the bag?
Options:
a. 6
b. 10
c. 18
d. 21

Answer: 21

Explanation:
The bag contains 6 white and 4 black beads, which are 6 + 4 = 10 beads in total and thus there are 10 favorable outcomes.
possible outcomes = 10
6 of the 10 beads are white and thus there are 6 favorable outcomes.
favorable outcomes = 6
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(white) = favorable outcomes/possible outcomes = 6/10 = 3/5 = 0.6 = 60%
The predicted number of selected white beads is then obtained by multiplying the number of repetitions by the probability.
Prediction = Number of repetitions × P(white)
35 × 6/10
= 210/10
= 21
Thus we predict that we removed 21 white beads from the bag.
Thus the correct answer is option D.

Question 10.
A set of cards includes 20 yellow cards, 16 green cards, and 24 blue cards. What is the probability that a blue card is chosen at random?
Options:
a. 0.04
b. 0.24
c. 0.4
d. 0.66

Answer: 0.4

Explanation:
There are 20 yellow, 16 green, and 24 blue cards, which are thus 20 + 16 + 21 = 60 cards and thus there are 60 possible outcomes.
possible outcomes = 60
24 of the 60 cards are blue and thus there are 24 favorable outcomes.
favorable outcomes = 24
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(blue) = favorable outcomes/possible outcomes = \(\frac{24}{60}\) = \(\frac{2}{5}\)
Thus the correct answer is option C.

Page No. 430

Question 11.
Jason, Erik, and Jamie are friends in art class. The teacher randomly chooses 2 of the 21 students in the class to work together on a project. What is the probability that two of these three friends will be chosen?
Options:
a. \(\frac{1}{105}\)
b. \(\frac{1}{70}\)
c. \(\frac{34}{140}\)
d. \(\frac{4}{50}\)

Answer: \(\frac{1}{70}\)

Explanation:
The probability that one of the friends is chosen as the first student is 3/21 = 1/7 since there are 3 friends and 21 total students.
The probability that a second friend is chosen is then 2/20 = 1/10 since there are 2 remaining friends and a total of 20 remaining students.
The probability that two friends is chosen is then (1/7)(1/10) = \(\frac{1}{70}\)
Thus the correct answer is option B.

Question 12.
Philip rolls a number cube 12 times. Which is the best prediction for the number of times that he will roll a number that is odd and less than 5?
Options:
a. 2
b. 3
c. 4
d. 6

Answer: 4

Explanation:
A number cube has 6 possible outcomes: 1, 2, 3, 4, 5, 6
possible outcomes = 6
2 of the 6 possible outcomes are odd and less than 5
favorable outcomes = 2
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(odd and less than 5) = favorable outcomes/possible outcomes = \(\frac{2}{6}\) = \(\frac{1}{3}\)
The predicted number of odd numbers less than 5 that is rolled is then obtained by multiplying the number of rolls by the probability.
Prediction = Number of rolls × P(odd and less than 5)
= 12 × \(\frac{1}{3}\)
= 4
Thus 4 of the trolls are expected to result in an odd number less than 5.
Thus the correct answer is option C.

Question 13.
A survey reveals that one airline’s flights have a 92% probability of being on time. Based on this, out of 4,000 flights in a year, how many flights would you predict will arrive on time?
Options:
a. 368
b. 386
c. 3680
d. 3860

Answer: 3680

Explanation:
Given,
P(on time) = 92% = 0.92
The predicted number of flights that arrive on time is then obtained by multiplying the number of flights by the probability.
Prediction = Number of flights × P(on time)
= 4000 × 0.92
= 3680
Thus we predict that about 3680 of the flights are on time.
Thus the correct answer is option C.

Question 14.
Matt’s house number is a two-digit number. Neither of the digits is 0 and the house number is even. What is the probability that Matt’s house number is 18?
Options:
a. \(\frac{1}{45}\)
b. \(\frac{1}{36}\)
c. \(\frac{1}{18}\)
d. \(\frac{1}{16}\)

Answer: \(\frac{1}{36}\)

Explanation:
The Fundamental Counting Principle (also called the counting rule) is a way to figure out the number of outcomes in a probability problem. Basically, you multiply the events together to get the total number of outcomes.
There are 9 digits excluding 0 (1, 2, 3, 4, 5, 6, 7, 8, 9) and there are 4 even digits excluding 0 (2, 4, 6, 8). By the fundamental counting principle, there are then 9 . 4 =36 two-digit numbers that do not contain a 0 and that are even. Thus there are 36 possible outcomes.
possible outcomes = 36
18 is 1 of the 36 possible outcomes and thus there is 1 favorable outcome.
favorable outcomes = 1
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(blue) = favorable outcomes/possible outcomes = \(\frac{1}{36}\)
Thus the correct answer is option B.

Mini-Tasks

Question 15.
Laura picked a crayon randomly from a box, recorded the color, and then placed it back in the box. She repeated the process and recorded the results in the table.

Find each experimental probability. Write your answers in the simplest form.
a. The next crayon Laura picks is red.
\(\frac{□}{□}\)

Answer:
There are 5 red, 6 blue, 7 yellow, and 2 green crayons, which are thus 5 + 6 + 7 + 2 = 20 crayons in total and thus there are 20 possible outcomes.
possible outcomes = 20
5 of the 20 crayons are red and thus there are 5 favorable outcomes.
favorable outcomes = 5
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(red) = favorable outcomes/possible outcomes = \(\frac{5}{20}\) = \(\frac{1}{4}\)

Question 15.
b. The next crayon Laura picks is not red.
\(\frac{□}{□}\)

Answer:
The sum of the probabilities of an event and its complement is always equal to 1.
P(not red) + P(red) = 1
Let us then determine the probability of picking the marble that is not marked with the number 5.
P(not red) = 1 – P(red)
= 1 – 1/4
= 3/4

Question 16.
For breakfast, Trevor has a choice of 3 types of bagels (plain, sesame, or multigrain), 2 types of eggs (scrambled or poached), and 2 juices (orange or apple).
a. Use the space below to make a tree diagram to find the sample space.
Type below:
_____________

Answer:
There are 3 types of bagels and thus we draw a root with 3 possible children labeled plain, sesame, and multigrain.
There are 2 types of eggs, thus we draw 2 children for each of the 3 previous children and label these two children as scrambled and poached.
There are 2 juices, thus we draw 2 children for each of the 2 previous children and label these two children as orange or apple.

Question 16.
b. If he chooses at random, what is the probability that Trevor eats a breakfast that has orange juice?
\(\frac{□}{□}\)

Answer: \(\frac{1}{2}\)

Explanation:
The bottom row of the tree diagram of the part (a) contains 12 elements and thus there are 12 possible outcomes.
possible outcomes = 12
6 of the labels in the bottom row are “orange” and thus there are 6 favorable outcomes.
favorable outcomes = 6
The probability is the number of favorable outcomes divided by the number of possible outcomes:
P(orange juice) = favorable outcomes/possible outcomes = \(\frac{6}{12}\) = \(\frac{1}{2}\)

Conclusion:

In this lesson, we will look into experimental probability and theoretical probability. The solutions prepared in this chapter are made by math experts. So, refer to the Go Math Grade 7 Answer Key Chapter 13 Theoretical Probability and Simulations while practicing for the exams. Follow our Go Math Answer Key to get the step-by-step explanation for all the chapters of 7th grade.

Go Math Grade 7 Answer Key Chapter 3 Rational Numbers consists of step by step explanations which help the students to score good marks in the exams. We provide the answers in such a way that all students can understand the concepts easily. Students who are facing issues in solving the problems can refer our Go Math Grade 7 Answer Key Chapter 3 Rational Numbers.

Go Math Grade 7 Answer Key Chapter 3 Rational Numbers

All concepts are covered in one place along with answers for Go Math Grade 7 Chapter 3 Rational Numbers. Follow the steps to Download HMH Go Math Chapter 3 Grade 7 Answer Key pdf to learn simple methods to solve the problems. The quick way of solving problems will help the students to save time. Hence, check the question and find out the complete answers and explanations for every problem.

Chapter 3 – Rational Numbers and Decimals

• Rational Numbers and Decimals – Guided Practice – Page No. 64
• Rational Numbers and Decimals – Independent Practice – Page No. 65
• Rational Numbers and Decimals – Page No. 66

Chapter 3 – Adding Rational Numbers

• Adding Rational Numbers – Guided Practice – Page No. 72
• Adding Rational Numbers – Independent Practice – Page No. 73
• Adding Rational Numbers – Independent Practice – Page No. 74

Chapter 3 – Subtracting Rational Numbers

• Subtracting Rational Numbers – Guided Practice – Page No. 79
• Subtracting Rational Numbers – Independent Practice – Page No. 80
• Subtracting Rational Numbers – Page No. 81
• Subtracting Rational Numbers – H.O.T – Page No. 82

Chapter 3 – Multiply Rational Numbers

• Multiply Rational Numbers – Guided Practice – Page No. 86
• Multiply Rational Numbers – Independent Practice – Page No. 87
• Multiply Rational Numbers – Page No. 88

Chapter 3 – Divide Rational Numbers

• Divide Rational Numbers – Guided Practice – Page No. 92
• Divide Rational Numbers – Independent Practice – Page No. 93
• Divide Rational Numbers – Page No. 94

Chapter 3 – Applying Rational Number Operations

• Applying Rational Number Operations – Guided Practice – Page No. 98
• Applying Rational Number Operations – Independent Practice – Page No. 99
• Applying Rational Number Operations – Page No. 100

Chapter 3 – Module Review

• Module Quiz – 3.1 Rational Numbers and Decimals – Page No. 101
• MODULE 3 MIXED REVIEW – Selected Response – Page No. 102
• Module 3 Review – Rational Numbers – Page No. 106

Chapter 3 – Performance Tasks

• Unit 1 Performance Tasks – Page No. 107
• Unit 1 Performance Tasks – Page No. 108

Chapter 3 – MIXED REVIEW

• Unit 1 MIXED REVIEW – Selected Response – Page No. 109
• Unit 1 MIXED REVIEW – Page No. 110

Rational Numbers and Decimals – Guided Practice – Page No. 64

Write each rational number as a decimal. Then tell whether each decimal is a terminating or a repeating decimal.

Question 1.
\(\frac{3}{5}\) =
___________ decimals

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.

\(\frac{3}{5}\) = 3 ÷ 5
3/5 = 0.6
The decimal is not repeating so it is a terminating decimal which is 0.6

Question 2.
\(\frac{89}{100}\) =
___________ decimals

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{89}{100}\) = 0.89
The decimal is not repeating so it is a terminating decimal which is 0.89

Lesson 3.1 Classifying Rational Numbers Answer Key Question 3.
\(\frac{4}{12}\) =
___________ decimals

Answer: repeating

Explanation:
To convert fraction decimals, we have to divide the numerator by the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{4}{12}\) = 4 ÷ 12
4/12 = 0. 333….
The quotient is a repeating decimal which is 0.33…

Question 4.
\(\frac{25}{99}\) =
___________ decimals

Answer: repeating

Explanation:
To convert fraction decimals, we have to divide the numerator by the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{25}{99}\) = 0.2525…
The quotient is a repeating decimal which is 0.2525…

Question 5.
\(\frac{7}{9}\) =
___________ decimals

Answer: repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{7}{9}\) = 0.77…
The quotient is a repeating decimal which is 0.77…

Question 6.
\(\frac{9}{25}\) =
___________ decimals

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{9}{25}\) = 0.36
The decimal is not repeating so it is a terminating decimal which is 0.36

Question 7.
\(\frac{1}{25}\) =
___________ decimals

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{1}{25}\) = 0.04
The decimal is not repeating so it is a terminating decimal which is 0.04

Question 8.
\(\frac{25}{176}\) =
___________ decimals

Answer: repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{25}{176}\) = 0.14204545454
The quotient is a repeating decimal which is 0.14204545454

Question 9.
\(\frac{12}{1000}\) =
___________ decimals

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{12}{1000}\) =0.012
The decimal is not repeating so it is a terminating decimal which is 0.012

Write each mixed number as a decimal.

Question 10.
11 \(\frac{1}{6}\) =
___________ decimals

Answer: repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
11 \(\frac{1}{6}\) = 11.1666666667
The quotient is a repeating decimal which is 11.1666666667

Math Accelerated Chapter 3 Operations with Rational Numbers Question 11.
2 \(\frac{9}{10}\) =
___________ decimals

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
First, convert the mixed fraction to the improper fraction.
2 \(\frac{9}{10}\) = \(\frac{29}{10}\) = 2.9
Thus, the decimal is not repeating so it is a terminating decimal which is 2.9

Question 12.
8 \(\frac{23}{100}\) =
___________ decimals

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator by the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
First, convert the mixed fraction to the improper fraction.
8 \(\frac{23}{100}\) = \(\frac{823}{100}\) = 8.23
Thus, the decimal is not repeating so it is a terminating decimal which is 8.23

Question 13.
7 \(\frac{3}{15}\) =
___________ decimals

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
First, convert the mixed fraction to the improper fraction.
7 \(\frac{3}{15}\) = \(\frac{108}{15}\) = 7.2
Thus, the decimal is not repeating so it is a terminating decimal which is 7.2

Question 14.
54 \(\frac{3}{11}\) =
___________ decimals

Answer: repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
First, convert the mixed fraction to the improper fraction.
54 \(\frac{3}{11}\) = \(\frac{597}{11}\) = 54.2727…
The quotient is a repeating decimal which is 54.2727…

Question 15.
3 \(\frac{1}{18}\) =
___________ decimals

Answer: repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
First, convert the mixed fraction to the improper fraction.
3 \(\frac{1}{18}\) = \(\frac{55}{18}\) = 3.055..
The quotient is a repeating decimal which is 3.055..

Question 16.
Maggie bought 3 \(\frac{2}{3}\) lb of apples to make some apple pies. What is the weight of the apples written as a decimal?
3 \(\frac{2}{3}\) =
___________ decimal

Answer: repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
First, convert the mixed fraction to the improper fraction.
3 \(\frac{2}{3}\) = \(\frac{11}{3}\) = 3.66..
The quotient is a repeating decimal which is 3.66..

