# Into Math Grade 7 Module 5 Lesson 2 Answer Key Multiply Rational Numbers

We included HMH Into Math Grade 7 Answer Key PDF Module 5 Lesson 2 Multiply Rational Numbers to make students experts in learning maths.

## HMH Into Math Grade 7 Module 5 Lesson 2 Answer Key Multiply Rational Numbers

I Can compute the products of signed rational numbers using the properties of numbers to simplify calculations.

Two submersibles, Atlas and Pisces, are exploring an ocean trench. Every hour, Atlas makes 5 descents of 10 meters each. Every hour, Pisces makes 2 descents of 50 meters each. What integer represents the total change in elevation for each submersible after 3 hours?

The integer represents the total change in elevation for each submersible after 3 hours is 180 meters.

Explanation:
Atlas makes 5 descents of 10 meters each per hour.
For 3 hours it is 30 meters.
Pisces makes 2 descents of 50 meters each per hour.
For 3 hours it is 150 meters.

Turn and Talk What mathematical operations could you use to solve the problem? Explain.

Build Understanding

1. The product 3(-8) represents the hourly change in elevation for the submersible Argo, as shown here. The total change in elevation after 4 hours is represented by the product 4(3)(- 8).

A. First multiply the first two factors. What is the sign of the product of the first two factors, 4 and 3? Explain.
___________________
___________________
___________________
The product of 4(3) = 12
The sign of the product is positive.

Explanation:
The multiplication of the first two factors is 4 Ã— 3 = 12
The sign of the product of the first two factors, 4 and 3 is positive. Because when we multiply the positive number with the positive number the result is always positive.

B. Now multiply the result by the third factor, – 8. What is the sign of the final product? Why is this reasonable?
___________________
___________________
-96.
The sign of the final product is negative.

Explanation:
The product is 12 and the third factor is -8.
Multiply the product with the third factor 12 Ã— -8 = -96
And the sign of the final product is negative.

C. Use the above method of multiplying pairs of factors to find each product and complete the table.

D. What relationship do you see between the number of negative factors and the sign of the corresponding product?
___________________
___________________

E. How could you apply the Associative Property of Multiplication to the second expression to make it easier to simplify?
___________________
The Associative Property of Multiplication is
(A Ã— B) Ã— C = A Ã— ( B Ã— C)

Turn and Talk How do the rules you learned previously for multiplying two numbers with the same sign and two numbers with different signs support your answer to Step D?
The rules are
(+) Ã— (+) = (+)
(-) Ã— (+) = (-)
The example is for multiplying two numbers with the same sign.
+(6) Ã— +(4) = 24
+(10) Ã— +(2) = 20
The two numbers with different signs are
-(7) Ã— +(4) = -28

Step It Out

2. Find the product. Identify any properties that you used.

A. (4)(-0.5)(-3) = (___)(-3) = ___
6.
Commutative Property of multiplication

Explanation:
(4)(-0.5)(-3) = (-2)(-3) = 6
The product is 6.
Commutative Property of multiplication.

I used the order of operations and multiplied from ___ to ___. The product is ____, because the number of negative factors is (even / odd).
B.

I used the ___ Property of Multiplication to switch the order of the factors, and then I used the ____ Property of Multiplication to group factors to make the multiplication easier. The product is ____, because the number of negative factors is (even / odd).
I used the Commutative Property of Multiplication to switch the order of the factors, and then I used the Property of Multiplication to group factors to make the multiplication easier. The product is -21 because the number of negative factors is odd.

Turn and Talk Why is it helpful to apply properties of operations when multiplying rational numbers?

Check Understanding

Question 1.
Jared writes a multiplication expression with eight rational factors. Half of the factors are positive and half are negative. Is the product positive or negative? Why? _________

Question 2.
The expression (8)(2)(-1.5) represents the change in the scuba diverâ€™s elevation after 8 minutes. Find the change in elevation.
-24

Explanation:
The expression is (8)(2)(-1.5)
8 Ã— 2 Ã— -1.5
8 Ã— -3
-24
So the elevation is -24.

