Lesson 12 Multiplying Fractions · Combine the fractions by multiplying the numerators, then the denominators. Convert the improper fraction to a mixed or whole number. Interpret

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Free Pre-Algebra Lesson 12 page 1

© 2010 Cheryl Wilcox

Lesson 12

Multiplying Fractions

Here’s a band with 6 people. Notice that the word “of” in natural language translates to a multiplication in the corresponding math sentence.

Natural Language Picture Math

3 of the bands have green hats. 18 people have green hats.

3 • 6 18

1/3 of the band has green hats. 2 people have green hats.

1

3• 6 2

How to Multiply a Fraction and a Whole Number

To find a fraction of a group, multiply the fraction by the number in the group. 1/3 of a group of 6 people is written

1

3• 6 .

Here are the steps for the multiplication:

Step 1 Step 2 Step 3 Step 4 Step 5 Write the problem. Rewrite the whole

number as a fraction with denominator 1.

Combine the fractions by multiplying the numerators, then the denominators.

Convert the improper fraction to a mixed or whole number.

Interpret the result in the context of the problem

1

3• 6

1

3•

6

1

1• 6

3 • 1

6

3

6 3 2r0

6

32

1/3 of 6 people is 2 people.

Multiplication is commutative, so note that 6 •

1

3 has the same numeric result as

1

3• 6 . However, the picture interpreting

it is different. 6 pieces, each 1/3 of a whole, makes two wholes: .

Example: What is 2/5 of a group of 20?

2

5• 20

2

5•

20

1

2 • 20

5 • 1

40

5

40 5 8r040

58

2/5 of 20 is 8.

2/5 of 20 is 8.

Free Pre-Algebra Lesson 12 ! page 2


If the number in the group is not divisible by the denominator of the fraction, the result will be a mixed number. This doesn’t make much sense in groups of people, but might make sense in other contexts where items in the group are themselves divisible.

Example: What is 1/2 of a group of 7 apples?

1

2• 7

=1

2•

7

1

=1• 7

2 • 1=

7

2

7 ÷ 2 = 3r17

2= 3

1

2

1/2 of 7 is 3 and 1/2

apples.

1/2 of 7 apples is 31/2 apples.

Example: What is 2/3 of 10?

2

3• 10

=2

3•

10

1

=2 • 10

3 • 1=

20

3

20 ÷ 3 = 6r220

3= 6

2

3

2/3 of 10 is 6 and 2/3.

2/3 of the 10 squares are green. 6 2/3 squares are green.

A Fraction Times a Fraction The “group” that you are finding a fraction of may itself be a fraction or mixed number. Just multiply as before.

Example: You have inherited 1/3 of a 1/2 acre parcel of land. How many acres have you inherited? (What is 1/3 of 1/2?)

1

3•

1

2

=1• 1

3 • 2=

1

6

1/3 of 1/2 is 1/6.

The whole rectangle is an acre. 1/2 acre is shaded. When the 1/2 acre is divided into 3rds, the whole acre is divided in 6ths.

1/3 of 1/2 acre is 1/6 acre. I’ve inherited 1/6 acre.



When multiplying fractions, you can cancel any common factors BEFORE multiplying. A factor in either numerator may be cancelled with a factor in either denominator. This is called cross-cancelling. It’s convenient because you won’t have to simplify your answer – the result will already be in lowest terms.

Example: What is 1/2 of 2/3?

Cancelling to Simplify After Multiplication: Cross-Cancelling Before Multiplication

1

2•

2

3

1• 2

2 • 3

2

1

63

1

3

1

2•

2

3

1

21

•2

1

3

1

3

1/2 of 2/3 is 1/3. This makes sense! Imagine 2/3 of a pizza. Then split it in half.


5

6•

2

3

5 • 2

6 • 3

5 • 2

1

63

• 3

5 • 1

3 • 3

5

9

5/6 of 2/3 is 5/9.


28

15•

20

21

28

4

153

•20

4

213

16

9

Note that 5 divides both 15 and 20, and 7 divides both 28 and 21. Cross-cancelling before multiplication saves a lot of work in multiplying large numbers. Without cross-cancelling, the

previous problem would have been:

28

15•

20

21

28 • 20

15 • 21

560 35

315 35

16

9.



A Fraction Times a Mixed Number

Example: You have inherited 1/2 of a 18 2/3 acre parcel of land. How many acres have you inherited? (What is 1/2 of 18 2/3?)