Topic 1 Use Positive Rational Numbers Answer Key Question 17.
Harry’s dog weighs 12 \(\frac{7}{8}\) pounds. What is the weight of Harry’s dog written as a decimal?
12 \(\frac{7}{8}\) =
___________ decimals

Answer: terminating

Explanation:
Given that,
Harry’s dog weighs 12 \(\frac{7}{8}\) pounds.
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
First, convert the mixed fraction to the improper fraction.
12 \(\frac{7}{8}\) = \(\frac{103}{8}\) = 12.875

Essential Question Check-In

Question 18.
Tom is trying to write \(\frac{3}{47}\) as a decimal. He used long division and divided until he got the quotient 0.0638297872, at which point he stopped. Since the decimal doesn’t seem to terminate or repeat, he concluded that \(\frac{3}{47}\) is not rational. Do you agree or disagree? Why?
___________

Answer: Disagree

Explanation:
We are given the number:
{0, 1, 2, 3, ……45, 46}
When dividing a number by 47 the possible remainders at each step are:
This means that after at most 47 steps, we get a remainder that repeats. This means that the process and which repeats. This means that the process stops and we get a repeating decimal.

Rational Numbers and Decimals – Independent Practice – Page No. 65

Use the table for 19–23. Write each ratio in the form \(\frac{a}{b}\) and then as a decimal. Tell whether each decimal is a terminating or a repeating decimal.

Question 19.
Basketball players to football players
___________ decimal

Answer: Repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
Since the item is asking us to write basketball players to football players, we write the number of basketball players (5) in the numerator and the number of football players (11) in the denominator.
5/11 = 0.4545..
This is a repeating decimal with 45 as the repeating digits.

Question 20.
Hockey players to lacrosse players
___________ decimal

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
Since the item is asking us to write hockey players to lacrosse players, we write the number of hockey players (6) in the numerator and the number of lacrosse players (10) in the denominator.
Now convert the fraction into the decimal
6/10 = 0.6
This is a terminating decimal which is 0.6.

Question 21.
Polo players to football players
___________ decimal

Answer: Repeating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
Since the item is asking us to write polo players to football players, we write the number of polo players (4) in the numerator and the number of football players (11) in the denominator.
Now we convert this as a decimal.
4/11 = 0.36..
This is a repeating decimal with 36 as the repeating digits.

Question 22.
Lacrosse players to rugby players
___________ decimal

Answer: Repeating

Explanation:
To convert fraction decimals, we have to divide the numerator by the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
Since the item is asking us to write lacrosse players to rugby players, we write the number of lacrosse players (10) in the numerator and the number of rugby players (15) in the denominator.
10/15 = 0.66..
This is a repeating decimal with 6 as the repeating digit.

Rational Numbers Practice Worksheet Question 23.
Football players to soccer players
___________ decimal

Answer: terminating

Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits
Since the item is asking us to write football players to soccer players, we write the number of football players (11) in the numerator and the number of soccer players (11) in the denominator.
11/11 = 1
This is a terminating decimal which is 1.

Question 24.
Look for a Pattern Beth said that the ratio of the number of players in any sport to the number of players on a lacrosse team must always be a terminating decimal. Do you agree or disagree? Why?
___________

Answer: Agree

Explanation:
The ratios of the number of players in any sport to the number of players on a lacrosse team are:
{9/10, 5/10, 11/10, 6/10, 10/10, 4/10, 15/10, 11/10}
All these ratios are terminating decimals as all numerators divided by 10 lead to a terminating decimal.

Question 25.
Yvonne bought 4 \(\frac{7}{8}\) yards of material to make a dress.
a. What is 4 \(\frac{7}{8}\) written as an improper fraction?
\(\frac{□}{□}\)

Answer:
To convert fraction decimals, we have to divide the numerator by the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
Convert from mixed fraction to improper fraction.
4 \(\frac{7}{8}\) = (8 × 4) + 7 = 32 + 7 = 39/8

Question 25.
b. What is 4 \(\frac{7}{8}\) written as a decimal?
______

Answer:
Remember that we need to add the whole number and just convert the fraction part to decimal.
7/8 = 0.875
The fraction is a terminating decimal. Combining the whole number and the decimal part we get,
4 + 0.875 = 4.875

Question 25.
c. Communicate Mathematical Ideas If Yvonne wanted to make 3 dresses that use 4 \(\frac{7}{8}\) yd of fabric each, explain how she could use estimation to make sure she has enough fabric for all of them.
Type below:
_____________

Answer:
Using estimation, we say that 4 \(\frac{7}{8}\) ≈ 5.
We can now multiply 3 by 5, and therefore, she needs 15 yards of fabric.

Rational Numbers and Decimals – Page No. 66

Question 26.
Vocabulary A rational number can be written as the ratio of one _______ to another and can be represented by a repeating or ______ decimal.
Type below:
_____________

Answer: A rational number can be written as the ratio of one integer to another and can be represented by a repeating or terminating decimal.

Question 27.
Problem-Solving
Marcus is 5 \(\frac{7}{24}\) feet tall. Ben is 5 \(\frac{5}{16}\) feet tall. Which of the two boys is taller? Justify your answer.
_____________

Answer:
Explanation:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
To determine who is taller, we convert both to decimals. Remember that we need to add the whole number and just convert the fraction part to decimal.
For Marcus:
7/24 = 0.29166..
Combine the whole number and the decimal part we get 5.29166..
For Ben:
5/16 = 0.3125
Combine the whole number and the decimal part we get 5.1325
Hence Ben is taller.

Rational Numbers Grade 7 Question 28.
Represent Real-World Problems If one store is selling \(\frac{3}{4}\) of a bushel of apples for 9 dollars, and another store is selling \(\frac{2}{3}\) of a bushel of apples for 9 dollars, which store has the better deal? Explain your answer.
_____________

Answer:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
To determine which store has a better deal, we convert both fractions to decimals.
For the first store:
3/4 = 0.75
For the second store:
2/3 = 0.666..
Since the first store offers 0.75 of a bushel of apples, this store has a better deal.

Question 29.
Analyze Relationships You are given a fraction in simplest form. The numerator is not zero. When you write the fraction as a decimal, it is a repeating decimal. Which numbers from 1 to 10 could be the denominator?
Type below:
_____________

Answer: {3, 6, 7, 9}

Explanation:
Since the only numbers that can be factors of the denominators lead to a terminating decimal are 1, 2, and 5 and combinations of them, it means that if the denominator has at least one of the other numbers at the denominator, the decimal form will be a repeating decimal.
Among the numbers from 1 to 10, the presence of any of these numbers in the denominator will lead to a repeating decimal:
{3, 6, 7, 9}

Question 30.
Communicate Mathematical Ideas Julie got 21 of the 23 questions on her math test correct. She got 29 of the 32 questions on her science test correct. On which test did she get a higher score? Can you compare the fractions \(\frac{21}{23}\) and \(\frac{29}{32}\) by comparing 29 and 21? Explain. How can Julie compare her scores?
_____________

Answer:
To convert fraction decimals, we have to divide the numerator to the denominator. If the quotient goes on and on, then it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
For the math test:
21/23 = 0.9130
For the science test:
29/32 = 0.9063
Therefore she got a higher score in her math test.
Julie got a higher score in her math test. We cannot compare the fractions by comparing the numerators. Instead, we can compare her scores if the denominators of the fractions are the same.

Question 31.
Look for a Pattern Look at the decimal 0.121122111222.… If the pattern continues, is this a repeating decimal? Explain.
_____________

Answer: The number is not a repeating decimal.

Adding Rational Numbers – Guided Practice – Page No. 72

Use a number line to find each sum.

Question 1.
−3 + (−1.5) =
______

Answer: -4.5

Explanation:
Remember if the number being added is positive, move the number of units going to the right and if the number being added is negative, move the number of units to the left.
Since we are adding a negative number, starting from -3, we move 1.5 units to the left. This results in -4.5.

Question 2.
1.5 + 3.5 =
______

Answer: 5

Explanation:
Remember if the number being added is positive, move the number of units going to the right and if the number being added is negative, move the number of units to the left.
Since we are adding a positive number, starting from 1.5 we move 3.5 units to the right. This results in 5.

Question 3.
\(\frac{1}{4}+\frac{1}{2}\) =
\(\frac{□}{□}\)

Answer: \(\frac{3}{4}\)

Explanation:
Remember if the number being added is positive, move the number of units going to the right and if the number being added is negative, move the number of units to the left.
Since we are adding a positive number, starting from 1/4, we move 1/2 or 2/4, units to the right. This results in 3/4.

Adding Rational Numbers Lesson 3.2 Answer Key Question 4.
−1 \(\frac{1}{2}\) + (−1 \(\frac{1}{2}\)) =
______

Answer: -3

Explanation:
Remember if the number being added is positive, move the number of units going to the right and if the number being added is negative, move the number of units to the left.
Since we are adding a negative number, starting from −1 \(\frac{1}{2}\), we move 1 1/2 units to the left. This is results in -3.

Question 5.
3 + (−5) =
______

Answer: -2

Explanation:
Remember if the number being added is positive, move the number of units going to the right and if the number being added is negative, move the number of units to the left.
Since we are adding a negative number, starting from 3 we move 5 units to the left. This results in -2.

Question 6.
(−1.5) + 4 =
______

Answer: 2.5

Explanation:
Remember if the number being added is positive, move the number of units going to the right and if the number being added is negative, move the number of units to the left.
Since we are adding a positive number, starting from 1.5 we move 4 units to the left. This results in 2.5

Question 7.
Victor borrowed 21.50 dollars from his mother to go to the theater. A week later, he paid her 21.50 dollars back. How much does he still owe her?
______

Answer: 0

Explanation:
We use positive numbers for the money he receives and negative numbers for the money he returns.
21.50 – 21.50 = 0
The result is zero. This means he doesn’t owe anything to his mother.

Question 8.
Sandra used her debit card to buy lunch for 8.74 on Monday. On Tuesday, she deposited 8.74 back into her account. What is the overall increase or decrease in her bank account?
______

Answer: 0

Explanation:
We use positive numbers for the money she deposits and negative numbers for the money she spends.
-8.74 + 8.74 = 0
The result is zero. This means her bank account didn’t increase or decrease.

Find each sum without using a number line.

Question 9.
2.75 + (−2) + (−5.25) =
______

Answer: -4.50

Explanation:
We are given the expression:
2.75 + (-2) + (-5.25)
We group numbers with the same sign using the associative property.
2.75 – 7.25 = -4.50

Question 10.
−3 + (1 \(\frac{1}{2}\)) + (2 \(\frac{1}{2}\)) =
______

Answer: 1

Explanation:
We are given the expression
-3 + (1 \(\frac{1}{2}\)) + (2 \(\frac{1}{2}\))
-3 + 1.5 + 2.5
We group numbers with the same sign using the associative property.
-3 + 4 = 1
The larger number has a positive sign so the sum is 1.

Question 11.
−12.4 + 9.2 + 1 =
______

Answer: -2.2

Explanation:
We are given the expression
-12.4 + 9.2 + 1
We group numbers with the same sign using the associative property.
-12.4 + 10.2 = -2.2
The larger number has a negative sign so the answer is -2.2.

Question 12.
−12 + 8 + 13 =
______

Answer: 9

Explanation:
We are given the expression|
-12 + 8 + 13
We group numbers with the same sign using the associative property.
-12 + 21 = 9
The larger number has the positive sign so the answer is 9.

Question 13.
4.5 + (−12) + (−4.5) =
______

Answer: -12

Explanation:
We are given the expression
4.5 + (-12) + (-4.5)
We group numbers with the same sign using the associative property.
0 – 12 = -12
The larger number has the negative sign so the answer is -12.

Question 14.
\(\frac{1}{4}\) + (− \(\frac{3}{4}\)) =
– \(\frac{□}{□}\)

Answer: -0.50

Explanation:
We are given the expression
\(\frac{1}{4}\) + (− \(\frac{3}{4}\))
Convert the fraction to Decimal.
0.25 – 0.75 = -0.50
The larger number has a negative sign so the sum is -0.50

Adding Rational Numbers Worksheet 7th Grade Pdf Question 15.
−4 \(\frac{1}{2}\) + 2 =
– \(\frac{□}{□}\)

Answer: -2.5

Explanation:
We  = are given the expression
−4 \(\frac{1}{2}\) + 2
Convert from fraction to decimal.
-4.5 + 2 = -2.5
The larger number has the negative sign so the sum is -2.5.

Question 16.
−8 + (−1 \(\frac{1}{8}\)) =
– \(\frac{□}{□}\)

Answer: -9.125

Explanation:
We are given the expression
−8 + (−1 \(\frac{1}{8}\))
Convert from fraction to decimal.
-8 + (-1.125) = – 9.125

Question 17.
How can you use a number line to find the sum of -4 and 6?
Type below:
____________

Answer: 6

Explanation:
Remember if the number being added is positive, move the number of units going to the right and if the number being added is negative, move the number of units to the left.
Since we are adding a positive number, starting from -4 we move 6 units to the right. This results in 2.

Adding Rational Numbers – Independent Practice – Page No. 73

Question 18.
Samuel walks forward 19 steps. He represents this movement with a positive 19. How would he represent the opposite of this number?
_______

Answer: -19
He would represent the opposite of 19 by a negative 19.

Question 19.
Julia spends 2.25 on gas for her lawn mower. She earns 15.00 mowing her neighbor’s yard. What is Julia’s profit?
_______

Answer: $12.75

Explanation:
We use positive numbers for the money she earns and negative numbers for the money she spends.
-2.25 + 15 = 12.75
Thus her profit is $12.75

Question 20.
A submarine submerged at a depth of -35.25 meters dives an additional 8.5 meters. What is the new depth of the submarine?
_______

Answer: In adding two integers with same sign add their absolute value and keep the common sign.
When adding two integers with opposite sign subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
Since the submarine dove 32.25 meters down this can be interrupted as -32.25. And because it dove an additional 8.5 meters down, we can add -8.5 meters to the previous distance.
Add 32.25 and 8.5 meters
32.25 + 8.5 = 43.75 meters
Thus the submarines new depth is 43.75 meters deep or -43.75 meters.