Find each product.

Question 3.
(-0.25)(-0.5)(4)(8) ____
4

Explanation:
Now let us solve the given expression
Given expression is (-0.25)(-0.5)(4)(8)
– $$\frac{25}{100}$$ Ã— – $$\frac{5}{10}$$ Ã— 4 Ã— 8
– $$\frac{1}{4}$$ Ã— –$$\frac{1}{2}$$ Ã— 4 Ã— 8
$$\frac{1}{8}$$ Ã— 4 Ã— 8
4

Question 4.
(10)(-3)(-$$-\frac{4}{5}$$) ($$-\frac{5}{6}$$) _____
20

Explanation:
Let us solve the given expression
(10)(-3)(-$$-\frac{4}{5}$$) ($$-\frac{5}{6}$$)
(10) Ã— (-3) Ã— (-$$-\frac{4}{5}$$) Ã— ($$-\frac{5}{6}$$)
-30 Ã— (-$$-\frac{4}{5}$$) Ã— ($$-\frac{5}{6}$$)
-30 Ã— – $$\frac{4}{6}$$
-30 Ã— – $$\frac{2}{3}$$
10 Ã— 2
20

Question 5.
Name the property illustrated and the value of the expressions.
[(3.95)(-0.2)](-5) = (3.95)[(-0.2)(-5)]
Associative property of multiplication.
The value of the expression is 3.95

Explanation:
The property illustrated in the question is the Associative property of multiplication.
[(3.95)(-0.2)](-5) = (3.95)[(-0.2)(-5)]
3.95 = 3.95

Question 6.
Every week, Estella has guitar lessons.
A. Complete the expression to represent the change in Estella’s account balance due to guitar lessons after 2 years.
___ (52)(-35)

____________________
____________________
The expression is 2 Ã— (52) Ã— (-35)

B. Find the product. Then explain what it represents.
____________________
____________________
-3640

Explanation:
The product is
2 Ã— (52) Ã— (-35)
104 Ã— (-35)
-3640
It represents a change in Estella’s account balance due to guitar lessons after 2 years.

Question 7.
DeMarcus multiplies all of the integers from – 10 to – 1, including – 10 and – 1. Should his answer be positive or negative? Explain your thinking.
____________________
____________________
____________________

Explanation:
If we multiply integers from -10 to -1 is always positive.
If we multiply one positive number with another positive number the answer will be always positive.
-10 Ã— -1 = 10
-9 Ã— -1 = 9
-9 Ã— -2 = 18

Identify the property illustrated by the statement. Find the value of the equivalent expressions.

Question 8.
(6)(-18)(5) = (-18)(6)(5)
____________________
The property used is commutative property.
Value is -540.

Explanation:
The property used for the given statement is commutative property.
The value of the equivalent expression is (6) Ã— (-18) Ã— (5) = -540.

Question 9.
[(-0.7)(-2)](5) = (-0.7)[(-2)(5)]
____________________
The property used is associative property.
(A Ã— B) Ã— C = A Ã— (B Ã— C)
The value of the expression is 7.

Explanation:
The property used here is associative property.
(A Ã— B) Ã— C = A Ã— (B Ã— C)
[((-0.7) Ã— (-2)] Ã— 5 = (-0.7) Ã— [(-2) Ã— (5)]
7 = 7

Find each product.

Question 10.
(-4)(-8)(-5)
____________________
-160

Explanation:
Let us solve the given expression
(-4)(-8)(-5)
-4 Ã— -8 Ã— -5
-160

Question 11.
(-4)(-2)(5)(3)
____________________
120

Explanation:
Let us solve the given expression
(-4)(-2)(5)(3)
-4 Ã— -2 Ã— 5 Ã— 3
8 Ã— 15
120

Question 12.
(-1 )(9)(-2)(5)(-1)(-1)
____________________
90

Explanation:
Let us solve the given expression
(-1 ) Ã— (9) Ã— (-2) Ã— (5) Ã— (-1) Ã— (-1)
-9 Ã— -10 Ã— 1
90