We know we need to multiply

1

2• 18

2

3. First let’s just think about it with pictures.

You have 18 acres and then some, and you’re going to take 1/2 of that. 1/2 of the 18 acres is 9 acres, so that part seems clear. Then all that’s left to do is to divide the 2/3 acre in half. This is just like a previous problem, 1/2 of 2/3, which we found was 1/3. The answer should be 9 1/3 acre.

I’ve inherited 9 and 1/3 acres.

Method 1: Just as we reasoned above. Multiply the fraction by each part of the mixed number separately, then add the results.

1

2• 18

2

3

1

2• 18

1

2•

2

3

18

2

1• 2

2 • 3

91

3 Method 2: Convert the mixed number to an improper fraction, then multiply. This method is more work, but it is easier in cases where the whole number is not divisible by the denominator of the fraction.

Step 1 Convert the mixed number to an improper fraction

Step 2 Multiply.

Step 3 Convert the result to a mixed number.

18

2

3

3 • 18 2

3

56

3

1

2•

56

3

1• 56

2 • 3

56

6

56 6 9r2

56

69

2

69

1

3

I’ve inherited 9 and 13 acres.

Example: You have inherited 1/5 of a 18 2/3 acre parcel of land. How many acres have you inherited? Since the whole number of acres, 18, is not divisible by the denominator of the fraction, 5, it is easier to use method 2.

18

2

3

3 • 18 2

3

56

3

1

5•

56

3

1• 56

5 • 3

56

15

56 15 3r11

56

153

11

15

I’ve inherited 3 and 11/15 acres.



Multiplying a Mixed Number by a Mixed Number Example: You’re making 2 1/2 times a recipe that calls for 1 3/4 cup of rice. How much rice should you add?

2

1

2 times

1

3

4 cup rice is how much rice?

Re-write both mixed numbers as improper fractions before you begin multiplying:

2

1

2

5

2

1

3

4

7

4

5

2•

7

4

5 • 7

2 • 4

35

8 (Although we have a 2 and a 4, they are both in the denominator. Nothing can be cancelled.)

Finally, convert 35/8 back to a mixed number:

35

84

3

8.

I should add 4 3/8 cups rice.



Lesson 12: Multiplying Fractions

Worksheet Name

1. Use the picture to show that 1/4 of a group of 12 is 3.

Write out the steps for the multiplication

1

4• 12 .

2. What is 1/2 of 11? What does this mean, in terms of a

group of 11?

What is 11 • 1/2? What does this mean, in terms of halves?

3. Draw a sketch to illustrate each multiplication.

a.

2

5• 15

b.

2

5• 16

4. Write out the steps for each multiplication.

a.

2

5• 15

b.

2

5• 16

5. Sketch a picture to show that 1/3 of 5/7 is 5/21.

6. Sketch a picture to show that 1/2 of 4/5 is 2/5.



7. Mulltiply the fractions. Use cross-cancelling if possible. Do not convert answers to mixed numbers.

a.

2

7•

7

3

b.

2

7•

14

3

c.

6

7•

14

3

d.

7

10•

5

21

8. Convert the mixed numbers to improper fractions before multiplying. Write the answer as a mixed number.

a.

2

7• 2

1

3

b. 3

1

8• 3

1

5

c. 11

1

4• 1

1

27

d. 1

1

4•

7

20

9. You are making 2 1/2 times a recipe with the following ingredients:

1/2 cup butter; 4 eggs; 3 1/2 cups flour

How much of each will you use?

10. When you multiply a whole number with a fraction, (as in

the problem

1

4• 20 ), is the answer greater than or less than

the whole number? Why?




Homework 12A Name

1. Simplify 6

3! 4 3 + 2 • 5( ) .

2. Draw a line 2 5/8 inches long. Convert 2 5/8 to an improper fraction. Show the eighths on the ruler.

3. List all the factor pairs of the numbers, then find the GCF.

68 102

Use the GCF to simplify the fractions:

68

102

102

68

4. Find the prime factorization of the numbers.

126

231

Use the prime factorizations to simplify the fractions.

126

231

231

126

5. Write 5 fractions equivalent to

11

6.

6. Simplify

8

10 to lowest terms.

Could the mixed number

58

10 be simplified to lowest

terms?

1 32



7. Change the mixed numbers to improper fractions.

a.

43

8

b.

124

5

c.

2317

100

8. Change the improper fractions to mixed numbers.

a.

72

12

b.

72

13

c.

72

14

9. Use the picture to answer the questions.

a. What is 1/2 of 12? b. What is 1/3 of 12?

c. What is 1/4 of 12? d. What is 1/6 of 12?

e. What is 1/12 of 12?