Question 21.
Renee hiked for 4 \(\frac{3}{4}\) miles. After resting, Renee hiked back along the same route for 3 \(\frac{1}{4}\) miles. How many more miles does Renee need to hike to return to the place where she started?
_______ \(\frac{□}{□}\)

Answer:
Given that
Renee hiked for 4 \(\frac{3}{4}\) miles. After resting, Renee hiked back along the same route for 3 \(\frac{1}{4}\) miles.
4 \(\frac{3}{4}\) + (-3 \(\frac{1}{4}\)) = 1 \(\frac{1}{2}\)
Thus Renee needs to hike to return to the place where she started is 1 \(\frac{1}{2}\) or 1.5 miles.

Question 22.
Geography
The average elevation of the city of New Orleans, Louisiana, is 0.5 m below sea level. The highest point in Louisiana is Driskill Mountain at about 163.5 m higher than New Orleans. How high is Driskill Mountain?
_______

Answer: 163 meters

Explanation:
We use the positive numbers for the elevation above the sea level and the negative numbers for the elevation below the sea level.
163.5 – 0.5 = 163 meters
Thus the height of the Driskill mountain is 163 meters.

Question 23.
Problem-Solving
A contestant on a game show has 30 points. She answers a question correctly to win 15 points. Then she answers a question incorrectly and loses 25 points. What is the contestant’s final score?
_______

Answer: 20

Explanation:
We use positive numbers for won points and negative numbers for lost points.
30 + 15 + (-25) = 20
Thus the final score is 20.

Financial Literacy

Use the table for 24–26. Kameh owns a bakery. He recorded the bakery income and expenses in a table.

Question 24.
In which months were the expenses greater than the income? Name the month and find how much money was lost.
Type below:
___________

Answer:
We count the balance for January
1205 + (-1290.60)  = -85.60
We count the balance for February
1183 + (-1345.44) = -162.44
January: $85.60
February: $162.44

Question 25.
In which months was the income greater than the expenses? Name the months and find how much money was gained.
Type below:
___________

Answer:
The income was greater than the expenses in the months:
We count the balance for june:
2413 + (-2106.23) = 306.77
We count the balance for july:
2260 + (-1958.50) = 301.5
We count the balance for august:
2183 + (-1845.12) = 337.88
June: $306.77 gained
July: $301.5 gained
August: $337.88 gained

Question 26.
Communicate Mathematical Ideas
If the bakery started with an extra $250 from the profits in December, describe how to use the information in the table to figure out the profit or loss of money at the bakery by the end of August. Then calculate the profit or loss.
Balance: $ _______

Answer: 948.71

Explanation:
If the bakery started with an extra $250 from the profits in December.
We will add this amount to January’s income.
250 + 1205 + 1183 + 1664 + 2413 + 2260 + 2183 = 11,158
We compute the expenses during the 6 months
(-1290) + (-1345.44) + (-1664) + (-2106.24) + (-1958.50) + (-1845.12) = -10209.29
11158 -10209.29 = 948.71
Since the result is a positive number, the bakery has a profit.

Adding Rational Numbers – Independent Practice – Page No. 74

Question 27.
Vocabulary
-2 is the ________ of 2.
__________

Answer: additive inverse

Explanation:
When the sum of two numbers with opposite signs is 0, then they are additive inverses of each other.
Therefore, -2 is the additive inverse of 2.

Solve Problems with Rational Numbers Iready Question 28.
The basketball coach made up a game to play where each player takes 10 shots at the basket. For every basket made, the player gains 10 points. For every basket missed, the player loses 15 points.
a. The player with the highest score sank 7 baskets and missed 3. What was the highest score?
_______ points

Answer: 25

Explanation:
We use the positive numbers for won points and the negative numbers for lost points.
We determine the highest score:
7(10) + 3(-15) = 70 + (-45) = 25

Question 28.
b. The player with the lowest score sank 2 baskets and missed 8. What was the lowest score?
_______ points

Answer: -100

Explanation:
We determine the lowest score:
2(10) + 8(-15) = 20 + (-120) = -100

Question 28.
c. Write an expression using addition to find out what the score would be if a player sank 5 baskets and missed 5 baskets.
Type below:
__________

Answer: -25

Explanation:
We determine the score for 5 baskets and 5 missed baskets:
5(10) + 5(-15) = 50 + (-75)
50 – 75 = -25

H.O.T

FOCUS ON HIGHER ORDER THINKING

Question 29.
Communicate Mathematical Ideas
Explain the different ways it is possible to add two rational numbers and get a negative number.
Type below:
__________

Answer:
The sum of two rational numbers is negative either if both numbers are negative, or they have different signs, but the negative number is the one with the greater absolute value.

Practice and Homework Lesson 3.1 Answer Key Question 30.
Explain the Error
A student evaluated -4 + x for x = -9 \(\frac{1}{2}\) and got an answer of 5 \(\frac{1}{2}\). What might the student have done wrong?
Type below:
__________

Answer:
We expect about 95% of all possible samples to have a 95% confidence interval that contains the population proportion who favor such an amendment.

Question 31.
Draw Conclusions
Can you find the sum [5.5 + (-2.3)] + (-5.5 + 2.3) without performing any additions?
_______

Answer:
Yes, we can find the sum without performing any computation if we notice that the two numbers from each set of brackets are the opposites of the numbers in the other set  of bracelets, thus the sum is zero:
[5.5 + (-2.3)] + (-5.5 + 2.3)
5.5 – 2.3 – 5.5 + 2.3 = 0

Subtracting Rational Numbers – Guided Practice – Page No. 79

Use a number line to find each difference.

Question 1.
5 − (−8) =
_______

Answer: 13

Explanation:
Remember if the number being subtracted is positive, move the number of units going to the left and if the number being subtracted is negative, move the number of units to the right.
Since we are subtracting a negative number, starting from 5, we move 8 units to the right. This results in 13.

Question 2.
−3 \(\frac{1}{2}\) − 4 \(\frac{1}{2}\) =
_______

Answer: -8

Explanation:
Remember if the number being subtracted is positive, move the number of units going to the left and if the number being subtracted is negative, move the number of units to the right.
Since we are subtracting a positive number, starting from −3 \(\frac{1}{2}\), we move 4 \(\frac{1}{2}\) units to the left. This results in -8.

Question 3.
−7 − 4 =
_______

Answer: -11

Explanation:
Remember if the number being subtracted is positive, move the number of units going to the left and if the number being subtracted is negative, move the number of units to the right.
Since we are subtracting a positive number, starting from -7, we move 4 units to the left. This results in -11.

Question 4.
−0.5 − 3.5 =
_______

Answer: -4

Explanation:
Remember if the number being subtracted is positive, move the number of units going to the left and if the number being subtracted is negative, move the number of units to the right.
Since we are subtracting a positive number, starting from -0.5, we move 3.5 units to the left. This results in -4

Find each difference.

Question 5.
−14 − 22 =
_______

Answer: -36

Explanation:
We have to determine the difference
-14 – 22 = (-14) + (-22) = -36
−14 − 22 = -36

Question 6.
−12.5 − (−4.8) =
_______

Answer: -7.7

Explanation:
-12.5 – (-4.8)
We convert subtraction into addition with the opposite number
-12.5 – (-4.8) = -12.5 + 4.8 = -7.7
So, the answer is -7.7

Question 7.
\(\frac{1}{3}\) − (−\(\frac{2}{3}\)) =
_______

Answer: 1

Explanation:
\(\frac{1}{3}\) − (−\(\frac{2}{3}\))
\(\frac{1}{3}\) + \(\frac{2}{3}\) = \(\frac{3}{3}\) = 1
The result is 1.

Question 8.
65 − (−14) =
_______

Answer: 79

Explanation:
We convert subtraction into addition with the opposite number
65 − (−14) = 65 + 14 = 79
The answer is 79.

Question 9.
− \(\frac{2}{9}\) − (−3) =
_______ \(\frac{□}{□}\)

Answer: 2 \(\frac{7}{9}\)

Explanation:
We convert subtraction into addition with the opposite number
− \(\frac{2}{9}\) − (−3) = − \(\frac{2}{9}\) + 3 = 2 \(\frac{7}{9}\)
The answer is 2 \(\frac{7}{9}\)

Question 10.
24 \(\frac{3}{8}\) − (−54 \(\frac{1}{8}\)) =
_______ \(\frac{□}{□}\)

Answer: 78 \(\frac{1}{2}\)

Explanation:
We convert subtraction into addition with the opposite number.
24 \(\frac{3}{8}\) − (−54 \(\frac{1}{8}\)) = 24 \(\frac{3}{8}\) + 54 \(\frac{1}{8}\) = 78 \(\frac{1}{2}\)
Thus the result is 78 \(\frac{1}{2}\).

Question 11.
A girl is snorkeling 1 meter below sea level and then dives down another 0.5 meter. How far below sea level is the girl?
_______

Answer: 1.5 meter

Explanation:
1 m below sea level is represented by the number -14. Since she is diving down 0.5 m, you must subtract -1 – 0.5 = -1.5 m
Thus the girl is 1.5 m long.

Question 12.
The first play of a football game resulted in a loss of 12 \(\frac{1}{2}\) yards. Then a penalty resulted in another loss of 5 yards. What is the total loss or gain?
_______

Answer: 17 \(\frac{1}{2}\) yards

Explanation:
The first play of a football game resulted in a loss of 12 \(\frac{1}{2}\) yards. Then a penalty resulted in another loss of 5 yards.
-12 \(\frac{1}{2}\) – 5 = -17 \(\frac{1}{2}\) yards
It is a loss of 17 \(\frac{1}{2}\) yards

Question 13.
A climber starts descending from 533 feet above sea level and keeps going until she reaches 10 feet below sea level. How many feet did she descend?
_______

Answer: 543 feet

Explanation:
Given,
A climber starts descending from 533 feet above sea level and keeps going until she reaches 10 feet below sea level.
533 feet + 10 feet = 543 feet

Question 14.
Eleni withdrew 45.00 dollars from her savings account. She then used her debit card to buy groceries for 30.15 dollars. What was the total amount Eleni took out of her account?
_______

Answer: $75.15

Explanation:
Given that,
Eleni withdrew 45.00 dollars from her savings account. She then used her debit card to buy groceries for 30.15 dollars.
$45 + $30.15 = $75.15
Thus Eleni took $75.15 out of her account.

Question 15.
Mandy is trying to subtract 4 – 12, and she has asked you for help. How would you explain the process of solving the problem to Mandy, using a number line?
Type below:
____________

Answer: Start at 4 on the number line. Then move 12 places to the left since you are subtracting. This gives -8.

Subtracting Rational Numbers – Independent Practice – Page No. 80

Question 16.
Science
At the beginning of a laboratory experiment, the temperature of a substance is -12.6 °C. During the experiment, the temperature of the substance decreases 7.5 °C. What is the final temperature of the substance?
_______

Answer: -20.1°C

Explanation:
Remember if the number being subtracted is positive, move the number of units going to the left and if the number being subtracted is negative, move the number of units to the right.
Since the temperature of the substance is -12.6 and it decreases further by 7.5, we can create the expression -12.6 – 7.5.
-12.6 – 7.5 = -20.1
Thus the final temperature is -20.1°C

Question 17.
A diver went 25.65 feet below the surface of the ocean, and then 16.5 feet further down, he then rose 12.45 feet. Write and solve an expression to find the diver’s new depth.
_______

Answer: -29.7 feet

Explanation:
Remember if the number being subtracted is positive, move the number of units going to the left and if the number being subtracted is negative, move the number of units to the right.
Since the diver went down 25.65 feet then dove again further by 16.5 feet then rose up by 12.45 feet, we can create the expression -25.65 – 16.5 + 12.45 = -29.7 feet
The diver’s new depth is -29.7 feet.

Question 18.
A city known for its temperature extremes started the day at -5 degrees Fahrenheit. The temperature increased by 78 degrees Fahrenheit by midday, and then dropped 32 degrees by nightfall.
a. What expression can you write to find the temperature at nightfall?
Type below:
____________

Answer:
The temperature started at -5 degrees then increased 78 degrees and then dropped 32 degrees.
The expression is -5 + 78 – 32

Question 18.
b. What expression can you write to describe the overall change in temperature? Hint: Do not include the temperature at the beginning of the day since you only
Type below:
____________

Answer: The overall change is the increase and decrease combined.
The expression is 78 – 32

Question 18.
c. What is the final temperature at nightfall? What is the overall change in temperature?
Type below:
____________

Answer:
Use the first expression -5 + 78 – 32 = 73 – 32 = 41 degrees
78 – 32 = 46 degrees

Adding and Subtracting Rational Numbers Worksheet Answer Key Question 19.
Financial Literacy
On Monday, your bank account balance was -$12.58. Because you didn’t realize this, you wrote a check for $30.72 for groceries.
a. What is the new balance in your checking account?
$ _______

Answer:
Subtract the check amount from the initial balance.
-$12.58 – $30.72 = -$43.30

Question 19.
b. The bank charges a $25 fee for paying a check on a negative balance. What is the balance in your checking account after this fee?
$ _______

Answer:
Subtract 25 from the balance from part a.
-$43.30 – $25 = -$68.30

Question 19.
c. How much money do you need to deposit to bring your account balance back up to $0 after the fee?
$ _______

Answer:
Since the account balance is -$68.30, a deposit of $68.30 is required to make the balance $0.

Astronomy

Use the table for problems 20–21.