Question 13.
(0.2)(50)(-0.9)
____________________
-9

Explanation:
Let us solve the given expression
(0.2) Ã— (50) Ã— (-0.9)
$$\frac{2}{10}$$ Ã— 50 Ã— – $$\frac{9}{10}$$
2 Ã— 5 Ã— – $$\frac{9}{10}$$
10 Ã— – $$\frac{9}{10}$$
Cancel 10 from numerator and denominator
-9

Question 14.
(0.1)(-0.2)(10)(- 10)
____________________
-2

Explanation:
Let us solve the given expression
(0.1)(-0.2)(10)(- 10)
-2

Question 15.
(-28)(-0.5)(-0.5)(-0.1)(10)
____________________
7

Explanation:
Let us solve the given expression
(-28) Ã— (-0.5) Ã— (-0.5) Ã— (-0.1) Ã— (10)
7

Question 16.
(-$$\frac{1}{3}$$)($$\frac{3}{5}$$)(-$$\frac{5}{7}$$)
0.14 or $$\frac{1}{7}$$

Explanation:
Let us solve the given expression
–$$\frac{1}{3}$$) Ã— ($$\frac{3}{5}$$) Ã— (-$$\frac{5}{7}$$
–$$\frac{1}{5}$$ Ã— –$$\frac{5}{7}$$
$$\frac{1}{7}$$
0.14

Question 17.
($$\frac{2}{7}$$)($$\frac{14}{15}$$)(-$$\frac{1}{2}$$)
– 0.133

Explanation:
Let us solve the given expression
($$\frac{2}{7}$$)($$\frac{14}{15}$$)(-$$\frac{1}{2}$$)
cancel all the common factors
$$\frac{4}{15}$$ Ã— –$$\frac{1}{2}$$
–$$\frac{2}{15}$$
-0.133

Question 18.
(-12)($$\frac{5}{6}$$)(-$$\frac{3}{4}$$)(-$$\frac{4}{5}$$)
-6

Explanation:
Let us solve the given expression
(-12)($$\frac{5}{6}$$)(-$$\frac{3}{4}$$)(-$$\frac{4}{5}$$)
-12 Ã— $$\frac{3}{6}$$
-12 Ã— $$\frac{1}{2}$$
-6

I’m in a Learning Mindset!

What is challenging about multiplying rational numbers? Do I need help, or can I work through it on my own?

Lesson 5.2 More Practice/Homework

Question 1.
As part of an experiment, scientists decrease the temperature in a laboratory freezer as shown.

A. Complete the expression to represent the change in the freezerâ€™s temperature at the end of 2 days.
2(24)(____) (-0.5)
2(24)(3) (-0.5)

Explanation:
The expression represents the change in the freezerâ€™s temperature at the end of 2 days is 2 (24) (3) (-0.5).

B. Find the product. Then explain what it represents.
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___________________
The product of 2(24)(3)(-0.5) is -72.

Explanation:
The product that represents the change in the freezerâ€™s temperature at the end of 2 days is -72.

___________________
___________________
___________________
By calculating the product given for the expression we would know the answer is reasonable.

Question 2.
Use Structure Complete the following to show how the expression can be simplified using the Commutative and Associative Properties of Multiplication.

7

Find each product.

Question 3.
(3)(5)(-3)(-2)(10)(-1)
-900

Explanation:
Let us solve the given expression
(3)(5)(-3)(-2)(10)(-1)
15 Ã— -3 Ã— -2 Ã— 10 Ã— -1
-45 Ã— -20 Ã— -1
-900

Question 4.
(-$$\left(\frac{3}{4}\right)$$)($$\left(\frac{1}{3}\right)$$)(-$$\left(\frac{4}{5}\right)$$)
0.2

Explanation:
Let us solve the given expression
(-$$\left(\frac{3}{4}\right)$$)($$\left(\frac{1}{3}\right)$$)(-$$\left(\frac{4}{5}\right)$$)
Cancel all the common multiples
0.2

