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.
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© 2010 Cheryl Wilcox
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.
Free Pre-Algebra Lesson 12 page 3
© 2010 Cheryl Wilcox
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.
Example: What is 5/6 of 2/3?
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.
Example: What is 28/15 of 20/21?
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.
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© 2010 Cheryl Wilcox
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.
Free Pre-Algebra Lesson 12 page 5
© 2010 Cheryl Wilcox
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.
Free Pre-Algebra Lesson 12 page 6
© 2010 Cheryl Wilcox
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.
Free Pre-Algebra Lesson 12 page 7
© 2010 Cheryl Wilcox
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?
Free Pre-Algebra Lesson 12 ! page 8
© 2010 Cheryl Wilcox
Lesson 12: Multiplying Fractions
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
Free Pre-Algebra Lesson 12 ! page 9
© 2010 Cheryl Wilcox
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 _____.
Free Pre-Algebra Lesson 12 page 10
© 2010 Cheryl Wilcox
Lesson 12: Multiplying Fractions
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
3. List all the factor pairs of the numbers, then find the GCF.
Use the GCF to simplify the fractions:
68
102
68 34
102 34
2
3
102
68
102 34
68 34
3
2
4. Find the prime factorization of the numbers.
126 2 • 3 • 3 • 7
231 3 • 7 • 11
Use the prime factorizations to simplify the fractions.
126
231
2 • 3 • 3 • 7
3 • 7 • 11
6
11
231
126
11
6
5. Write 5 fractions equivalent to
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
Free Pre-Algebra Lesson 12 page 11
© 2010 Cheryl Wilcox
7. Change the mixed numbers to improper fractions.
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
9. Use the picture to answer the questions.
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
10. Multiply the fractions. Use cross-cancelling if possible.
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
11. Write the answer. Then write the four related equations.
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
Free Pre-Algebra Lesson 12 ! page 12
© 2010 Cheryl Wilcox
Lesson 12: Multiplying Fractions
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.
3. List all the factor pairs of the numbers, then find the GCF.
100 175
Use the GCF to simplify the fractions:
100
175
175
100
4. Find the prime factorization of the numbers.
165
110
Use the prime factorizations to simplify the fractions.
165
110
110
165
5. Write 5 fractions equivalent to
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
Free Pre-Algebra Lesson 12 ! page 13
© 2010 Cheryl Wilcox
7. Change the mixed numbers to improper fractions.
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
9. Use the picture to answer the questions.
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?
10. Multiply the fractions. Use cross-cancelling if possible.
a.
5
8 of 8
b.
5
8•
80
7
c.
5
8• 8
4
5
d.
5
8 of 16
11. Write the answer. Then write the four related equations.
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 _____.