Texas Go Math Grade 7 Lesson 1.6 Answer Key Dividing Rational Numbers

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Texas Go Math Grade 7 Lesson 1.6 Answer Key Dividing Rational Numbers

Texas Go Math Grade 7 Lesson 1.6 Explore Activity 1 Answer Key 

A diver needs to descend to a depth of 100 feet below sea level. She wants to do it in 5 equal descents. How far should she travel in each descent?

A. To solve this problem, you can set up a division problem: \(\frac{-100}{}\) =?
B. Rewrite the division problem as a multiplication problem. Think: Some number multiplied by 5 equals -100.
_______ × ? = -100
C. Remember the rules for integer multiplication. If the product is negative, one of the factors must be negative. Since ________ ¡s positive, the unknown factor must be [Positive/negative.]
D. You know that 5 × _________ = 100. So, using the rules for integer multiplication you can say that 5 × ____ 100.
The diver should descend ________ feet in each descent.

Reflect

Question 1.
What do you notice about the quotient of two rational numbers with different signs?
Answer:
The quotient of two rational numbers with different signs will have a negative sign.

Texas Go Math Grade 7 Pdf Lesson 1.6 Answer Key Question 2.
What do you notice about the quotient of two rational numbers with the same sign? Does it matter if both signs are positive or both are negative?
Answer:
The quotient of two rational numbers with the same sign will have a positive sign. It does not matter if both signs are positive or both signs are negative.

Write two equivalent expressions for each quotient.

Question 3.
\(\frac{14}{-7}\) __________, __________
Answer:
\(\frac{-14}{7}\), – (\(\frac{14}{7}\))

Question 4.
\(\frac{-32}{-8}\) __________, ___________
Answer:
\(\frac{32}{8}\), -(\(\frac{-32}{8}\))

Your Turn

Find each quotient.

Question 5.
\(\frac{2.8}{-4}\) = ____________
Answer:
The quotient will be negative because signs are different.
Write a decimal as fraction: \(\frac{\frac{28}{10}}{-4}\)
Write complex fraction as division: \(\frac{28}{10}\) ÷ (-4)
Rewrite using multiplication:
\(\frac{28}{10} \times \frac{-1}{4}=\frac{-28}{40}\)
= \(\frac{-7}{10}\)

Question 6.
\(\frac{-\frac{5}{8}}{-\frac{6}{7}}\) = ____________
Answer:
The quotient will be positive because the signs are the same.
Write complex fractions as division:
–\(\frac{5}{8}\) ÷ (-\(\frac{6}{7}\))
Rewrite using multiplication:
–\(\frac{5}{8}\) × (-\(\frac{7}{6}\)) = \(\frac{35}{48}\)

Texas Go Math Grade 7 Pdf Dividing Rational Numbers Question 7.
– \(\frac{5.5}{0.5}\) = ___________
Answer:
The quotient will be negative because signs are different.
Write decimal numbers as fractions:
\(-\frac{\frac{55}{10}}{\frac{5}{10}}\)
Write complex fractions as division:
–\(\frac{55}{10}\) ÷ \(\frac{5}{10}\)
Rewrite using multiplication:
–\(\frac{55}{10}\) × \(\frac{10}{5}\) = -11

Texas Go Math Grade 7 Lesson 1.6 Guided Practice Answer Key 

Find each quotient. (Explore Activity 1 and 2, Example 1)

Question 1.
\(\frac{0.72}{-0.9}\) = ____________
Answer:
The quotient will be negative because signs are different.
Write decimal numbers as fraction:
\(\frac{\frac{72}{100}}{\frac{-9}{10}}\)
Write complex fraction as division:
\(\frac{72}{100}\) ÷ \(\frac{-9}{10}\)
Rewrite using multiplication:
\(\frac{72}{100}\) × \(\frac{10}{-9}\) = \(\frac{8}{-10}\)
= –\(\frac{4}{5}\)