There are 12 inches in a foot.

f. What fraction of a foot is 6 inches?

g. What fraction of a foot is 3 inches?

h. What fraction of a foot is 2 inches?

i. What fraction of a foot is 1 inch?

10. Multiply the fractions. Use cross-cancelling if possible.

a.

7

10 of 64

b.

7

10•

80

9

c.

7

10• 8

4

7

d.

7

10 of 10

11. Write the answer. Then write the four related equations.

a. The product of 4 and 2 is ____.

b. The difference of 9 and 3 is _____.

c. The sum of 18 and 6 is _____.

d. The quotient of 20 and 5 is _____.




Homework 12A Answers

1. Simplify 63 4 3 2 • 5 .

216 4 3 2 • 5

216 4 3 10

216 4 13

216 52

164

2. Draw a line 2 5/8 inches long. Convert 2 5/8 to an improper fraction. Show the eighths on the ruler.

2

5

8

21

8



68

102

68 34

102 34

2

3

102

68

102 34

68 34

3

2


126 2 • 3 • 3 • 7

231 3 • 7 • 11


126

231

2 • 3 • 3 • 7

3 • 7 • 11

6

11

231

126

11

6


11

6.

11

6

22

12

33

18

44

24

55

30

66

36

6. Simplify

8

10 to lowest terms.

8

10

4

5

Could the mixed number 5

8

10 be simplified to lowest

terms?

5

8

105

4

5




a. 4

3

8

32

8

3

8

35

8

b. 12

4

5

60

5

4

5

64

5

c. 23

17

100

2300

100

17

100

2317

100

8. Change the improper fractions to mixed numbers.

a.

72

1272 12 6r0

72

126

b.

72

1372 13 5r7

72

135

7

13

c.

72

1472 14 5r2

72

145

2

145

1

7


a. What is 1/2 of 12? 6 b. What is 1/3 of 12? 4

c. What is 1/4 of 12? 3 d. What is 1/6 of 12? 2

e. What is 1/12 of 12? 1

There are 12 inches in a foot.

f. What fraction of a foot is 6 inches? 1/2

g. What fraction of a foot is 3 inches? 1/4

h. What fraction of a foot is 2 inches? 1/6

i. What fraction of a foot is 1 inch? 1/12


a.

7

10 of 64

7

105

•64

32

1

224

544

4

5

b.

7

10•

80

9

7

101

•80

8

9

56

96

2

9

c.

7

10• 8

4

78

4

7

56

7

4

7

60

7

7

10•

60

7

7

1

101

•60

6

71

6

16

d.

7

10 of 10

7

10•

10

1

7

17


a. The product of 4 and 2 is __8__.

4 • 2 8 8 2 4

2 • 4 8 8 4 2

b. The difference of 9 and 3 is __6___.

9 3 6 6 3 9

9 6 3 3 6 9

c. The sum of 18 and 6 is __24___.

18 6 24 24 6 18

6 18 24 24 18 6

d. The quotient of 20 and 5 is __4___.

20 5 4 4 • 5 20

20 4 5 5 • 4 20




Homework 12B Name

1. Simplify 4

6! 3 12 + 2

3( )2

.

2. Draw a line 1 3/4 inches long. Convert 1 3/4 to an improper fraction. Show the fourths on the ruler.


100 175


100

175

175

100


165

110


165

110

110

165


5

14.

6.Change

6

5 to a mixed number.

The mixed number

36

5 is not in the correct format,

because 6/5 is an improper fraction. What would be the correct way to write it?

1 32




a.

54

9

b.

157

12

c.

351

3

8. Change the improper fractions to mixed numbers. Simplify the improper fraction before you begin or simplify the fraction part of the result if not in lowest terms.

a.

96

22

b.

96

23

c.

96

24


There are 60 minutes in one hour.

a. How many minutes in 1/2 hour? What is 1/2 of 60?

b. What is 1/3 of 60? c. What is 1/4 of 60?

d. What is 1/6 of 60? e. What is 1/12 of 60?

f. What fraction of an hour is 6 minutes?

g. What fraction of a hour is 20 minutes?

h. What fraction of an hour is 15 minutes?

i. What fraction of an hour is 45 minutes?


a.

5

8 of 8

b.

5

8•

80

7

c.

5

8• 8

4

5

d.

5

8 of 16


a. The difference of 18 and 2 is ____.

b. The quotient of 15 and 3 is _____.

c. The product of 8 and 4 is _____.

d. The sum of 20 and 5 is _____.

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Lesson 12 Multiplying Fractions · Combine the fractions by multiplying the numerators, then the denominators. Convert the improper fraction to a mixed or whole number. Interpret