Question 20.
How much deeper is the deepest canyon on Mars than the deepest canyon on Venus?
_______

Answer: -16,500 feet deper

Explanation:
Subtract the lowest elevations of Mars and Venus.
-26,000 – (-9500) = -16,500

Question 21.
Persevere in Problem-Solving
What is the difference between Earth’s highest mountain and its deepest ocean canyon? What is the difference between Mars’ highest mountain and its deepest canyon? Which difference is greater? How much greater is it?
Type below:
____________

Answer:
Subtract the highest elevation and the lowest elevation on Earth.
29,035 – (-36,198) = 65,233
Subtract the highest elevation and the lowest elevation on Mars.
96,000 – 65,233 = 30,767
96,000 is greater than 65,233 so the difference for Mars is greater. subtract these two numbers to get how much greater.

Subtracting Rational Numbers – Page No. 81

Question 22.
Pamela wants to make some friendship bracelets for her friends. Each friendship bracelet needs 5.2 inches of string.
a. If Pamela has 20 inches of string, does she have enough to make bracelets for 4 of her friends?
a. _______

Answer: no

Explanation:
Each bracelet needs 5.2 inches so multiply 4 and 5.2 inches to see how many total inches she needs this is greater than 20 so she does not have enough.
4 × 5.2 = 20.8 inches

Question 22.
b. If so, how much string would she had left over? If not, how much more string would she need?
_______ in.

Answer: She needs 0.8 inches more

Question 23.
Jeremy is practicing some tricks on his skateboard. One trick takes him forward 5 feet, then he flips around and moves backwards 7.2 feet, then he moves forward again for 2.2 feet.
a. What expression could be used to find how far Jeremy is from his starting position when he finishes the trick?
Type below:
___________

Answer: 5 – 7.2 + 2.2

Explanation:
He moves 5 feet forward, back 7.2 feet, and then forward 2.2 feet.

Question 23.
b. How far from his starting point is he when he finishes the trick? Explain
_______ ft.

Answer: 0 ft

Explanation:
Since the distance just pulls hi back and forth at the same amount of distance.
5 – 7.2 + 2.2 = 0 ft

Question 24.
Esteban has $20 from his allowance. There is a comic book he wishes to buy that costs $4.25, a cereal bar that costs $0.89, and a small remote control car that costs $10.99.
a. Does Esteban have enough to buy everything?
a. _______

Answer:
Find the total amount of money he wants to spend this is less than 20 so he has enough
4.25 + 0.89 + 10.99 = 16.13
Thus Esteban had enough money.

Question 24.
b. If so, how much will he have left over? If not, how much does he still need?
$ _______

Answer:
Subtract the amount he wants to spend from the amount he has to find how much he has left.
20 – 16.13 = 3.87
Thus $3.87 left.

Subtracting Rational Numbers – H.O.T – Page No. 82

Focus on Higher Order Thinking

Question 25.
Look for a Pattern
Show how you could use the Commutative Property to simplify the evaluation of the expression \(-\frac{7}{16}-\frac{1}{4}-\frac{5}{16}\).
_______

Answer:
\(-\frac{7}{16}-\frac{1}{4}-\frac{5}{16}\)
-12/16 – 1/4
= -3/4 -1/4
= -4/4 = -1

Question 26.
Problem Solving
The temperatures for five days in Kaktovik, Alaska, are given below.
-19.6 °F, -22.5 °F, -20.9 °F, -19.5 °F, -22.4 °F
Temperatures for the following week are expected to be twelve degrees lower every day. What are the highest and lowest temperatures expected for the corresponding 5 days next week?
Type below:
____________

Answer:
The highest temperature for the first five days was -19.5 degrees so the highest temperature the following week is 12 degrees less than that. the lowest temperature the first week was -22.9 degree so the lowest temperature the second week is 12 degree below that
high: -19.5 – 12 = -31.5°F
low: -22.5 – 12 = -34.5°F

Question 27.
Make a Conjecture
Must the difference between two rational numbers be a rational number? Explain.
_______

Answer:
Yes, the difference between two rational numbers must be rational. Subtracting two fractions equals a fraction of an integer. Integers are rational numbers so even if the answer isn’t a fraction, it is still a rational number.

Question 28.
Look for a Pattern
Evan said that the difference between two negative numbers must be negative. Was he right? Use examples to illustrate your answer.
_______

Answer:
He is not correct. The difference between -2 and -5 is -2- (-5) = -2 + 5 = 3
which is not negative.

Multiply Rational Numbers – Guided Practice – Page No. 86

Use a number line to find each product.

Question 1.
5(−\(\frac{2}{3}\)) =
_______ \(\frac{□}{□}\)

Answer: -3 \(\frac{1}{3}\)

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying −\(\frac{2}{3}\) by 5, starting from 0, we move \(\frac{2}{3}\) units to the left five times. This results in -3 \(\frac{1}{3}\)

Question 2.
3(−\(\frac{1}{4}\)) =
\(\frac{□}{□}\)

Answer: –\(\frac{3}{4}\)

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying −\(\frac{1}{4}\) by 3, starting from 0, we move −\(\frac{1}{4}\) units to the left three times. This results in –\(\frac{3}{4}\).

Lesson 3 Extra Practice Rational Numbers Answer Key Question 3.
−3(−\(\frac{4}{7}\)) =
_______ \(\frac{□}{□}\)

Answer: 1 \(\frac{5}{7}\)

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying −\(\frac{4}{7}\) by -3, let us first multiply −\(\frac{4}{7}\) by 3. Starting from 0, we move \(\frac{4}{7}\) units to the left three times.
This results in -1 \(\frac{5}{7}\)
Therefore the opposite of this is 1 \(\frac{5}{7}\).

Question 4.
−\(\frac{3}{4}\)(−4) =
______

Answer: 3

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying −\(\frac{3}{4}\) by -4, let us first multiply −\(\frac{3}{4}\) by 4. Starting from 0, we move \(\frac{3}{4}\) units to the left three times. This results in -3. Therefore the opposite of this is 3.

Question 5.
4(−3) =
______

Answer: -12

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying -3 by 4, starting from 0, we move 3 units to the left four times. This results in -12.

Question 6.
(−1.8)5 =
______

Answer: -9

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying -1.8 by 5, starting from 0, we move 1.8 units to the left five times. This results in -9.

Question 7.
−2(−3.4) =
______

Answer: 6.8

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying -2 by -3.4, starting from 0, starting from 0, we move 3.4 units to the left two times. This results in -6.8. Therefore, the opposite of this is 6.8.

Question 8.
0.54(8) =
______

Answer: 4.32

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying 0.54 by 8, starting from 0, we move 0.54 units to the right eight times. This results in 4.32.

Question 9.
−5(−1.2) =
______

Answer: 6

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying -1.2 by -5, Starting from 0, we move 1.2 units to the left five times. This results in -6. Therefore the opposite of this is 6.

Question 10.
−2.4(3) =
______

Answer: -7.2

Explanation:
Remember if the number being multiplied is positive, starting from zero move the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Since we are multiplying -2.4 by 3, starting from 0, we move 2.4 units to the left three times. This results in -7.2

Multiply.

Question 11.
\(\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4}\) = □ × \(\frac{3}{4}\) =
\(\frac{□}{□}\)

Answer: \(\frac{1}{4}\)

Explanation:
\(\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4}\) = □ × \(\frac{3}{4}\)
1/3 × 3/4 = 1/4
\(\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4}\) = □ × \(\frac{3}{4}\) = \(\frac{1}{4}\)

Question 12.
\(-\frac{4}{7}\left(-\frac{3}{5}\right)\left(-\frac{7}{3}\right)\) = □ × \(\left(-\frac{7}{3}\right)\) =
\(\frac{□}{□}\)

Answer: – \(\frac{4}{5}\)

Explanation:
Multiply the first two fractions by multiplying the top numbers together and multiply the bottom numbers together.
Remember that two negatives make a positive so the product of the first two fractions is positive.
\(-\frac{4}{7}\left(-\frac{3}{5}\right)\left(-\frac{7}{3}\right)\) = □ × \(\left(-\frac{7}{3}\right)\)
12/35 × -7/3 = -4/5

Question 13.
\(-\frac{1}{8} \times 5 \times \frac{2}{3}\) =
\(\frac{□}{□}\)

Answer: –\(\frac{5}{12}\)

Explanation:
Use the commutative property to switch the order of the first two fractions.
\(-\frac{1}{8} \times 5 \times \frac{2}{3}\) = –\(\frac{1}{8}\) × \(\frac{2}{3}\) × 5
–\(\frac{1}{12}\) × 5 = –\(\frac{5}{12}\)

How to Multiply Fractions with a Negative Question 14.
\(-\frac{2}{3}\left(\frac{1}{2}\right)\left(-\frac{6}{7}\right)\) =
\(\frac{□}{□}\)

Answer: \(\frac{2}{7}\)

Explanation:
Multiply the first two fractions by canceling the 2s.
\(-\frac{2}{3}\left(\frac{1}{2}\right)\left(-\frac{6}{7}\right)\) = –\(\frac{1}{3}\)(-\(\frac{6}{7}\))
Multiply by canceling the 3 and 6 to get a 2 in the numerator two negatives make a positive.
So the answer is \(\frac{2}{7}\)

Question 15.
The price of one share of Acme Company declined $3.50 per day for 4 days in a row. What is the overall change in the price of one share?
$ _______

Answer: -$14

Explanation:
Given that,
The price of one share of Acme Company declined $3.50 per day for 4 days in a row.
-$3.50 × 4 = -$14.00
Thus the overall change in the price of one share is -$14.

Question 16.
In one day, 18 people each withdrew $100 from an ATM machine. What is the overall change in the amount of money in the ATM machine?
$ _______

Answer: The overall change in the amount of money in the ATM machine is the product of the amount people withdrew times the number of people. This gives -100(18) = -1800.
Therefore the overall change in the amount of money in the ATM machine is -$1800.

Question 17.
Explain how you can find the sign of the product of two or more rational numbers.
Type below:
____________

Answer: If the product has an even number of negative signs, then the product is positive. If the product has an odd number of negative signs, then the product is negative.

Multiply Rational Numbers – Independent Practice – Page No. 87

Question 18.
Financial Literacy
Sandy has $200 in her bank account.
a. If she writes 6 checks for exactly $19.98, what expression describes the change in her bank account?
_______

Answer: The change in her bank account is equal to the product of the check amounts and the number of checks.
This gives the expression 6(-19.98)

Question 18.
b. What is her account balance after the checks are cashed?
$ _______

Answer: She started with $200 and her account balance changes by 6(-19.98) dollars so her account balance is 200 – 6(-19.98) = 200 – 119.88 = 80.12

Question 19.
Communicating Mathematical Ideas
Explain, in words, how to find the product of -4(-1.5) using a number line. Where do you end up?
Type below:
____________

Answer:
First, find the value of -4(-1.5) by starting at 0 on the number line and moving 1.5 units left four times.
This gives a value of 4(-1.5) = -6
Since -4(-1.5) is the opposite of 4(-1.5), the answer is 6.

Question 20.
Greg sets his watch for the correct time on Wednesday. Exactly one week later, he finds that his watch has lost 3 \(\frac{1}{4}\) minutes. If his watch continues to lose time at the same rate, what will be the overall change in time after 8 weeks?
_______ minutes

Answer: 26 minutes

Explanation:
Given,
Greg sets his watch for the correct time on Wednesday.
Exactly one week later, he finds that his watch has lost 3 \(\frac{1}{4}\) minutes.
8(3 \(\frac{1}{4}\)) = 8 \(\frac{13}{4}\)
= 2 × 13 = 26 minutes.
Therefore the overall change in time after 8 weeks is 26 minutes.

Question 21.
A submarine dives below the surface, heading downward in three moves. If each move downward was 325 feet, where is the submarine after it is finished diving?
_______ feet

Answer: 975

Explanation:
Moving downward is represented by a negative number. Multiply the distance traveled down by the number of moves.
3 × -325 feet = -975
The submarine is 975 feet below the surface.

Question 22.
Multistep
For Home Economics class, Sandra has 5 cups of flour. She made 3 batches of cookies that each used 1.5 cups of flour. Write and solve an expression to find the amount of flour Sandra has left after making the 3 batches of cookies.
_______ cups

Answer: 0.5 cups

Explanation:
Sandra has a total of 5 cups of flour. Since she used 1.5 cups per batch of the cookie, and there are 3 batches, we can subtract the product of the cups and the number of batches
1.5 × 3 = 4.5
Therefore the expression should be 5 – 4.5 = 0.5
Thus Sandra has 0.5 cups of flour left.

Question 23.
Critique Reasoning
In class, Matthew stated,“I think that a negative is like the opposite. That is why multiplying a negative times a negative equals a positive. The opposite of negative is positive, so it is just like multiplying the opposite of a negative twice, which is two positives.”
Do you agree or disagree with his reasoning? What would you say in response to him?
_______

Answer: I agree with him. The product of two negatives is positive because the product of two positives is positive and negatives are opposites of positives.

Question 24.
Kaitlin is on a long car trip. Every time she stops to buy gas, she loses 15 minutes of travel time. If she has to stop 5 times, how late will she be getting to her destination?
_______ minutes

Answer: 75 minutes

Explanation:
Multiply the number of stops by the length of each stop to find the time she will be late.
5 × 15 = 75
Thus Kaitlin will be 75 minutes late to reaching her destination.

Multiply Rational Numbers – Page No. 88

Question 25.
The table shows the scoring system for quarterbacks in Jeremy’s fantasy football league. In one game, Jeremy’s quarterback had 2 touchdown passes, 16 complete passes, 7 incomplete passes, and 2 interceptions. How many total points did Jeremy’s quarterback score?

_______ pts

Answer: 13.5 points

Explanation:
Write the expression for the total number of points
2(6) + 16(0.5) + 7(-0.5) + 2(-1.5)
= 12 + 8 – 3.5 – 3
= 20 – 6.5
= 13.5
Thus Jeremy’s quarterback scored 13.5 points.