Question 2.
\(\left(-\frac{\frac{1}{5}}{\frac{7}{5}}\right)\) = ____________
Answer:
The quotient will be negative because signs are different.
Write complex fraction as division:
–\(\frac{1}{5}\) ÷ \(\frac{7}{5}\)
Rewrite using multiplication:
–\(\frac{1}{5}\) × \(\frac{5}{7}\) = –\(\frac{1}{7}\)

Question 3.
\(\frac{56}{-7}\) = _____________
Answer:
The quotient will be negative because the signs are different.
\(\frac{56}{-7}\) = -8

Question 4.
\(\frac{251}{4} \div\left(-\frac{3}{8}\right)\) = ____________
Answer:
The quotient will be negative because the complex fraction is negative.
Rewrite using multiplication:
– \(\frac{251}{4}\) × \(\frac{8}{3}\) = –\(\frac{502}{3}\)

Texas Go Math Grade 7 Answer Key Pdf Lesson 1.6 Question 5.
\(\frac{75}{-\frac{1}{5}}\) = ____________
Answer:
The quotient will be negative because the signs are different
Write complex fractions as division:
-75 ÷ \(\frac{1}{5}\)
Rewrite using multiplication:
-75 × 5 = -375

Question 6.
\(\frac{-91}{-13}\) = ____________
Answer:
The quotient will be positive because the signs are the same.
\(\frac{-91}{-13}\) = \(\frac{91}{13}\)
= 7

Question 7.
\(\frac{-\frac{3}{7}}{\frac{9}{4}}\) = _____________
Answer:
The quotient will be negative because the signs are different.
Write complex fraction as division:
–\(\frac{3}{7}\) ÷ \(\frac{9}{4}\)
Rewrite using multiplication:
–\(\frac{3}{7}\) × \(\frac{4}{9}\) = –\(\frac{4}{21}\)

Question 8.
–\(\frac{12}{0.03}\) = ____________
Answer:
The quotient will be negative because the fraction has a negative sign.
Write decimal numbers as fraction:
–\(\frac{12}{\frac{3}{100}}\)
Write complex fraction as division:
-12 ÷ \(\frac{3}{100}\)
Rewrite using multiplication:
-12 × \(\frac{100}{3}\) = -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? (Example 1)
Answer:
Use a negative number to represent spiLLage of water
Find \(\frac{-3.5}{4}\).
The quotient will be negative because signs are different.
Write decimal numbers as fraction:
\(-\frac{\frac{35}{10}}{4}\)
Write complex fraction as division:
– \(\frac{35}{10}\) ÷ 4
Rewrite using multiplication:
–\(\frac{35}{10}\) × \(\frac{1}{4}\) = – \(\frac{35}{40}\)
= –\(\frac{7}{8}\)
The average change in water volume each day is –\(\frac{7}{8}\) liters.

Question 10.
The price of one share of ABC Company declined a total of $45.75 in 5 days. What was the average change of the price of one share per day? (Example 1)
Answer:
Use a negative number to represent decline in share price.
Find \(\frac{-45.75}{5}\)
The quotient will be negative because signs are different
Write decimal numbers as fraction:
\(-\frac{\frac{4575}{100}}{5}\)
Write complex fraction as division:
–\(\frac{4575}{100}\) ÷ 5
Rewrite using multiplication:
–\(\frac{915}{100}\) × \(\frac{1}{5}\) = –\(\frac{915}{100}\)
= –\(\frac{183}{25}\)
The average change of the price of one share per day is –\(\frac{183}{25}\)

Question 11.
To avoid a storm, a passenger jet pilot descended 0.44 mile in 0.8 minute. What was the plane’s average change of altitude per minute? (Example 1)
Answer:
Use a negative number to represent descent
Find \(\frac{-0.44}{0.8}\).
UL
The quotient will be negative because signs are different.
Write decimal numbers as fraction:
\(\frac{1}{2}\)
Write complex fraction as division:
–\(\frac{44}{100}\) ÷ \(\frac{8}{10}\)
Rewrite using multiplication:
–\(\frac{44}{100}\) × \(\frac{10}{8}\) = –\(\frac{11}{20}\)
The average change of altitude per minute is –\(\frac{11}{20}\) miles.