Question 5.
($$\frac{2}{3}$$)(-$$\frac{9}{8}$$)(-$$\frac{4}{5}$$)(-1)
15

Explanation:
Let us solve the given expression
($$\frac{2}{3}$$)(-$$\frac{9}{8}$$)(-$$\frac{4}{5}$$)(-1)
Now cancel all the common multiples
15

Question 6.
(-9)($$\frac{7}{12}$$)(-$$\frac{4}{7}$$)(-$$\frac{1}{3}$$)
1

Explanation:
Let us solve the given expression
(-9)($$\frac{7}{12}$$) Ã— (-$$\frac{4}{7}$$) Ã— (-$$\frac{1}{3}$$)
Cancel the common multiples

Question 7.
(0.5)(-0.5)(-4)(-4)(-1)
4

Explanation:
Let us solve the given expression
(0.5)(-0.5)(-4)(-4)(-1)
$$\frac{5}{10}$$ Ã— –$$\frac{5}{10}$$) Ã— -4 Ã— -4 Ã— -1
4

Question 8.
(-0.01)(-50)(-0.5)(-0.2)(-10)
0.5

Explanation:
Let us solve the given expression
(-0.01)(-50)(-0.5)(-0.2)(-10)
– $$\frac{1}{100}$$ Ã— -50 Ã— –$$\frac{5}{10}$$ Ã— –$$\frac{2}{10}$$ Ã— -10
0.5

Test Prep

Question 9.
Which product has a negative value?
A. (-$$\frac{1}{3}$$) (-$$\frac{3}{4}$$)($$\frac{4}{5}$$)
B. (-$$\frac{1}{2}$$) (-$$\frac{2}{3}$$)(-$$\frac{3}{4}$$)
C. (-$$\frac{2}{3}$$) (-$$\frac{3}{4}$$)(-$$\frac{4}{7}$$)(-$$\frac{7}{8}$$)
D. (-$$\frac{1}{2}$$) ($$\frac{2}{5}$$)($$\frac{5}{6}$$)(-$$\frac{6}{7}$$)
B. (-$$\frac{1}{2}$$) (-$$\frac{2}{3}$$)(-$$\frac{3}{4}$$)

Explanation:
The product that has a negative value is (-$$\frac{1}{2}$$) (-$$\frac{2}{3}$$)(-$$\frac{3}{4}$$).

Question 10.
Tyrell is exploring a cave. Every hour, he makes 4 vertical descents of 3.5 meters. The expression 3(4)(- 3.5) represents his change in elevation in meters after 3 hours. What is his change in elevation?
A. -42 meters
B. -4.2 meters
C. 4.2 meters
D. 42 meters
-4.2 meters

Explanation:
Every hour, he makes 4 vertical descents of 3.5 meters
3 hours = 12 vertical
The expression 3(4)(- 3.5) = 12 Ã— -3.5
= -4.2 meters

Question 11.
Match the product to its value.

Spiral Review

Question 12.
Callie bought 4 pies from a bakery for a holiday dinner. The total cost was $75.80. If each pie cost the same, how much did one pie cost? ___________________ Answer: 18.95 18$$\frac{19}{20}$$ Explanation: Based on the given question find 75.8 Ã· 4 Convert the decimal to fraction (758 Ã· 10) Ã· 4 (758 Ã· 10) Ã— (1 Ã· 4) (379 Ã· 5) Ã— (1 Ã· 4) 379 Ã·(5 Ã— 4) 379 Ã· 20 18.95 or 18$$\frac{19}{20}$$ Question 13. A graphic designer charges a fee of$1,600 to design a poster for a jazz festival. The designer also receives an 8% commission on sales of the poster at the festival. What is the designer’s total income from this work, assuming that sales of the poster bring in \$620 during the festival?
1649.6

Explanation:
Based on the given question, find
1600 + (620 Ã— 8%)
1600 + 49.6
Calculate the sum
1649.6

Question 14.
The temperature at 6:00 a.m. on a winter day is – 6 Â°F. The temperature rises by 7 Â°F by noon. Use the number line to represent the situation. Then complete the equation.

– 6 Â°F + 7 Â°F = ___Â°F