H.O.T

Focus On Higher Order Thinking

Question 26.
Represent Real-World Problems
The ground temperature at Brigham Airport is 12 °C. The temperature decreases by 6.8 °C for every increase of 1 kilometer above the ground. What is the overall change in temperature outside a plane flying at an altitude of 5 kilometers above Brigham Airport?
_______ °C

Answer: -22°C

Explanation:
Remember if the number being multiplied is positive, starting from zero moves the number of units by how many times it is multiplied going to the right and if the number being multiplied is negative, starting from zero, move the number of units by how many times it is multiplied going to the left.
Note that the ground temperature is 12°C. Since the temperature decreases by 6.8°C for every kilometer above ground, and the given height of the plane is 5 kilometers,
We can subtract the product of the temperature and the distance 5(6.8) from the ground temperature.
Therefore the expression should be 12 – 5(6.8)
= 12 – 34
= -22
Thus the temperature outside a plane flying at an altitude of 5 kilometers above Brigham Airport is -22°C

Question 27.
Identify Patterns
The product of four numbers, a, b, c, and d, is a negative number. The table shows one combination of positive and negative signs of the four numbers that could produce a negative product. Complete the table to show the seven other possible combinations.

Type below:
_____________

Answer:
In multiplying numbers, an odd number of negative signs produces a negative product.

Question 28.
Reason Abstractly
Find two integers whose sum is -7 and whose product is 12. Explain how you found the numbers.
Type below:
_____________

Answer: -3 and -4

Explanation:
Let x and y be the two numbers. Write the equations using the given information
x + y = -7
xy = 12
Since the two numbers multiply to a positive and add to a negative the two numbers must be negative. Find the pairs of negative numbers that multiply to 12.
-1 and -12, -2 and -6 and -3 and -4.
Thus the pairs that have a sum of -7 and a product of 12 are -3 and -4.

Divide Rational Numbers – Guided Practice – Page No. 92

Find each quotient.

Question 1.
\(\frac{0.72}{-0.9}\) =
_______

Answer: -0.8

Explanation:
We have to find the quotient:
\(\frac{0.72}{-0.9}\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
\(\frac{0.72}{-0.9}\) = -0.8

Question 2.
\(\left(-\frac{\frac{1}{5}}{\frac{7}{5}}\right)\) =
\(\frac{□}{□}\)

Answer: – \(\frac{1}{7}\)

Explanation:
We have to find the quotient:
\(\left(-\frac{\frac{1}{5}}{\frac{7}{5}}\right)\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
\(\left(-\frac{\frac{1}{5}}{\frac{7}{5}}\right)\) = – \(\frac{5}{35}\) = – \(\frac{1}{7}\)

Question 3.
\(\frac{56}{-7}\) =
_______

Answer: -8

Explanation:
We have to find the quotient:
\(\frac{56}{-7}\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
7 divides 56 eight times.
Thus the quotient of \(\frac{56}{-7}\) = -8

Question 4.
\(\frac{251}{4} \div\left(-\frac{3}{8}\right)\) =
\(\frac{□}{□}\)

Answer: –\(\frac{502}{3}\)

Explanation:
We have to find the quotient:
\(\frac{251}{4} \div\left(-\frac{3}{8}\right)\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
Rewrite using multiplication by multiplying with the reciprocal.
\(\frac{251}{4}\) × –\( \frac{8}{3}\) = –\(\frac{2008}{12}\)
–\(\frac{2008}{12}\) = –\(\frac{502}{3}\)
The quotient of \(\frac{251}{4} \div\left(-\frac{3}{8}\right)\) is –\(\frac{502}{3}\)

Multiplying and Dividing Rational Numbers Worksheet Answer Key Question 5.
\(\frac{75}{-\frac{1}{5}}\) =
_______

Answer: -375

Explanation:
We have to find the quotient:
\(\frac{75}{-\frac{1}{5}}\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
75 ÷ 1/5
75 × -5 = -375
Thus the quotient of \(\frac{75}{-\frac{1}{5}}\) is -375.

Question 6.
\(\frac{-91}{-13}\) =
_______

Answer: 7

Explanation:
We have to find the quotient:
\(\frac{-91}{-13}\)
We determine the sign of the quotient.
The quotient will be positive because the numbers have the same signs.
13 divides 91 seven times.
\(\frac{-91}{-13}\) = 7
Thus the quotient is 7.

Question 7.
\(\frac{-\frac{3}{7}}{\frac{9}{4}}\) =
\(\frac{□}{□}\)

Answer: –\(\frac{4}{21}\)

Explanation:
We have to find the quotient:
\(\frac{-\frac{3}{7}}{\frac{9}{4}}\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
\(\frac{-\frac{3}{7}}{\frac{9}{4}}\) = -3/7 × 4/9 = -12/63
-12/63 = -4/21
\(\frac{-\frac{3}{7}}{\frac{9}{4}}\) = –\(\frac{4}{21}\)

Question 8.
– \(\frac{12}{0.03}\) =
_______

Answer: -400

Explanation:
We have to find the quotient:
– \(\frac{12}{0.03}\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
– \(\frac{12}{0.03}\) = -400
So the quotient is -400.

Question 9.
A water pail in your backyard has a small hole in it. You notice that it has drained a total of 3.5 liters in 4 days. What is the average change in water volume each day?
_______ liters per day

Answer: -0.875 litres/day

Explanation:
Given that,
A water pail in your backyard has a small hole in it. You notice that it has drained a total of 3.5 liters in 4 days.
The average change of water volume each day is the quotient.
So, divide -3.5 by 4.
The quotient will be negative because the numbers have different signs.
-3.5/4 = -0.875
Thus the water volume diminishes by 0.875 liters each day.

Question 10.
The price of one share of ABC Company declined a total of $45.75 in 5 days. What was the average change in the price of one share per day?
$ _______

Answer: -$9.15

Explanation:
The price of one share of ABC Company declined a total of $45.75 in 5 days.
We use negative numbers for the price going down.
The average change in the price of one share per day is the quotient.
-45.75/5
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
-45.75/5 = -9.15
Thus the price of one share diminishes by $9.15 per day.

Rational Numbers Grade 7 Pdf Question 11.
To avoid a storm, a passenger jet pilot descended 0.44 miles in 0.8 minutes. What was the plane’s average change of altitude per minute?
_______

Answer: -0.55 miles/min

Explanation:
We use negative numbers for the altitude going down.
The plane’s average change of altitude per minute is the quotient:
-0.44/0.8
We determine the sign of the quotient
The quotient will be negative because the numbers have different signs.
-0.44/0.8 = -0.55
Therefore the plane descends by 0.55 miles per minute.

Essential Question Check-In

Question 12.
Explain how you would find the sign of the quotient \(\frac{32 \div(-2)}{-16 \div 4}\).
Type below:
___________

Answer: Positive

Explanation:
Given,
\(\frac{32 \div(-2)}{-16 \div 4}\)
Since all the operations are of multiplication and division, the sign is given by the number of negative signs.
If the number of negative signs is even, the quotient is positive while if the number of negative signs is odd, the quotient is negative.
In this case, the number of negative signs is 2, therefore even, so the quotient is positive.
\(\frac{32 \div(-2)}{-16 \div 4}\) = -16/-4 = 4
Thus the solution is positive.

Divide Rational Numbers – Independent Practice – Page No. 93

Question 13.
\(\frac{5}{-\frac{2}{8}}\) =
_______

Answer: -20

Explanation:
We are given the expression:
\(\frac{5}{-\frac{2}{8}}\)
The quotient will be negative because the numbers have different signs.
\(\frac{5}{-\frac{2}{8}}\) = 5 ÷ (-2/8)
We rewrite using the multiplication by multiplying with the reciprocal:
5 × -8/2 = 5 × -4 = -20
Thus the quotient for \(\frac{5}{-\frac{2}{8}}\) is -20.

Question 14.
\(5 \frac{1}{3} \div\left(-1 \frac{1}{2}\right)\) =
\(\frac{□}{□}\)

Answer: – \(\frac{32}{9}\)

Explanation:
We have to find the quotient:
\(5 \frac{1}{3} \div\left(-1 \frac{1}{2}\right)\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
16/3 ÷ -3/2
16/3 × -2/3 = -32/9
Thus the quotient for \(5 \frac{1}{3} \div\left(-1 \frac{1}{2}\right)\) is – \(\frac{32}{9}\)

Question 15.
\(\frac{(-120)}{(-6)}\) =
_______

Answer: 20

Explanation:
We have to find the quotient:
\(\frac{(-120)}{(-6)}\)
We determine the sign of the quotient.
The quotient will be positive because the numbers have the same signs.
6 divides 120 twenty times.
\(\frac{(-120)}{(-6)}\) = 20
Thus the quotient for \(\frac{(-120)}{(-6)}\) is 20.

Question 16.
\(\frac{-\frac{4}{5}}{-\frac{2}{3}}\) =
\(\frac{□}{□}\)

Answer: \(\frac{6}{5}\)

Explanation:
We have to find the quotient:
\(\frac{-\frac{4}{5}}{-\frac{2}{3}}\)
We determine the sign of the quotient.
The quotient will be positive because the numbers have the same signs.
(-4/5) × (-3/2) = 12/10 = 6/5
Thus the quotient for \(\frac{-\frac{4}{5}}{-\frac{2}{3}}\) is \(\frac{6}{5}\)

Question 17.
1.03 ÷ (−10.3) =
_______

Answer: -0.1

Explanation:
We have to find the quotient:
1.03 ÷ (−10.3)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
1.03 ÷ (-10.3) = -0.1

7th Grade Math Multiplication and Division of Rational Numbers Question 18.
\(\frac{(-0.4)}{80}\) =
_______

Answer: -0.005

Explanation:
We have to find the quotient:
\(\frac{(-0.4)}{80}\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
\(\frac{(-0.4)}{80}\) = -0.005
Thus the quotient for \(\frac{(-0.4)}{80}\) is -0.005.

Question 19.
\(1 \div \frac{9}{5}\) =
\(\frac{□}{□}\)

Answer: \(\frac{5}{9}\)

Explanation:
We have to find the quotient:
\(1 \div \frac{9}{5}\)
We determine the sign of the quotient.
The quotient will be positive because the numbers have the same signs
\(1 \div \frac{9}{5}\) = 1 × 5/9 = 5/9
Thus the quotient for \(1 \div \frac{9}{5}\) is \(\frac{5}{9}\)

Question 20.
\(\frac{\frac{-1}{4}}{\frac{23}{0.4}}\) =
\(\frac{□}{□}\)

Answer: –\(\frac{6}{23}\)

Explanation:
We have to find the quotient:
\(\frac{\frac{-1}{4}}{\frac{23}{0.4}}\)
We determine the sign of the quotient.
The quotient will be negative because the numbers have different signs.
\(\frac{\frac{-1}{4}}{\frac{23}{0.4}}\) = (-1/4) . (24/23) = -24/92
-24/92 = -6/23
Thus the quotient for \(\frac{\frac{-1}{4}}{\frac{23}{0.4}}\) is –\(\frac{6}{23}\)

Question 21.
\(\frac{-10.35}{-2.3}\) =
_______

Answer: 4.5

Explanation:
We have to find the quotient:
\(\frac{-10.35}{-2.3}\)
We determine the sign of the quotient.
The quotient will be positive because the numbers have the same signs
\(\frac{-10.35}{-2.3}\) = 4.5
So, the quotient for \(\frac{-10.35}{-2.3}\) is 4.5

Question 22.
Alex usually runs for 21 hours a week, training for a marathon. If he is unable to run for 3 days, describe how to find out how many hours of training time he loses, and write the appropriate integer to describe how it affects his time.
_______ hours

Answer: -9

Explanation:
Alex usually runs for 21 hours a week, training for a marathon.
If he runs 21 hours a week, he runs 21/3 = 3 hours.
If he doesn’t run for 3 days, then he is losing 3(3) = 9 hours of training time.
Since he is losing hours, the integer is negative so the answer is -9.

Question 23.
The running back for the Bulldogs football team carried the ball 9 times for a total loss of 15 \(\frac{3}{4}\) yards. Find the average change in field position on each run.
\(\frac{□}{□}\) yards per run

Answer: 1 \(\frac{3}{4}\) yards per run

Explanation:
The running back for the Bulldogs football team carried the ball 9 times for a total loss of 15 \(\frac{3}{4}\) yards
Convert from mixed fractions to improper fractions.
15 \(\frac{3}{4}\) = \(\frac{63}{4}\)
\(\frac{63}{4}\) × \(\frac{1}{9}\)
Divide 63 and 9 by 9 and then multiply the remaining factors.
\(\frac{7}{4}\) × 1 = \(\frac{7}{4}\)
Rewrite as a mixed fraction.
\(\frac{7}{4}\) = 1 \(\frac{3}{4}\) yards per run.

Question 24.
The 6:00 a.m. temperatures for four consecutive days in the town of Lincoln were -12.1 °C, -7.8°C, -14.3°C, and -7.2°C. What was the average 6:00 a.m. temperature for the four days?
_______ °C

Answer: -10.35°C

Explanation:
The average is the sum of the temperatures divided by the number of temperatures.
(-12.1 – 7.8 – 14.3 – 7.2)/4
= 41.4/4 = -10.35°C

Question 25.
Multistep
A seafood restaurant claims an increase of $1,750.00 over its average profit during a week where it introduced a special of baked clams.
a. If this is true, how much extra profit did it receive per day?
$ _______ per day

Answer:
There are 7 days in a week so divide the total profit of $1750 by 7 to find the extra profit per day.
1750/7 = 250
Thus he receives $250 extra profit per day.

Question 25.
b. If it had, instead, lost $150 per day, how much money would it have lost for the week?
$ _______

Answer:
Multiply the daily loss of $150 by 7 to get the weekly loss.
150 × 7 = $1050

Question 25.
c. If its total loss was $490 for the week, what was its average daily change?
$ _______ per day

Answer:
Since the company lost $490, its income changed by -$490.
Divide the change in income by 7 to find the average daily change.
-$490/7 = -$70
Thus the average daily change is -$70 per day.

Question 26.
A hot air balloon descended 99.6 meters in 12 seconds. What was the balloon’s average rate of descent in meters per second?
_______ m/s

Answer: 8.3 meters per second

Explanation:
Given that,
A hot air balloon descended 99.6 meters in 12 seconds.
99.6/12 = 8.3 meters per second.
Thus the balloon’s average rate of descent is 8.3 meters per second.