Essential Question Check-In

Question 12.
Explain how you would find the sign of the quotient \(\frac{32 \div(-2)}{-16 \div 4}\).
Answer:
I would first find the sign of the numerator and denominator separately, and then the sign of the whole fraction.
Numerator: Negative, because signs are different
Denominator: Negative, because signs are different.
Whole fraction: Positive, because signs are the same.

Texas Go Math Grade 7 Lesson 1.6 Independent Practice Answer Key  

Question 13.
\(\frac{5}{-\frac{2}{8}}\) = __________
Answer:
The quotient will be negative because the signs are different
Write complex fraction as division: -5 ÷ \(\frac{2}{8}\)
Rewrite using multiplication:
-5 × \(\frac{8}{2}\) = -5 × 4
= -20

7th Grade Go Math Answer Key Lesson 1.6 Question 14.
\(5 \frac{1}{3} \div\left(-1 \frac{1}{2}\right)\) = __________
Answer:
Write mixed fractions as proper fractions:
\(\frac{16}{3}\) ÷ (-\(\frac{3}{2}\))
The quotient will be negative because the complex fraction is negative
Rewrite using multiplication:
– \(\frac{16}{3}\) × \(\frac{2}{3}\) = –\(\frac{32}{9}\)

Question 15.
\(\frac{-120}{-6}\) = ___________
Answer:
The quotient will be positive because the signs are the same.
\(\frac{-120}{-6}\) = \(\frac{120}{6}\)
= 20

Question 16.
\(\frac{-\frac{4}{5}}{-\frac{2}{3}}\) = _____________
Answer:
The quotient will be positive because the signs are the same.
Write complex fraction as division:
\(\frac{4}{5}\) ÷ \(\frac{2}{3}\)
Rewrite using multiplication:
\(\frac{4}{5}\) × \(\frac{3}{2}\) = \(\frac{6}{5}\)

Question 17.
1.03 ÷ (-10.3) = _____________
Answer:
Write decimal numbers as fractions.
\(\frac{103}{100}\) ÷ (-\(\frac{103}{10}\))
The quotient will be negative because the signs are different.
Rewrite using multiplication:
–\(\frac{103}{100}\) × \(\frac{10}{103}\) = –\(\frac{1}{10}\)

Question 18.
\(\frac{-0.4}{80}\) = ____________
Answer:
The quotient will be negative because signs are different.
Write decimal numbers as fraction:
\(\frac{-\frac{4}{10}}{80}\)
Write complex fraction as division:
–\(\frac{4}{10}\) ÷ 80
Rewrite using multiplication:
–\(\frac{4}{10}\) × \(\frac{1}{80}\) = –\(\frac{1}{200}\)

Question 19.
1 ÷ \(\frac{9}{5}\) = ___________
Answer:
The quotient will be positive because the signs are the same.
Rewrite using multiplication:
1 × \(\frac{5}{9}\) = \(\frac{5}{9}\)

Question 20.
\(\frac{\frac{-1}{4}}{\frac{23}{24}}\) = _____________
Answer:
The quotient will be negative because the signs are different
Write complex fractions as division:
–\(\frac{1}{4}\) ÷ \(\frac{23}{24}\)
Rewrite using multipLication:
–\(\frac{1}{4}\) × \(\frac{24}{23}\) = –\(\frac{6}{23}\)

Lesson 1.6 Go Math 7th Grade Dividing Rational Numbers Answer Key Question 21.
\(\frac{-10.35}{-2.3}\) = ___________
Answer:
The quotient will be positive because the signs are the same.
Write decimal numbers as fractions:
\(\frac{-\frac{1035}{100}}{-\frac{23}{10}}\)
Write complex fractions as division:
\(\frac{1035}{100}\) ÷ \(\frac{23}{10}\)
Rewrite using muLtiplication:
\(\frac{1035}{100}\) × \(\frac{10}{23}\) = \(\frac{45}{10}\)
= \(\frac{9}{2}\)

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.
Answer:
If Alex runs 21 hours for a week, that means he runs \(\frac{21}{7}\) = 3 hours per day. If he is unable to run for 3 days, that means he loses 3 × 3 = 9 hours.

