Lesson 6
Multiply and Divide Rational Numbers
Name: Multiply Positive and Negative Integers
Study the example problem showing how to use repeated
addition to multiply a positive and a negative integer. Then
solve problems 1–6.
Example
Dues for the art club are $2 per week. Noriko pays $2 each
week from her bank account for her art club dues. At the end
of 5 weeks, what is the change in the amount of money in
Noriko’s account?
You can think of this as 5 groups of (22).
5 groups of (22) 5 (22) 1 (22) 1 (22) 1 (22) 1 (22).
You can start at 0 and make 5 jumps of 22.
22
212 211 210 29 28
22
22
22
27
26 25
24 23
22
22
21
0
(22) 1 (22) 1 (22) 1 (22) 1 (22) 5 210
So, Noriko’s account changes by 2$10.
1 Complete the table and describe the pattern
in the products.
2 Rewrite the repeated addition expression as a
multiplication expression and find the product.
a.(22) 1 (22) 1 (22) 1 (22) 1 (22) 5
• (22) 5
b.(23) 1 (23) 1 (23) 1 (23) 1 (23) 1 (23) 5
Multiplication
Product
22 ? 3
26
22 ? 2
24
22 ? 1
22
12
12
22 ? 0
22 ? (21)
22 ? (22)
22 ? (23)
c.(25) 1 (25) 1 (25) 5
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Lesson 6 Multiply and Divide Rational Numbers
49
Solve.
3 Write and simplify a multiplication expression for each
problem. Explain what the product means.
a.The temperature dropped 8 degrees every hour.
What was the change in temperature after 4 hours?
b.Mae loses 3 points for each wrong answer on a test.
How does her score change if she has 7 wrong answers?
4 Carlo is playing a game with the cards shown. He draws
cards at random and multiplies the numbers.
22
0
23
24
8
a.Give an example of two cards Carlo could draw that
have a positive product. Find the product.
b.Give an example of two cards Carlo could draw that
have a negative product. Find the product.
5 Each month Marla pays a $30 cell phone bill from
her back account. At the end of one year, what is the
change in the amount of money in Marla’s account
because of her cell phone bill? Write a multiplication
equation to represent the problem. Explain what the
product represents.
6 How many integers must be negative for the product of
three integers to be negative? Explain. Find two different
groups of three integers whose product is 212.
Show your work.
Solution:
50
Lesson 6 Multiply and Divide Rational Numbers
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Lesson 6
Name: Multiply Rational Numbers
Study the example problem showing how to multiply
rational numbers. Then solve problems 1–8.
Example
The temperature of a liquid in an experiment is 0°C at 9 am.
The experiment calls for the temperature to change 2​ 5 ​ °C
4
··
every hour. What will the temperature be at noon?
The model shows that the temperature will be 2​ 15  ​°C, or
4
··
23​  3 ​ °C, at noon.
4
··
2 54
26
25
24
23
2 54
22
2 54
21
0
You can also use multiplication to solve:
3 ? 1​ 2​ 5 ​  2​ 5 2​ 15  ​ 5 23​ 3 ​
4
··
4
··
4
··
1 If the liquid continues to cool at the same rate, what will
the temperature be at 3 pm? Explain.
2 If the experiment called for the temperature of the liquid
to change by 2​ 3 ​°C each hour, what would the
4
··
temperature be at noon? Explain.
3 In a different experiment, a liquid must be cooled
6 times as fast as the liquid in the example, but it must
still start at 0°C.
a.What will the change in temperature be each hour?
Write and evaluate an expression to explain.
b.What will the temperature be after 4 hours? Write
and evaluate an expression to explain.
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Lesson 6 Multiply and Divide Rational Numbers
51
Solve.
4 Evaluate the products.
a.4 3 1​ 2​ 3 ​2​5
8
··
b.(24) 3 1​ 2​ 3 ​2​5
8
··
5 How can you use the rules for multiplying positive and
negative numbers to check that your answers to
problem 4 have the correct sign?
6 A deep sea diver changes his position relative to the
surface of the water by 2​ 7 ​ meters per minute. Write and
2
··
evaluate an expression to show his position after
3 minutes if he starts at the surface of the water.
7 Evaluate the products. Write your answers in simplest form.
a. 2​ 4 ​ 3 1​ 2​ 5 ​  2​ 5
5
··
8
··
b. ​
1 2​  3  ​   2​ 3 1​ ​ 2 ​  2​ 5
10
··
7
··
8 The height of a 9-inch candle changes at a rate of
2​ 2 ​ inch per hour when it is burning. How long will it
3
··
take the candle to burn completely?
Show your work.
Solution:
52
Lesson 6 Multiply and Divide Rational Numbers
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Lesson 6
Name: Divide Rational Numbers
Study the example problem showing how to divide rational
numbers. Then solve problems 1–6.