Divide Rational Numbers – Page No. 94

Question 27.
Sanderson is having trouble with his assignment. He’s shown work is as follows:
\(\frac{-\frac{3}{4}}{\frac{4}{3}}=-\frac{3}{4} \times \frac{4}{3}=-\frac{12}{12}=-1\)
However, his answer does not match the answer that his teacher gave him. What is Sanderson’s mistake? Find the correct answer.
\(\frac{□}{□}\)

Answer: – \(\frac{9}{16}\)

Explanation:
Sanderson made the mistake of not flipping the bottom fraction when he rewrote the problem as multiplication. The correct work is
\(\frac{-\frac{3}{4}}{\frac{4}{3}}\) = -3/4 × 4/3 = – \(\frac{9}{16}\)

Question 28.
Science
Beginning in 1996, a glacier lost an average of 3.7 meters of thickness each year. Find the total change in its thickness by the end of 2012.
_______ meters

Answer: 59.2 meters

Explanation:
Beginning in 1996, a glacier lost an average of 3.7 meters of thickness each year.
1996 to 2012 is 16 years so the total change in thickness is 16(3.7) = 59.2 meters.

H.O.T

Focus On Higher Order Thinking

Question 29.
Represent Real-World Problems
Describe a real-world situation that can be represented by the quotient -85 ÷ 15. Then find the quotient and explain what the quotient means in terms of the real-world situation.
Quotient: _______

Answer: -5.67

Explanation:
A possible real-world situation could be:
Sam has withdrawn $85 from his bank account over a period of 15 days. Find the average change in his account balance per day.
Answer: -85/15 = -5.67
So, the average rate of change in his account balance is -$5.67 per day.

Question 30.
Construct an Argument
Divide 5 by 4. Is your answer a rational number? Explain.
_______

Answer: The quotient is a rational number because it is a fraction.

Question 31.
Critical Thinking
Should the quotient of an integer divided by a nonzero integer always be a rational number? Why or why not?
_______

Answer:
Remember that in dividing and simplifying rational numbers, the quotient is positive if the signs of the numbers are the same, and negative if the signs of the numbers are different.
The quotient should be a rational number. This is because since the integers can be expressed as a quotient of two integers, then it is a rational number.

Applying Rational Number Operations – Guided Practice – Page No. 98

Question 1.
Mike hiked to Big Bear Lake in 4.5 hours at an average rate of 3 \(\frac{1}{5}\) miles per hour. Pedro hiked the same distance at a rate of 3 \(\frac{3}{5}\) miles per hour. How long did it take Pedro to reach the lake?
_______ hours

Answer: 4 hours

Explanation:
Given that,
Mike hiked to Big Bear Lake in 4.5 hours at an average rate of 3 \(\frac{1}{5}\) miles per hour. Pedro hiked the same distance at a rate of 3 \(\frac{3}{5}\) miles per hour.
4.5h × 3 \(\frac{3}{5}\) miles per hour = 4.5 × 3.2 miles = 14.4 miles
Plug in the distance you found in step 1 and the given rate in the problem to find the number of hours for Pedro.
14.4 miles ÷ 3 \(\frac{3}{5}\) miles per hour = 14.4 ÷ 3.6 hours = 4 hours

Operations with Rational Numbers Answer Key Question 2.
Until this year, Greenville had averaged 25.68 inches of rainfall per year for more than a century. This year’s total rainfall showed a change of −2 \(\frac{3}{8}\)% with respect to the previous average. How much rain fell this year?
_______ inches

Answer: 25.0701 inches

Explanation:
Greenville had averaged 25.68 inches of rainfall per year for more than a century.
This year’s total rainfall showed a change of −2 \(\frac{3}{8}\)% with respect to the previous average.
−2 \(\frac{3}{8}\)% = -2.375% = -0.02375
25.68 × 0.02375 ≈ 0.6099 inches
Find this year’s total rainfall
25.68 inches – 0.6099 inches = 25.0701 inches

Essential Question Check-In

Question 3.
Why is it important to consider using tools when you are solving a problem?
Type below:
___________

Answer: It is important to consider using tools, such as a calculator, when solving problems because some problems involve multiplying and dividing decimals that are too time-consuming to do by hand.

Applying Rational Number Operations – Independent Practice – Page No. 99

Solve, using appropriate tools.

Question 4.
Three rock climbers started a climb with each person carrying 7.8 kilograms of climbing equipment. A fourth climber with no equipment joined the group. The group divided the total weight of climbing equipment equally among the four climbers. How much did each climber carry?
_______ kilograms

Answer: 5.85 kilograms

Explanation:
Given,
Three rock climbers started a climb with each person carrying 7.8 kilograms of climbing equipment.
A fourth climber with no equipment joined the group.
3 × 7.8 = 23.4
The group divided the total weight of climbing equipment equally among the four climbers.
23.4/4 = 5.85 kilograms
Thus each climber carry 5.85 kilograms

Question 5.
Foster is centering a photo that is 3 \(\frac{1}{2}\) inches wide on a scrapbook page that is 12 inches wide. How far from each side of the page should he put the picture?
________ \(\frac{□}{□}\) inches

Answer: 4 \(\frac{1}{4}\) inches

Explanation:
Given,
Foster is centering a photo that is 3 \(\frac{1}{2}\) inches wide on a scrapbook page that is 12 inches wide.
Let x be how far the photo is from each side of the page.
Since the photo is 3 \(\frac{1}{2}\) inches wide, then the total width of the page is
x + 3 \(\frac{1}{2}\) + x = 2x + 3 \(\frac{1}{2}\)
2x + 3 \(\frac{1}{2}\) = 12
2x + 7/2 = 12
2x = 17/2
x = 17/4
Convert the fraction to the mixed fraction.
x = 4 \(\frac{1}{4}\) inches

Applying Rational Number Operations Question 6.
Diane serves breakfast to two groups of children at a daycare center. One box of Oaties contains 12 cups of cereal. She needs \(\frac{1}{3}\) cup for each younger child and \(\frac{3}{4}\) cup for each older child. Today’s group includes 11 younger children and 10 older children. Is one box of Oaties enough for everyone? Explain.
________

Answer: Yes

Explanation:
11 × \(\frac{1}{3}\) + 10 × \(\frac{3}{4}\)
\(\frac{11}{3}\) + \(\frac{15}{2}\)
\(\frac{22}{6}\) + \(\frac{45}{6}\) = \(\frac{67}{6}\)
= 11 \(\frac{1}{6}\)

Question 7.
The figure shows how the yard lines on a football field are numbered. The goal lines are labeled G. A referee was standing on a certain yard line as the first quarter ended. He walked 41 \(\frac{3}{4}\) yards to a yard line with the same number as the one he had just left. How far was the referee from the nearest goal line?
1
________ \(\frac{□}{□}\)

Answer: 29 \(\frac{1}{8}\)

Explanation:
The American football field is 100 yds long, 53 1/3 yards wide, and has 10-yard touchdown zones at each end of the field.
Let x = distance of the referee at the end of the quarter from the nearest goal.
The distance between the same yard lines on either side of the centerline is
100 – 2x
This distance is the 41 3/4 yards that the referee walked. Therefore
100 – 2x = 41.75
-2x = 41.75 – 100 = -58.25
x = 29.125 yd
Convert from decimal to fraction.
x = 29 \(\frac{1}{8}\) yards

In 8–10, a teacher gave a test with 50 questions, each worth the same number of points. Donovan got 39 out of 50 questions right. Marci’s score was 10 percentage points higher than Donovan’s.

Question 8.
What was Marci’s score? Explain.
________ %

Answer: 88 %

Explanation:
39/50 = 78/100
78/100 + 10/100 = 88/100 = 44/50
88/100 = 88%

Question 9.
How many more questions did Marci answer correctly? Explain.
________ questions

Answer: 5 questions

Explanation:
Marci got 44 correct and Donovan got 39 correct so she got 44 – 39 = 5 more questions correct.

Question 10.
Explain how you can check your answers for reasonableness.
Type below:
_____________

Answer:
You can check your answers for reasonableness by using estimates.
Donovan scored 39/50 which is about 40/50 = 80/100 = 80%
Ten percentage points higher is then 80% + 10% = 90% = 90/100 = 45/50.
Since Marci’s score was 44/50, it is a reasonable answer.

Applying Rational Number Operations – Page No. 100

For 11–13, use the expression 1.43 × \(\left(-\frac{19}{37}\right)\)

Question 11.
Critique Reasoning
Jamie says the value of the expression is close to −0.75. Does Jamie’s estimate seem reasonable? Explain.
_______

Answer: Yes

Explanation:
Jamie is correct. 1.43 is about 1.5 and -19/37 ≈ 1/2.
Since 1.5 × – 1/2 = -0.75
Jamie’s estimation is reasonable.

Operations with Rational Numbers Worksheet 7th Grade Pdf Question 12.
Find the product. Explain your method.
_______

Answer:
Using a calculator, you get that 1.43 × (-19/37) ≈ -0.734

Question 13.
Does your answer to Exercise 12 justify your answer to Exercise 11?
_______

Answer: Yes

Explanation:
-0.734 is close to the estimate of -0.75 so the answer to Exercise 12 justifies the answer to Exercise 11.

H.O.T

Focus On Higher Order Thinking

Question 14.
Persevere in Problem-Solving
A scuba diver dove from the surface of the ocean to an elevation of −79 \(\frac{9}{10}\) feet at a rate of -18.8 feet per minute. After spending 12.75 minutes at that elevation, the diver ascended to an elevation of −28 \(\frac{9}{10}\) feet. The total time for the dive so far was 19 \(\frac{1}{8}\) minutes. What was the rate of change in the diver’s elevation during the ascent?
_______ ft/min

Answer: 24 ft/min

Explanation:
Given that,
A scuba diver dove from the surface of the ocean to an elevation of −79 \(\frac{9}{10}\) feet at a rate of -18.8 feet per minute.
After spending 12.75 minutes at that elevation, the diver ascended to an elevation of −28 \(\frac{9}{10}\) feet. The total time for the dive so far was 19 \(\frac{1}{8}\) minutes.
−79 \(\frac{9}{10}\) ÷ -18.8 = 4.25 minutes
Find the time it took to ascend by subtracting the descent time and time spent at the descent elevation from the total dive time.
19 \(\frac{1}{8}\) – 4.25 – 12.75 = 2 1/8 minutes
-28 \(\frac{9}{10}\) – (-−79 \(\frac{9}{10}\)) = 51 feet
Find the rate of change by dividing the distance in feet divided by the time.
51/2 1/8 = 24 feet per minute

Question 15.
Analyze Relationships
Describe two ways you could evaluate 37% of the sum of 27 \(\frac{3}{5}\) and 15.9. Tell which method you would use and why
Type below:
___________

Answer:
Method 1:
Rewrite numbers in fraction form and evaluate algebraically.
37% (27 \(\frac{3}{5}\) + 15.9)
37/100 (27 3/5 + 15 9/10)
37/100 (138/5 + 159/10)
37/100 (435/10)
37/100 × 87/2 = 3219/200 = 16.095
Method 2:
Rewrite numbers in decimal form and evaluate with a calculator
37% (27 \(\frac{3}{5}\) + 15.9)
0.37(27.6 + 15.9)
0.37 × 43.5 = 16.095

Question 16.
Represent Real-World Problems
Describe a real-world problem you could solve with the help of a yardstick and a calculator.
Type below:
___________

Answer:
Finding the perimeter of the table. Using the yardstick you can get the side length of the table and add these measurements to get the perimeter.

Module Quiz – 3.1 Rational Numbers and Decimals – Page No. 101

Write each mixed number as a decimal.

Question 1.
4 \(\frac{1}{5}\) =
_______

Answer: 4.2

Explanation:
To convert fractions to decimals, simply divide the numerator to the denominator. If the quotient goes on and on, it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{1}{5}\) = 0.2
4 + 0.2 = 4.2
4 \(\frac{1}{5}\) = 4.2

Question 2.
12 \(\frac{14}{15}\) =
_______

Answer: 12.933..

Explanation:
To convert fractions to decimals, simply divide the numerator to the denominator. If the quotient goes on and on, it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{14}{15}\) = 0.933..
12 + 0.933 = 12.933..
12 \(\frac{14}{15}\) = 12.933..

Rational Number Operations Unit Study Guide Answer Key Question 3.
5 \(\frac{5}{32}\) =
_______

Answer: 5.15625

Explanation:
To convert fractions to decimals, simply divide the numerator to the denominator. If the quotient goes on and on, it is a repeating decimal, and to write this as a decimal, put a bar on top of the repeating digits.
\(\frac{5}{32}\) = 0.15625
5 + 0.15625 = 5.15625
5 \(\frac{5}{32}\) = 5.15625

3.2 Adding Rational Numbers

Find each sum.

Question 4.
4.5 + 7.1 =
_______

Answer: 11.6

Explanation:
To add or subtract numbers, make sure to align the digits vertically before doing the operation.
Make sure to align ones, tens, hundreds, and thousands of digits before adding
4.5
+7.1
 11.6

Question 5.
\(5 \frac{1}{6}+\left(-3 \frac{5}{6}\right)\) =
_______ \(\frac{□}{□}\)

Answer: 1 \(\frac{1}{3}\)

Explanation:
\(5 \frac{1}{6}+\left(-3 \frac{5}{6}\right)\) =
\(5 \frac{1}{6}\) – [/latex]3 \frac{5}{6}\right)[/latex] = 4 7/6 – 3 5/6
1 2/6 = 1 1/3
Thus \(5 \frac{1}{6}+\left(-3 \frac{5}{6}\right)\) = 1 \(\frac{1}{3}\)

3.3 Subtracting Rational Numbers

Find each difference.

