We included **H****MH 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.

**Spark Your Learning**

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?

Answer:

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.

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Answer:

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?

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Answer:

-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.

Answer:

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

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E. How could you apply the Associative Property of Multiplication to the second expression to make it easier to simplify?

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Answer:

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?

Answer:

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) = ___

Answer:

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.

Answer:

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? _________

Answer:

The answer is positive.

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.

Answer:

-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) ____

Answer:

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)(-\(-\frac{4}{5}\)) (\(-\frac{5}{6}\)) _____

Answer:

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)]

Answer:

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

**On Your Own**

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)

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Answer:

The expression is 2 Ã— (52) Ã— (-35)

B. Find the product. Then explain what it represents.

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Answer:

-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.

C. **Construct Arguments** Explain how you know your answer is reasonable.

Answer:

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.

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Answer:

The answer is positive.

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)

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Answer:

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)]

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Answer:

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)

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Answer:

-160

Explanation:

Let us solve the given expression

(-4)(-8)(-5)

-4 Ã— -8 Ã— -5

-160

Question 11.

(-4)(-2)(5)(3)

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Answer:

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)

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Answer:

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)

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Answer:

-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)

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Answer:

-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)

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Answer:

7

Explanation:

Let us solve the given expression

(-28) Ã— (-0.5) Ã— (-0.5) Ã— (-0.1) Ã— (10)

7

Question 16.

(-\(\frac{1}{3}\))(\(\frac{3}{5}\))(-\(\frac{5}{7}\))

Answer:

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.

(\(\frac{2}{7}\))(\(\frac{14}{15}\))(-\(\frac{1}{2}\))

Answer:

– 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)(\(\frac{5}{6}\))(-\(\frac{3}{4}\))(-\(\frac{4}{5}\))

Answer:

-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)

Answer:

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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Answer:

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.

C. Explain how you know your answer is reasonable.

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Answer:

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.

Answer:

7

**Find each product.**

Question 3.

(3)(5)(-3)(-2)(10)(-1)

Answer:

-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.

(-\(\left(\frac{3}{4}\right)\))(\(\left(\frac{1}{3}\right)\))(-\(\left(\frac{4}{5}\right)\))

Answer:

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)

Answer:

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)(\(\frac{7}{12}\))(-\(\frac{4}{7}\))(-\(\frac{1}{3}\))

Answer:

1

Explanation:

Let us solve the given expression

(-9)(\(\frac{7}{12}\)) Ã— (-\(\frac{4}{7}\)) Ã— (-\(\frac{1}{3}\))

Cancel the common multiples

The answer is 1.

Question 7.

(0.5)(-0.5)(-4)(-4)(-1)

Answer:

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)

Answer:

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. (-\(\frac{1}{2}\)) (\(\frac{2}{5}\))(\(\frac{5}{6}\))(-\(\frac{6}{7}\))

Answer:

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

Answer:

-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.

Answer:

**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?

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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?

Answer:

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

Answer:

– 6 Â°F + 7 Â°F = 1 Â°F

1 Â°F

Explanation: