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18    Topic 8

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8-1Multiplying a Fraction and a Whole NumberFind 12 3 1 _ 4 . Find 3 _ 5 of 15, or 3 _ 5 3 15.

12 3 1 _ 4 is the same as dividing 15 4 5 5 3, so 1 _ 5 3 15 5 3

12 by 4. Since 3 _ 5 is 3 times 1 _ 5 ,

12 4 4 5 3 3 _ 5 3 15 5 3 3 _ 1 _ 5 3 15+ 5 3 3 3 5 9.

12 3 1 _ 4 5 3 3 _ 5 3 15 5 9

Find each product.

1. 4 _ 5 3 20 5 2. 6 _ 7 of 14 5 3. 24 3 3 _ 4 5

4. 2 _ 5 of 15 5 5. 400 3 3 _ 8 5 6. 7 __ 10 of 80 5

7. Reasoning Can you use division and mental math to find 2 _ 3 of 24? Why or why not?

The chart shows the average high temperatures for different months in Phoenix, Arizona.

Phoenix WeatherMonth Average High

February 70°F

May 93°F

July 105°F

8. What is 4 _ 5 the average temperature in July?

9. What is 1 _ 2 the average temperature in February?

10. What is 2 _ 3 the average temperature in May?

16 12 18

6 150 56

Yes, because 3 is the greatest common factor of 3 and 24. 2 _ 3 of 24 5 16

84°F

35°F

62°F

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Topic 8 19

Practice 8-1

Practice

8-1Multiplying a Fraction and a Whole Number

Find each product.

1. 3 _ 4 � 16 � 2. 5 _

6 � 30 � 3. 42 � 5 _

6 �

4. 1 _ 8 of 72 � 5. 900 � 2 _

3 � 6. 13

__ 20

of 100 �

7. Reasoning Without multiplying, tell which is greater, 5 _ 6 of 81 or 9 __

10 of 81. Explain.

Driving DistancesDeparture City Destination City Distance

Pittsfield, Massachusetts Providence, Rhode Island 132 mi

Reno, Nevada Wendover, Utah 400 mi

8. Mike drove 1 _ 3 of the distance between

Pittsfield, Massachusetts, and Providence, Rhode Island. How far did he drive?

9. Bimal drove 3 _ 5 of the distance between

Reno, Nevada, and Wendover, Utah. How far did he drive?

10. Estimation How many more miles does Bimal have to drive to get to Wendover, Utah?

11. There are 25 students in Mr. Fitch’s sixth-grade class. If 3 _ 5 of the

students are girls, how many girls are in Mr. Fitch’s class?

A 5 girls B 10 girls C 15 girls D 20 girls

12. Writing to Explain Explain how you would find the product of 36 and 2 _ 3 .

12 25 35

9 600 65

9 __ 10 of 81 because 9 __ 10 is greater than 5 _ 6 .

44 mi

240 mi

160 mi

Write the problem as 36 � 2 _ 3 .36 � 3 � 12, so 1 _ 3 � 36 � 12. 2 _ 3 � 36 � 2 � ( 1 _ 3 � 36) � 2 � 12 � 24.

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24 Topic 8

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8-2Estimating ProductsWhen you are working with fractions and mixed numbers, you can estimate using rounding, compatible numbers, or compatible benchmark fractions.

Estimate 3 __ 10

� 21 using a whole number that is compatible with the denominator.

Estimate 3 __ 10

� 12 using a compatible benchmark fraction.

3 __ 10

� 21

3 __ 10

� 20 � 6

3 __ 10

� 21 ≈ 6

Change 21 to the nearest whole number that is compatible with 10.

Think: 20 ÷ 10 � 2.

3 � 2 � 6.

3 __ 10

� 12

1 _ 4 � 12 � 3

3 __ 10

� 12 ≈ 3

Round 3 __ 10

to a compatiblebenchmark fraction. Since 3 __

10 is

close to 1 _ 4 and 4 is a factor of

twelve, use 1 _ 4 .

Think: 12 � 4 � 3.

1 � 3 � 3.

Estimate each product by using compatible numbers or benchmark fractions.

1. 1 _ 6 � 19 � 2. 4 _

7 � 10 � 3. 5 _

8 � 23 �

4. 31 � 2 _ 5 � 5. 7 __

12 � 18 � 6. 9 __

16 � 90 �

7. 43 � 2 _ 7 � 8. 35 � 5 __

12 � 9. 16 � 4 _

9 �

Estimate each product by rounding each factor to the nearest whole number.

10. 6 2 _ 3 � 5 1 _

8 ‡ Round 6 2 _

3 : Round 5 1 _

8 : Multiply:

11. 10 2 _ 9 � 4 5 _

6 � 12. 2 7 _

8 � 3 3 _

4 = 13. 4 1 _

5 � 2 4 __

10 �

14. Reasonableness Carlotta estimated that 3 _ 7 � 20 is about 3 _

7 � 14 � 6. Is her estimate

reasonable? Why or why not?

15. Critical Thinking Why are the estimates of 6 __ 10

� 18 shown below different? Is one estimate better than the other?

6 __

10 � 18 ≈ 6 __

10 × 20 � 12 6 __

10 × 18 ≈ 1 _

2 � 18 � 9.

‚ ‚ ‚ ‚

3 5 15 12 9 45 12 15 8

7 5 35 50 12 8

No, a better choice for a number compatible to 20 is 21: 3 __ 7 � 21 � 9.

Sample answer: The first estimate uses a compatible whole number to estimate and the second uses a compatible benchmark fraction. Both are reasonable ways to estimate, and in this case, one method is not better than the other.

Sample answers are shown.

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Topic 8 25

Practice 8-2

Practice

8-2Estimating Products Estimate each product.

1. 4 5 _ 8 � 1 _

3 � 2. 3 � 2 1 _

5 � 3. 6 __

10 � 5 3 _

4 �

4. 2 7 _ 9 � 4 2 _

5 � 5. 6 1 _

2 � 2 1 _

3 � 6. 7 _

8 � 2 3 _

8 �

7. 38 � 3 _ 8 � 8. 1 _

4 � 17 � 9. 3 _

5 � 51 �

10. 7 4 _ 9 � 5 6 _

7 � 11. 12 __

25 � 8 � 12. 11 � 1 _

2 �

13. 8 _ 9 � 6 4 __

10 � 14. 7 1 __

7 � 2 2 _

3 � 15. 5 __

12 � 13 �

16. Show three ways to estimate 3 _ 5 � 5 3 _

4 . Identify each method you use.

17. Explain It Mr. Simpson lives 11 3 __ 10

miles from his office. He estimates that he commutes 11 � 2 � 5, or 110 miles each week. Is his estimate an overestimate or an underestimate? Explain.

18. Which benchmark fraction could you use to estimate the product of 38 � 7 __ 12

?_______

19. Geometry Which is the best estimate for the area of a square with sides equal to 3 1 _ 5

inches?

A 3 sq in.

B 6 sq in.

C 9 sq in.

D 16 sq in.

20. Joyce and Marianne have money jars. Joyce has 54 dimes in her jar. Marianne has 9 __ 10

as many dimes as Joyce. Estimate the number of dimes that Marianne has in her jar.

A 60 dimes

B 45 dimes

C 6 dimes

D 5 dimes

Sample answers are shown. 2 6 3 12 14 2 15 4 30 42 4 6 6 21 5

Sample answer: Compatible numbers: 3 _ 5 � 5 � 3; Rounding: 1 � 6 � 6; Compatible benchmark: 1 _ 2 � 6 � 3.

Sample answer: 110 is an underestimate because Mr. Simpson rounded the number of miles for each trip to the lower whole number before he multiplied. 1 _ 2

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30 Topic 8

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8-3Multiplying FractionsFind 3 _

4 � 2 _

5 .

Use thedenominators to determine the number of squares: 5 tall and 4 wide.

Draw a picture. Shade the squares. There are 20 squares in all. 6 squares have overlapping shading. 3 _

4 � 2 _

5 � 6 __

20 .

Simplify: 6 __ 20

� 3 __ 10

Multiply the numerators and the denominators. Simplify if possible.

3 _ 4 � 2 _

5 � (3 � 2)

_____ (4 � 5)

� 6 __ 20

� 3 __ 10

Simplify first. Divide a numerator and a denominator by their GCF. Then multiply.

3 _ 4 � 2 _

5 � 3

_____ (4 � 2)

� (2 � 2) _____

5

� 3 _ 2 � 1 _

5

� 3 __ 10

Draw a picture to solve.

1. 1 _ 3 � 4 _

5 = 2. 3 _

4 � 3 _

9 = 3. 1 _

2 � 7 _

8 =

Write an equation for each picture.

4. 5.

Find each product. Simplify if possible.

6. 5 _ 7 � 3 __

10 = 7. 1 _

2 � 6 __

15 = 8. 4 _

7 � 1 _

2 =

9. 5 _ 6 � 3 _

8 = 10. 6 _

7 � 5 __

12 = 11. 8 � 3 _

4 =

12. Number Sense Can you simplify before multiplying 14 � 4 _ 7 ? Explain.

4 __ 15 1 _ 4 7 __ 16

1 _ 3 � 1 _ 2 � 1 _ 6 3 _ 4 � 2 _ 3 � 1 _ 2

3 __ 14 1 _ 5 2 _ 7 5 __ 16 5 __ 14 6

Yes. Rewrite 14 as a fraction with 1 in the denominator. Then simplify the 14 and the 7 using their GCF of 7: 14 __ 1 � 4 _ 7 � 2 _ 1 � 4 _ 1 � 8 _ 1 � 8.

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Topic 8 31

Practice

8-3Practice 8

-3

Multiplying Fractions Write an equation for each picture.

1. 2.

Find each product. Simplify if possible.

3. 7 __ 10 3 13 __ 14 5 4. 4 _ 5 3 7 _ 8 5 5. 3 _ 7 3 4 _ 9 5

6. 3 _ 4 3 16 5 7. 2 _ 5 3 3 __ 10 5 8. 5 _ 6 3 42 5

9. 3 _ 5 3 17 __ 21 5 10. 1 _ 8 3 72 5 11. 15 __ 9 3 24 __ 25 5

12. 13 __ 20 3 100 5 13. 3 _ 8 3 4 _ 9 5 14. 1 __ 2 3 13 __ 16 5

Pamela spent 2 _ 3 of an hour doing homework. She solved math problems for 2 _ 5 of that time and read her science book for 3 _ 5 of that time. What fraction of one hour did Pamela spend:

15. solving math problems? 16. reading her science book?

17. Of the students in Mr. Moore’s room, 7 __ 13 live within a mile of school. Of those students, 4 _ 7 live within half a mile of school. What fraction of all students in Mr. Moore’s class live within half a mile of school?

A 3 __ 13

B 4 __ 13

C 3 __ 15

D 4 __ 15

18. Writing to Explain Without multiplying, tell which is greater: 55 __ 6 3 81 or 9 __ 10 3 81. Explain.

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3 _ 8 3 4 _ 7 5 12 __ 56 5 3 __ 14 2 _ 9 3 1 _ 3 5 2 __ 27

13 __ 20 7 __ 10 4 __ 21

12 3 __ 25 35 17 __ 35 9 8 _ 5 5 1 3 _ 5 65 1 _ 6 13 __ 32

4 __ 15 2 _ 5

55 __ 6 3 81 is greater because 55 __ 6 is more than 9

and 9 __ 10 is less than 1. Eighty-one times 9 is

greater than 81 3 1.

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36 Topic 8

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8-4Multiplying Mixed Numbers

How to find the product of two mixed numbers: Find 3 2 _ 3 � 4 1 _

2 .

Step 1 Estimate the product by rounding.

Step 2 Write each mixed number as an improper fraction.

Look for common factors and simplify.

Step 3 Multiply the numerators and denominators.

Write the product as a mixed number.

Round 3 2 _ 3 to 4 and 4 1 _

2 to 5:

4 � 5 � 20

3 2 _ 3 � 11 __

3 and 4 1 _

2 � 9 _

2

3 2 _ 3 � 4 1 _

2 � 11 __

3 × 9 _

2 � 11 __

1 × 3 _

2

11 __ 1 � 3 _

2 � 33

__ 2

33 __

2 � 16 1 _

2

16 1 _ 2 is close to 20, so the answer is reasonable.

Find each product. Simplify if possible.

1. 2 3 _ 4 � 3 1 _

2 2. 2 1 _

5 � 2 2 _

3 3. 6 � 3 1 _

4

4. 1 2 _ 5 � 3 1 _

4 5. 4 1 _

2 � 16 6. 1 3 _

8 � 2 1 _

2

Evaluate each expression for K � 2 1 _ 3 .

7. 12K 8. 1 3 _ 4 K 9. 2 2 _

3 K

10. Reasonableness What is a reasonable estimate for 7 3 _ 4 � 2 2 _

3 ?

Explain.

11. The cups of mushrooms in a recipe is 2 1 _ 2 times the cups of onions.

The cups of onions is 1 1 _ 2 . Solve c � 1 1 _

2 � 2 1 _

2 to find c, the cups of

mushrooms.

1

3

//

9 5 _ 8 5 13 __ 15 19 1 _ 2 4 11 __ 20 72 3 7 __ 16

28 4 1 __ 12 6 2 _ 9

24, since 7 3 _ 4 rounds to 8 and 2 2 _ 3 rounds to 3 and 8 � 3 � 24.

3 3 _ 4 c of mushrooms

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Topic 8 37

Practice 8-4

Practice

8-4Multiplying Mixed NumbersFind each product. Simplify if possible.

1. 3 1 _ 2 � 1 2 _

3 2. 1 1 _

8 � 2 1 _

3 3. 7 � 1 1 _

4

4. 2 1 _ 6 � 1 1 _

5 5. 3 1 _

6 � 18 6. 1 1 _

8 � 2 1 _

2

7. 1 2 _ 3 � 2 1 _

4 8. 10 � 1 1 _

3 9. 2 4 _

5 � 3 1 _

3

Evaluate each expression for S � 1 4 _ 5 .

10. 2 1 _ 3 S 11. 3 3 _

4 S 12. 5 1 _

6 S

Use the table to answer the questions.

13. If Berkeley receives 11 _ 4 times its average January

rainfall, how much rain will it receive?

14. How much rain will Berkeley receive if it is 2 1 _ 3 times

the October average?

15. Which month has about twice the rainfall as April?

16. Jessie stacked photographs of 6 zoo animals on top of each other to create a display. Each photo is 14 1 _

4 in. high. How high is

the display?

A 84 2 _ 3 in.

B 85 1 _ 2 in.

C 86 3 _ 4 in.

D 87 in.

17. Writing to Explain Explain how you would find 2 � 2 1 _ 3 using the

Distributive Property.

Average Rainfall in Berkeley, California

January 3 7 __ 10

in.

April 1 4 _ 5 in.

October 1 1 _ 2 in.

5 5 _ 6 2 5 _ 8 8 3 _ 4 2 3 _ 5 57 2 13 __

16

3 3 _ 4 13 1 _ 3 9 1 _ 3

4 1 _ 5 6 3 _ 4 9 3 __ 10

4 5 _ 8 in.

3 1 _ 2 in. January

Multiply 2 � 2 and 2 � 1 _ 3 , and then add the products to get 4 � 2 _ 3 or 4 2 _ 3 .

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8-5Problem Solving: Multiple-Step ProblemsSome word problems have hidden questions that must be answered before you can solve the problem.