Example
The position of a submersible water vehicle,
relative to sea level, changes by 20.5 mile
each hour. After how many hours will the
vehicle’s position have changed by
22.5 miles?
Time (h)
Position
0
0
1
20.5
2
21.0
You can use a table to understand the
problem.
3
21.5
4
22.0
You can also use division to understand the
problem.
5
22.5
20.5
20.5
20.5
20.5
20.5
(22.5) 4 (20.5) 5 225 4 (25) 5 5
1 You can also use repeated addition to evaluate the
quotient (22.5) 4 (20.5). Complete.
22.5 5 (20.5) 1
There are
It will take
hours for the submersible’s position
to change by 22.5 miles.
groups of 20.5 in
.
2 Mariella’s bank statement shows a change of 2$2.50 in
her account each week.
a.Complete the table to see how many weeks it will
be before the change in Mariella’s account is 2$20.
Number of Weeks
Change in Account ($)
1
2
3
4
5
6
7
8
22.5 25.0 27.5
b.Why are the amounts in the table negative?
c.How many weeks will it take for the the amount
deducted to total 2$20?
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Lesson 6 Multiply and Divide Rational Numbers
53
Solve.
3 A pitcher contains 28 fluid ounces of juice. How many
4-ounce servings can you pour before the pitcher is
empty? Complete the repeated subtraction to solve.
228 5 (24) 1
I can pour
four-ounce glasses of juice.
4 A small inflatable pool holds 10 gallons of water. A tiny
leak causes the amount of water in the pool to change
by 20.05 gallon each hour until the pool is empty.
a. Write the numbers that represent each quantity.
change in amount of water per hour 5
total change in amount of water 5
b.Will you multiply or divide to find how many hours it
will take for the pool to be empty? Will the answer
be positive or negative?
c.How many hours will it take for the pool to be
empty?
hours
5 The temperature at 6 pm was 0°F. At 10 pm the
temperature was 211.2°F. Write an expression that you
can use to find the average change in temperature per
hour during that time. Then evaluate the expression.
6 Marlene is solving a number puzzle involving three
rational numbers a, b, and c. Here are her clues:
54
• The quotient of a and c is 8.
• The product of a and c is 2.
• The quotient of a and b is 216.
• The product of a and b is 21.
• The quotient of b and c is ­20.5.
What are the three numbers in Marlene’s puzzle?
Lesson 6 Multiply and Divide Rational Numbers
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Lesson 6
Name: Multiply and Divide Rational Numbers
Solve the problems.
1 Imamu pays $30 each month for his gym membership.
How many months
are in a year?
What is the change in the amount of money in his
account after ​ 2 ​of a year?
3
··
A 2$120
C 2$270
B 2$240
D 2$360
ob chose A as the correct answer. How did he get that
B
answer?
2 Trevon, Leah, and Beth are playing a computer game.
Trevon scored 28. Beth’s score was ​ 3 ​of Trevon’s score,
4
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1
and Leah’s score was ​   ​of Beth’s score. What was Leah’s
4
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Will Leah’s score be
positive or negative?
score?
A 2​  2 ​
C ​ 2 ​
2​ 3 ​
2
··
D ​ 3 ​
3
··
B
3
··
2
··
3 Paula cut ribbon from a spool for a craft project. Each
piece of ribbon changed the length of the ribbon
remaining on the spool by 21.2 feet. When Paula
finished, the length of ribbon on the spool had changed
Can writing both
numbers as either
fractions or decimals
help you?
by 29​  3 ​feet. How many pieces of ribbon did Paula cut?
5
··
Show your work.
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Lesson 6 Multiply and Divide Rational Numbers
55
Solve.
4 Craig is making 3 recipes for his party. He has a container
of flour on his kitchen scale. The table shows the total
change in weight of the flour on the scale after each of
the 3 recipes. What is the average change in the amount
of flour, in ounces, used in the 3 recipes?
Flour Used
After Recipe 1
Change in Weight (oz)
23 ​  1 ​
2
··
24 ​  1 ​
2
··
C 23​ 1 ​
2
··
3
··
1
D 27​   ​
2
··
22 ​  1 ​ 
2
··
B
After Recipe 2 After Recipe 3
22
A 2​ 3 ​
5 On the number line, a and b are rational numbers. Tell
How can you tell
whether a product or
quotient is positive or
negative?
whether each statement is True or False.
b
How can you find an
average?
a
0
a. a ? b is negative.
b. a 4 b is greater than 0.
c. b 4 a is greater than a 4 b.
d. a ? b 5 b ? a
u True
u True
u True
u True
u False
u False
u False
u False
6 Martin chose two of the cards below. When he found the
quotient of the numbers, his answer was 2​  16  ​. Write the
9
··
Did Martin divide
2 negative numbers?
division problem that Martin solved.
2
3
9
24
23
8
25 1
2
Show your work.
Solution: 56
Lesson 6 Multiply and Divide Rational Numbers
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