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.
Answer:
Use negative number to represent loss of yards
Find \(\frac{-15 \frac{3}{4}}{9} .\)
Write mixed fractions as proper fractions:
\(\frac{-\frac{63}{4}}{9}\)
The quotient will be negative because the signs are different.
Write complex fraction as division:
–\(\frac{63}{4}\) ÷ 9
Rewrite using multiplication:
–\(\frac{63}{4}\) × \(\frac{1}{9}\) = –\(\frac{7}{4}\)
Averange change in field position on each run is –\(\frac{7}{4}\) yards.

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?
Answer:
First we need to add the temperatures up.
12.1 + (- 7.8) + (- 14.3) + (- 7.2) = 19.9 + (- 14.3) + (- 7.2)
= 34.2 + (- 7.2)
= 41.4
Now, we need to divide the result with the count of temperature measurements.
Find \(\frac{-41.4}{4}\).
The quotient will be negative because signs are different.
Write decimal numbers as fraction:
\(\frac{-\frac{414}{10}}{4}\)
Write complex fraction as division:
–\(\frac{414}{10}\) ÷ 4
Rewrite using multiplication:
–\(\frac{414}{10}\) × \(\frac{1}{4}\) = –\(\frac{207}{20}\)
The average 6.00 a.m temperature for four days was –\(\frac{207}{20}\) degrees Celsius.

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?
Answer:
Find \(\frac{1750}{7}\).
\(\frac{1750}{7}\) = 250
They received $250 extra profit per day.

b. If it had, instead, lost $150 per day, how much money would it have lost for the week?
Answer:
Find -150 × 7
-150 × 7 = -1050
They would have lost $1050 for the week.

c. If its total loss was $490 for the week, what was its average daily change?
Answer:
Find \(\frac{-490}{7}\)
\(\frac{-490}{7}\) = -70
The average daily change was -$70.

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?
Texas Go Math Grade 7 Lesson 1.6 Answer Key 1
Answer:
Use a negative number to represent descent.
Find \(\frac{-99.6}{12}\)
The quotient will be negative because signs are different.
Write decimal numbers as fraction:
\(-\frac{\frac{996}{10}}{12}\)
Write complex fraction as division:
–\(\frac{996}{10}\) ÷ 12
Rewrite using multiplication:
–\(\frac{996}{10}\) × \(\frac{1}{12}\) = –\(\frac{83}{10}\)
= -8.3
The average rate of descent is 8.3 meters per second

Question 27.
Sanderson is having trouble with his assignment. His 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.
Answer:
Sanderson jumped over one step. He should have written complex fractions using division.
\(\frac{3}{4}\) ÷ \(\frac{4}{3}\)
And then rewrite it using multiplication.
\(\frac{3}{4}\) × \(\frac{3}{4}\)

Go Math Answer Key Grade 7 Lesson 1.6 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.
Answer:
First, find out how many years have passed in the period 1996-2012.
2012 – 1996 = 16
Find -3.7 × 16.
-3.7 × 16 = -59.2
The total change in thickness by the end of 2012 is -59.2 inches.

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.
Answer:
A group of 15 people lost 85 dollars. If every person lost the same amount of dollars, how many dollars have each person lost?
–\(\frac{85}{15}\) = –\(\frac{17}{3}\)
Each person lost –\(\frac{17}{3}\)

Question 30.
Construct an Argument Divided 5 by 4. Is your answer a rational number? Explain.
Answer:
Yes, it is a Quotient of dividing 5 by 4 is a fraction, and every fraction is a rational number.

Question 31.
Critical Thinking Is the quotient of an integer divided by a nonzero integer always a rational number? Explain.
Answer:
Yes, it is. A quotient of any two integers can be written as a fraction, the denominator being a nonzero integer. Thus, it is a rational. number.

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