Question 6.
\(-\frac{1}{8}-\left(6 \frac{7}{8}\right)\) =
_______

Answer: -7

Explanation:
Both numbers are negative. so add the opposites of each number and write the answer as a negative.
\(-\frac{1}{8}-\left(6 \frac{7}{8}\right)\)
-(\(\frac{1}{8}+\left(6 \frac{7}{8}\right)\))
= – 6 \(\frac{8}{8}\) = -6 – 1 = -7

Question 7.
14.2 − (−4.9) =
_______

Answer: 19.1

Explanation:
14.2 − (−4.9)
= 14.2 + 4.9 = 19.1

3.4 Multiplying Rational Numbers

Multiply.

Lesson 3.4 Multiplying Rational Numbers Answers Key Question 8.
\(-4\left(\frac{7}{10}\right)\) =
\(\frac{□}{□}\)

Answer: –\(\frac{14}{5}\)

Explanation:
Multiply the whole number with the numerator. write this product in the numerator and keep the same denominator.
\(-4\left(\frac{7}{10}\right)\) = –\(\frac{14}{5}\)

Question 9.
−3.2(−5.6)(4) =
_______

Answer: 71.68

Explanation:
Multiply the first two numbers. There are two negative signs so the answer will be positive.
−3.2(−5.6)(4) = 17.92 × 4 = 71.68

3.5 Dividing Rational Numbers

Find each quotient.

Question 10.
\(-\frac{19}{2} \div \frac{38}{7}\) =
\(\frac{□}{□}\)

Answer: –\(\frac{7}{4}\)

Explanation:
\(-\frac{19}{2} \div \frac{38}{7}\)
-19/2 × 7/38 = -7/2 × 1/2
= –\(\frac{7}{4}\)

Question 11.
\(\frac{-32.01}{-3.3}\) =
_______

Answer: 9.7

Explanation:
Given,
\(\frac{-32.01}{-3.3}\)
Remember that dividing two negatives gives a positive answer.
-32.01 ÷ -3.3 = 9.7

3.6 Applying Rational Number Operations

Question 12.
Luis bought the stock at $83.60. The next day, the price increased by 15.35 dollars. This new price changed by −4 \(\frac{3}{4}\)% the following day. What was the final stock price? Is your answer reasonable? Explain.
$ _______

Answer: $94.25

Explanation:
83.60 + 15.35 = 98.95
98.95 × −4 \(\frac{3}{4}\)% = 98.95 × -0.0475 = 4.70
98.95 – 4.70 = $94.25

Essential Question

Question 13.
How can you use negative numbers to represent real-world problems?
Type below:
___________

Answer:
Negative numbers can be used in real-world problems to represent decreases or values that are below a level considered to be 0.

MODULE 3 MIXED REVIEW – Selected Response – Page No. 102

Question 1.
What is −7 \(\frac{5}{12}\) written as a decimal?
Options:
a. -7.25
b. -7.333…
c. -7.41666…
d. -7.512

Answer: -7.41666…

Explanation:
Given,
−7 \(\frac{5}{12}\)
Convert from fraction to decimal.
5 ÷ 12 = 0.4166..
−7 \(\frac{5}{12}\) = -7.4166….
Thus the correct answer is option C.

Rational Numbers Worksheet Grade 7 with Answers Pdf Question 2.
Glenda began the day with a golf score of -6 and ended with a score of -10. Which statement represents her golf score for that day?
Options:
a. -6 – (-10) = 4
b. -10 – (-6) = -4
c. -6 + (-10) = -16
d. -10 + (-6) = -16

Answer: -10 – (-6) = -4

Explanation:
Given,
Her golf score for the day can be found by subtracting her ending score and her beginning score which gives
-10 – (-6) = -10 + 6 = -4
So, the correct answer is option B.

Question 3.
A submersible vessel at an elevation of -95 feet descends to 5 times that elevation. What is the vessel’s new elevation?
Options:
a. -475 ft
b. -19 ft
c. 19 ft
d. 475 ft

Answer: -475 ft

Explanation:
Given,
A submersible vessel at an elevation of -95 feet descends to 5 times that elevation.
-95 feet × 5 = -475 feet
Thus the correct answer is option A.

Question 4.
The temperature at 7 P.M. at a weather station in Minnesota was -5 °F. The temperature began changing at the rate of -2.5 °F per hour. What was the temperature at 10 P.M.?
Options:
a. -15 °F
b. -12.5 °F
c. 2.5 °F
d. 5 °F

Answer: -12.5 °F

Explanation:
Find the total change in temperature by multiplying the rate of change per hour times the number of hours from 7 pm to 10 pm.
-5 + (-7.5) = -12.5°F
Thus the correct answer is option B.

Question 5.
What is the sum of -2.16 and -1.75?
Options:
a. 0.41
b. 3.91
c. -0.41
d. -3.91

Answer: -3.91

Explanation:
Both numbers are negative so add their opposites and make the answer negative.
-2.16 + (-1.75) = -(2.16 + 1.75) = -3.91
So, the correct answer is option D.

Question 6.
On Sunday, the wind chill temperature reached -36 °F. On Monday, the wind chill temperature only reached \(\frac{1}{4}\) of Sunday’s wind chill temperature. What was the lowest wind chill temperature on Monday?
Options:
a. -9 °F
b. -36 \(\frac{1}{4}\) °F
c. -40 °F
d. -144 °F

Answer: -9 °F

Explanation:
Given that,
On Sunday, the wind chill temperature reached -36 °F.
On Monday, the wind chill temperature only reached \(\frac{1}{4}\) of Sunday’s wind chill temperature.
-36 × \(\frac{1}{4}\) = -9°F
Thus the correct answer is option A.

Question 7.
The level of the lake was 8 inches below normal. It decreased 1 \(\frac{1}{4}\) inches in June and 2 \(\frac{3}{8}\) inches more in July. What was the new level with respect to the normal level?
Options:
a. -11 \(\frac{5}{8}\) in.
b. -10 \(\frac{5}{8}\) in.
c. -9 \(\frac{1}{8}\) in.
d. -5 \(\frac{3}{8}\) in.

Answer: -11 \(\frac{5}{8}\) in.

Explanation:
The level of the lake was 8 inches below normal. It decreased 1 \(\frac{1}{4}\) inches in June and 2 \(\frac{3}{8}\) inches more in July.
The initial level is below normal so it is represented by a negative number. The level continued to decrease in June and July so those changes are also represented by negative numbers.
Find the sum of these values to find what the new level was with respect to the normal level.
-8 – 1 \(\frac{1}{4}\) – 2 \(\frac{3}{8}\)
= -8 – \(\frac{5}{4}\) – \(\frac{19}{8}\)
= – \(\frac{93}{8}\)
= – 11 \(\frac{5}{8}\)
Thus the correct answer is option A.

Mini-Task

Question 8.
The average annual rainfall for a town is 43.2 inches.
a. What is the average monthly rainfall?
________

Answer:
If the average rainfall is 43.2 inches then the monthly rainfall is 43.2/12 = 3.6 inches since there are 12 months in a year.

Question 8.

b. The difference of a given month’s rainfall from the average monthly rainfall is called the deviation. What is the deviation for each month shown?
May: ___________ inch
June: ___________ inches
July: ___________ inches

Answer:
The deviation for May is 2 3/5 – 3.6 = 2.6 – 3.6 = -1 inches.
The deviation for June is 7/8 – 3.6 = -2.725 inches.
The deviation for July 4 1/4 – 3.6 = 0.65 inches.

Question 8.
c. The average monthly rainfall for the previous 9 months was 4 inches. Did the town exceed its average annual rainfall? If so, by how much?
________

Answer:
It is rained 4 inches for 9 months, the total amount of rain over the 12-month period is then 9(4) + 2 3/5 + 7/8 + 4 1/4
= 36 + 2.6 + 0.875 + 4.25 = 43.725.
Since this is greater than the average annual rainfall of 43.2, the town did exceed its average annual rainfall.
The difference of 43.725 and 43.2 is
43.725 – 43.2 = 0.525
so, it exceeded it by 0.525 inches.

Module 3 Review – Rational Numbers – Page No. 106

EXERCISES

Write each mixed number as a whole number or decimal. Classify each number according to the group(s) to which it belongs: rational numbers, integers, or whole numbers.

Question 1.
\(\frac{3}{4}\)
________

Answer: 0.75, rational

Explanation:
Write as a decimal by dividing 3 by 4. A shortcut with fourths is to think of the fractions in terms of money. 4 quarters make a dollar and 3 quarters is $0.75 so three-fourths is 0.75 in decimal form.
Since \(\frac{3}{4}\) could not be written as a whole number or integer, it is a rational number.

Question 2.
\(\frac{8}{2}\)
________

Answer: 4

Explanation:
\(\frac{8}{2}\) = 4
Since 4 doesn’t have a decimal and is positive, it is a whole number. All whole numbers are also integers and rational numbers so 4 is a rational number, integer, and a whole number.

Operations with Rational Numbers Question 3.
\(\frac{11}{3}\)
________

Answer: 3.66

Explanation:
Rewrite as a mixed number and then divide 2 by 3 to get the decimal part of the number.
\(\frac{11}{3}\) = 3 2/3 = 3.666…
Since 3.66.. has a decimal, it is not an integer or whole number. Therefore it is a rational number only.

Question 4.
\(\frac{5}{2}\)
________

Answer: 2.5

Explanation:
Write as a mixed number and then divide 1 by 2 to get the decimal part of the number a shortcut is to think of the fraction in terms of money. Half a dollar is $0.50 so one half equals 0.50 = 0.50
Since 2.5 has a decimal, it is not an integer or whole number. Therefore 2.5 is a rational number only.

Find each sum or difference.

Question 5.
−5 + 9.5
________

Answer: 4.5

Explanation:
Rewrite as subtraction and then subtract.
-5 + 9.5 = 4.5

Question 6.
\(\frac{1}{6}\) + (−\(\frac{5}{6}\))
\(\frac{□}{□}\)

Answer: –\(\frac{2}{3}\)

Explanation:
Rewrite as subtraction and then subtract.
\(\frac{1}{6}\) + (−\(\frac{5}{6}\))
\(\frac{1}{6}\) −\(\frac{5}{6}\)
= –\(\frac{4}{6}\) = –\(\frac{2}{3}\)

Question 7.
−0.5 + (−8.5)
________

Answer: -9

Explanation:
Both numbers are negative so add their opposites and write the answers as a negative.
−0.5 + (−8.5) = -(0.5 + 8.5) = -9

Question 8.
−3 − (−8)
________

Answer: 5

Explanation:
Rewrite as addition since subtracting a negative is the same as adding a positive.
−3 − (−8) = -3 + 8 = 5

Question 9.
5.6 − (−3.1)
________

Answer: 8.7

Explanation:
Rewrite as addition since subtracting a negative is the same as adding a positive.
5.6 − (−3.1) = 5.6 + 3.1 = 8.7

Question 10.
3 \(\frac{1}{2}\) − 2 \(\frac{1}{4}\)
\(\frac{□}{□}\)

Answer: 1 \(\frac{1}{4}\)

Explanation:
Get common denominator.
3 \(\frac{1}{2}\) − 2 \(\frac{1}{4}\)
3 \(\frac{2}{4}\) − 2 \(\frac{1}{4}\) = 1 \(\frac{1}{4}\)

Find each product or quotient

Question 11.
−9 × (−5)
________

Answer: 45

Explanation:
Multiply two negative numbers to make a positive number.
−9 × (−5) = 45

Question 12.
0 × (−7)
________

Answer: 0

Explanation:
Any number multiplied by 0 will be zero.
So, the product is 0.

Question 13.
−8 × 8
________

Answer: -64

Explanation:
Multiply since there is only one negative the answer is negative.
-8 × 8 = -64

Question 14.
\(\frac{-56}{8}\)
________

Answer: -7

Explanation:
Divide since there is only one negative the answer is negative.
8 divides 56 seven times.
\(\frac{-56}{8}\) = -7

Question 15.
\(\frac{-130}{-5}\)

Answer: 26

Explanation:
Divide since there are two negative signs the answer is positive.
\(\frac{-130}{-5}\) = 26

Question 16.
\(\frac{34.5}{1.5}\)
________

Answer: 23

Explanation:
Divide since both the numbers are positive the answer will be positive.
\(\frac{34.5}{1.5}\) = 23
1.5 divides 34.5 23 times.
So, the quotient is 23.

Question 17.
\(-\frac{2}{5}\left(-\frac{1}{2}\right)\left(-\frac{5}{6}\right)\)
\(\frac{□}{□}\)

Answer: –\(\frac{1}{6}\)

Explanation:
Multiply by canceling the 2s and 5s an odd number of negatives makes a negative so the answer is negative.
\(-\frac{2}{5}\left(-\frac{1}{2}\right)\left(-\frac{5}{6}\right)\) = –\(\frac{1}{6}\)

Question 18.
\(\frac{1}{5}\left(-\frac{5}{7}\right)\left(\frac{3}{4}\right)\)
\(\frac{□}{□}\)

Answer: –\(\frac{3}{28}\)

Explanation:
\(\frac{1}{5}\left(-\frac{5}{7}\right)\left(\frac{3}{4}\right)\)
multiply by cancelling the 5s
\(\frac{1}{5}\left(-\frac{5}{7}\right)\left(\frac{3}{4}\right)\) = – 3/7×4 = -3/28
Thus \(\frac{1}{5}\left(-\frac{5}{7}\right)\left(\frac{3}{4}\right)\) = –\(\frac{3}{28}\)

Question 19.
Lei withdrew $50 from her bank account every day for a week. What was the change in her account in that week?
$ ________

Answer: -$350

Explanation:
Lei withdrew $50 from her bank account every day for a week.
Convert from week to days
1 week = 7 days
7 × -50 = -350
The change in her account is -$350.

Question 20.
Dan is cutting 4.75-foot lengths of twine from a 240-foot spool of twine. He needs to cut 42 lengths and says that 40.5 feet of twine will remain. Show that this is reasonable.
Type below:
__________

Answer:
The estimation of 4.75 is 5 and 42 is 40.
5 × 40 = 200
So he will be using about 200 feet.
He has 240 feet so he will have about 240-200 = 40 feet remaining.
Since 40 ≈ 40.5
The answer is reasonable.