A paved trail is 8 miles long. Rita runs 3 _ 8 of the length of the trail and walks the rest of the way. How many miles of the trail does Rita walk?

What do you know? Rita runs 3 _ 8 of an 8-mile trail.What are you asked to find? How many miles of the trail that Rita

walks.How can you find the distance that Rita walks?

Subtract the distance Rita ran from the length of the trail.

What is the hidden question? The hidden question will help you find data you need to solve the problem.

How many miles did Rita run?

To answer, find 3 _ 8 3 8 5 3.

Use the data to solve: 8 2 3 5 5, so Rita walked 5 of the 8 miles.

Write and answer the hidden question(s) in each problem. Then solve the problem.

1. April surfed for 1 _ 3 of the 6 hours she was at the beach. She spent the rest of the time building a sand castle. How many hours did she spend building the castle?

Hidden question:

Solution:

2. Bill put gasoline in 2 of his 5-gallon cans and 4 of his 2-gallon cans. He filled all the cans to the exact capacity. How many gallons of gasoline did he buy?

Hidden question 1:

Hidden question 2:

Solution:

3. It costs Le Stor $20 to buy a shirt. The store sells the shirt for 2 1 _ 2 times its cost. What is the profit for 100 of these shirts? Hint: Profit equals sales minus cost.

Hidden question 1:

Hidden question 2:

Solution:

How many hours did April surf? 1 _ 3 3 6 5 26 2 2 5 4; 4 hours

How many gallons are in two 5-gallon cans? 2 3 5 5 10How many gallons are in four 2-gallon cans? 4 3 2 5 810 1 8 5 18; 18 gallons of gas

What is the selling price of the shirt? 2 1 _ 2 3 $20 5 $50What is the profit for one shirt? $50 2 $20 5 $30100 3 $30 5 $3,000

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Topic 8 43

Practice 8-5

Practice

8-5Problem Solving: Multiple-Step ProblemsWrite and answer the hidden question(s) in each problem. Then solve the problem.

1. Tiwa spent 1 1 _ 2 hours setting up her computer. It took her 3 times as long to install the

software. How long did it take Tiwa to set up the computer and install software?

Hidden question(s):

Solution:

2. Lon bought 40 ounces of sliced ham. He used 3 _ 4 of the ham to make sandwiches for

his friends and 1 _ 5 of the ham in an omelet. How many ounces of ham were left?

Hidden question(s):

Solution:

3. Lionel cut off 1 _ 6 of a 48-inch piece of rope. Marsha cut off 1 _

4 of a 36-inch piece of

rope. They compared their cut pieces. Whose piece is longer? How much longer?

Hidden question(s):

Solution:

4. Melanie bought 3 CDs. The country music CD cost $15. The rock music CD cost 2 _ 3

as much as the country music CD. The platinum edition CD cost twice as much as the rock CD. What was the cost of the three CDs?

Hidden question(s):

Solution:

5. Writing to Explain Choose one of the problems above. Explain how you determined the hidden question and why it was necessary to answer that question in order to solve the problem.

How many hours did it take to install software? 3 � 1 1 _ 2 � 4 1 _ 2 1 1 _ 2 � 4 1 _ 2 � 6; 6 hours

How many ounces of ham were used to make sandwiches? 3/4 × 40 = 30; How many ounces of ham were used in the omelet? 40 � 1 _ 5 � 840 � 30 � 8 � 2; 2 ounces

How long is Lionel’s cut piece? 1 _ 6 � 48 � 8; How long is Marsha’s cut piece? 1 _ 4 � 36 � 9Marsha’s rope is 1 inch longer: 9 � 8 � 1.

How much did the rock CD cost? 2 _ 3 � $15 � $10; How much did the platinum CD cost? 2 � $10 � $20$15 � $10 � $20 � $45

Sample answer: In Problem 1, the question asks for the time spent on two tasks, so I needed to add the two times to answer the question. Only the time for one task was given in the problem, so the hidden question had to be to find the other time.