Unit 1 Performance Tasks – Page No. 107

Question 1.
Armand is an urban planner, and he has proposed a site for a new town library. The site is between City Hall and the post office on Main Street.

The distance between City Hall and the post office is 612 miles. City Hall is 114 miles closer to the library site than it is to the post office.
a. Write 6 \(\frac{1}{2}\) miles and 1 \(\frac{1}{4}\) miles as decimals
6 \(\frac{1}{2}\) = __________
1 \(\frac{1}{4}\) = __________

Answer:
Write as decimal by dividing 1 by 2 and dividing 1 by 4 A shortcut is to think about ur in terms of money. Half of a dollar is $0.50 and a quarter is $0.25
So 1/2 = 0.50 = 0.5
1/4 = 0.25
6 1/2 = 6.5 and 1 1/4 = 1.25

Question 1.
b. Let d represent the distance from City Hall to the library site. Write an expression for the distance from the library site to the post office.
__________

Answer:
The library is closer to City Hall than the post office is so d is the difference between the distance from City Hall to the Post Office and the distance between City Hall and the Library Site.
d = 6 1/2 – 1 1/4

Question 1.
c. Write an equation that represents the following statement: The distance from City Hall to the library site plus the distance from the library site to the post office is equal to the distance from City Hall to the post office.
Type below:
__________

Answer:
The distance from the City Hall to the library is d, the distance from the library to the post office is 1 1/4 since the library is 1 1/4 miles closer to City Hall than the post office is, the distance from City Hall to the Post Office is 6 1/4
d + 1 1/4 = 6 1/4

Question 1.
d. Solve your equation from part c to determine the distance from City Hall to the library site, and the distance from the post office to the library site.
City Hall to library site: __________ miles
Library site to post office: __________ miles

Answer:
d = 6 1/2 – 1 1/4
d = 6 2/4 – 1 1/4
d = 5 1/4
Thus the distance is 5 1/4 miles.

Question 2.
Sumaya is reading a book with 288 pages. She has already read 90 pages. She plans to read 20 more pages each day until she finishes the book.
a. Sumaya writes the equation 378 = -20d to find the number of days she will need to finish the book. Identify the errors that Sumaya made.
Type below:
__________

Answer:
She made the mistake of using -20 in the equation instead of a positive 20. The negative can’t be used since she is not reading a negative number of pages per day.
She also made the mistake of adding 90 to 288 instead of subtracting.
Since she has already read 90 pages she has less than 288 pages left to read, not more.
288 – 90 = 198
The correct equation is 198 = 20d

Question 2.
b. Write and solve an equation to determine how many days Sumaya will need to finish the book. In your answer, count part of a day as a full day. Show that your answer is reasonable.
______ days

Answer:

198 = 20d is dividing both sides by 20 gives d = 198/20 = 9.9
Rounding this up gives 10 days.
This answer is reasonable since the book is about 300 pages and she has read about 100 pages of the book leaving about 200 pages left to read.
She is reading 20 pages per day and 20 × 10 = 200
So it would take 10 days to read about 200 pages.

Question 2.
c. Estimate how many days you would need to read a book about the same length as Sumaya’s book. What information did you use to find the estimate?
Type below:
__________

Answer:
Sumaya’s book is about 300 pages. Reading 20 pages a day would mean it would take about 300/20 = 15 days to read the book.

Unit 1 Performance Tasks – Page No. 108

Question 3.
Jackson works as a veterinary technician and earns $12.20 per hour.
a. Jackson normally works 40 hours a week. In a normal week, what is his total pay before taxes and other deductions?
$ ______

Answer: $488

Explanation:
Jackson works as a veterinary technician and earns $12.20 per hour.
Jackson normally works 40 hours a week.
40 × $12.20 = $488
Thus the total pay before taxes and other deductions is $488.

Question 3.
b. Last week, Jackson was ill and missed some work. His total pay before deductions was $372.10. Write and solve an equation to find the number of hours Jackson worked.
______ hours

Answer: 30.5 hours

Explanation:
Jackson works as a veterinary technician and earns $12.20 per hour.
His total pay before deductions was $372.10.
$12.20h = $372.10
h = 372.10/12.20
h = 30.5 hours

Question 3.
c. Jackson records his hours each day on a time sheet. Last week when he was ill, his time sheet was incomplete. How many hours are missing? Show your work. Then show that your answer is reasonable.

______ hours

Answer: 6.75 hours

Explanation:
8 + 7.25 + 8.5 = 23.75
30.5 – 23.75 = 6.75 hours

Question 3.
d. When Jackson works more than 40 hours in a week, he earns 1.5 times his normal hourly rate for each of the extra hours. Jackson worked 43 hours one week. What was his total pay before deductions? Justify your answer.
$ __________________

Answer: $542.90

Explanation:
When Jackson works more than 40 hours in a week, he earns 1.5 times his normal hourly rate for each of the extra hours.
Jackson worked 43 hours one week.
40 × 12.20 + 3 × 1.5 × 12.20 = $488 + $54.90 = $542.90

Question 3.
e. What is a reasonable range for Jackson’s expected yearly pay before deductions? Describe any assumptions you made in finding your answer.
$ __________________

Answer:

Assuming he works between 40 and 45 hours per week, his weekly pay range is between 40 × 12.20 = $488
40 × 12.20 + 5 × 1.5 × 12.20 = 488 + 91.50 = $579.50
Since there are 52 weeks in a year, his yearly pay is between 52 × 488 ≈ $25,000
and 52 × $579.50 ≈ $30,000.

Unit 1 MIXED REVIEW – Selected Response – Page No. 109

Question 1.
What is −6 \(\frac{9}{16}\) written as a decimal?
Options:
a. -6.625
b. -6.5625
c. -6.4375
d. -6.125

Answer: -6.5625

Explanation:
−6 \(\frac{9}{16}\)
Divide 9 by 16 to get 9/16 = 0.5625.
6 \(\frac{9}{16}\) = 6 + 0.5625 = 6.5625
−6 \(\frac{9}{16}\) = -6.5625
Thus the correct answer is option B.

Lesson 1 Rational Numbers Answer Key Question 2.
Working together, 6 friends pick 14 \(\frac{2}{5}\) pounds of pecans at a pecan farm. They divide the pecans equally among themselves. How many pounds does each friend get?
Options:
a. 20 \(\frac{2}{5}\) pounds
b. 8 \(\frac{2}{5}\) pounds
c. 2 \(\frac{3}{5}\) pounds
d. 2 \(\frac{2}{5}\) pounds

Answer: \(\frac{2}{5}\) pounds

Explanation:
Divide the number of pounds by the number of friends to get the number of pounds each friend gets.
14 \(\frac{2}{5}\)/6 = 14.4/6 = 2.4 pounds.
2.4 = 2 \(\frac{2}{5}\) pounds
Thus the correct answer is option D.

Question 3.
What is the value of (−3.25)(−1.56)?
Options:
a. -5.85
b. -5.07
c. 5.07
d. 5.85

Answer: 5.07

Explanation:
Multiply two negatives make a positive.
So the answer is positive.
(−3.25)(−1.56) = 5.07
The answer is option C.

Question 4.
Mrs. Rodriguez is going to use 6 \(\frac{1}{3}\) yards of material to make two dresses. The larger dress requires 3 \(\frac{2}{3}\) yards of material. How much material will Mrs. Rodriguez have left to use on the smaller dress?
Options:
a. 1 \(\frac{2}{3}\) yards
b. 2 \(\frac{1}{3}\) yards
c. 2 \(\frac{2}{3}\) yards
d. 3 \(\frac{1}{3}\) yards

Answer: 2 \(\frac{2}{3}\) yards

Explanation:
Subtract the yards of material for the larger dress from the total yards of material.
6 \(\frac{1}{3}\) yards – 3 \(\frac{2}{3}\) yards = 2 \(\frac{2}{3}\) yards
Thus the correct answer is option C.

Question 5.
Jaime had $37 in his bank account on Sunday. The table shows his account activity for the next four days. What was the balance in Jaime’s account after his deposit on Thursday?

Options:
a. $57.49
b. $59.65
c. $94.49
d. $138.93

Answer: $94.49

Explanation:
Add up all the deposits and withdrawals to his original balance and make sure deposits are represented by positive numbers and withdrawals are represented by negative numbers.
37 + 17.42 – 12.60 – 9.62 + 62.29 = 94.49
Thus the correct answer is option C.

Question 6.
A used motorcycle is on sale for $3,600. Erik makes an offer equal to \(\frac{3}{4}\) of this price. How much does Erik offer for the motorcycle?
Options:
a. $4800
b. $2700
c. $2400
d. $900

Answer: $2700

Explanation:
Given that,
A used motorcycle is on sale for $3,600. Erik makes an offer equal to \(\frac{3}{4}\) of this price.
\(\frac{3}{4}\) × 3600 = 2700
Thus the correct answer is option B.

Question 7.
Ruby ate \(\frac{1}{3}\) of a pizza, and Angie ate \(\frac{1}{5}\) of the pizza. How much of the pizza did they eat in all?
Options:
a. 1 \(\frac{1}{5}\) of the pizza
b. \(\frac{1}{8}\) of the pizza
c. \(\frac{3}{8}\) of the pizza
d. \(\frac{8}{15}\) of the pizza

Answer: \(\frac{8}{15}\) of the pizza

Explanation:
Ruby ate \(\frac{1}{3}\) of a pizza, and Angie ate \(\frac{1}{5}\) of the pizza.
\(\frac{1}{3}\) = \(\frac{1}{5}\) = \(\frac{5}{15}\) + \(\frac{3}{15}\) = \(\frac{8}{15}\)
Thus the correct answer is option D.

Unit 1 MIXED REVIEW – Page No. 110

Question 8.
Winslow buys 1.2 pounds of bananas. The bananas cost $1.29 per pound. To the nearest cent, how much does Winslow pay for the bananas?
Options:
a. $1.08
b. $1.20
c. $1.55
d. $2.49

Answer: $1.55

Explanation:
Winslow buys 1.2 pounds of bananas. The bananas cost $1.29 per pound.
1.2 × $1.29 = $1.548 ≈ $1.55
Thus the correct answer is option C.

Question 9.
The temperature was -10 °F and dropped by 16 °F. Which statement represents the resulting temperature in degrees Fahrenheit?
Options:
a. -10 – (-16) = -6
b. -10 – 16 = -26
c. 10 – (-16) = 26
d. -10 + 16 = 6

Answer: -10 – 16 = -26

Explanation:
The temperature was -10 °F and dropped by 16 °F.
-10 + (-16) = -26°F.
So, the correct answer is option B.

Question 10.
A scuba diver at a depth of -12 ft (12 ft below sea level), dives down to a coral reef that is 3.5 times the diver’s original depth. What is the diver’s new depth?
Options:
a. -420 ft
b. -42 ft
c. 42 ft
d. about 3.4 ft

Answer: -42 ft

A scuba diver at a depth of -12 ft, dives down to a coral reef that is 3.5 times the diver’s original depth.
-12 × 3.5 = -42 ft
So, the correct answer is option B.

Question 11.
The school Spirit Club spent $320.82 on food and took in 643.59 selling the food. How much did the Spirit Club make?
Options:
a. -$322.77
b. -$964.41
c. $322.77
d. $964.41

Answer: $322.77

Explanation:
The school Spirit Club spent $320.82 on food and took in 643.59 selling the food.
$643.59 – $320.82 = $322.77
So, the answer is option C.

Question 12.
Lila graphed the points -2 and 2 on a number line. What does the distance between these two points represent?
Options:
a. the sum of -2 and 2
b. the difference of 2 and -2
c. the difference of -2 and 2
d. the product of -2 and 2

Answer: the difference between 2 and -2

Explanation:
Distance is found by subtracting the larger number and the smaller number so it is the difference of 2 and -2.
Thus the correct answer is option B.

Question 13.
What is a reasonable estimate of −3 \(\frac{4}{5}\) + (−5.25) and the actual value?
Options:
a. -4 + (-5) = -9; −9 \(\frac{1}{20}\)
b. -3 + (-5) = -8; −8 \(\frac{1}{20}\)
c. -4 + (-5) = -1; −8 \(\frac{9}{20}\)
d. -3 + (-5) = 8; 8 \(\frac{1}{20}\)

Answer: -4 + (-5) = -9; −9 \(\frac{1}{20}\)

Explanation:
−3 \(\frac{4}{5}\) + (−5.25)
−3 \(\frac{4}{5}\) ≈ -4
−5.25 ≈ -5
So the sum is about -4 + -5 = -9.
The estimated answer is -9.

Mini-Task

Grade 7 Math Rational Numbers Question 14.
Juanita is watering her lawn using the water stored in her rainwater tank. The water level in the tank drops \(\frac{1}{3}\) inch every 10 minutes she waters.
a. What is the change in the tank’s water level after 1 hour?
______ inches

Answer: -2 inches

Explanation:
Juanita is watering her lawn using the water stored in her rainwater tank.
There are six 10-minute intervals in 1 hour so change is
6 × –\(\frac{1}{3}\) = -2 inches
Therefore, the tank’s water level after 1 hour is -2 inches.

Question 14.
b. What is the expected change in the tank’s water level after 2.25 hours?
______ inches

Answer: -4.5 inches

Explanation:
Since the water level drops 2 inches every hour, in 2.25 hours the water level change will be -2 × 2.25 = -4.5 inches
Thus the expected change in the tank’s water level after 2.25 hours is -4.5 inches.

Question 14.
c. If the tank’s water level is 4 feet, how many days can Juanita water if she waters for 15 minutes each day?
______ days

Answer: 96 days

Explanation:
15 minutes is 1/4 of an hour so in 15 minutes the water level will have dropped by 2 × 1/4 = 1/2 inches.
Since the water level is initially 4 feet = 48 inches
She can water for 48/1/2 = 48 × 2 = 96 days
It takes 96 days if she waters for 15 minutes.

Conclusion:

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