Download Lesson 12: Multiplying Fractions

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Lesson 12  page 1
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 • 6 18
3 of the bands have green
hats. 18 people have green
hats.
1/3 of the band has green
hats. 2 people have green
hats.
1
•6
3
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
Here are the steps for the multiplication:
Step 1
Write the problem.
Step 2
Rewrite the whole
number as a fraction
with denominator 1.
1
•6
3
1 6
•
3 1
Multiplication is commutative, so note that 6 •
Step 3
Combine the fractions
by multiplying the
numerators, then the
denominators.
1• 6
3•1
6
3
Step 4
Convert the improper
fraction to a mixed or
whole number.
6 3 2r 0
6
3
1
•6.
3
Step 5
Interpret the result in
the context of the
problem
2
1/3 of 6 people
is 2 people.
1
1
has the same numeric result as • 6 . However, the picture interpreting
3
3
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
• 20
5
2/5 of 20 is 8.
© 2010 Cheryl Wilcox
2 20
•
5 1
2 • 20
5•1
40
5
40 5
8r 0
40
5
8
2/5 of 20
is 8.
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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
•7
2
1• 7 7
=
=
2•1 2
7 ÷ 2 = 3r 1
2 • 10 20
=
=
3•1
3
20 ÷ 3 = 6r 2
1 7
= •
2 1
7
1
=3
2
2
1/2 of 7
is 3 and 1/2
apples.
20
2
=6
3
3
2/3 of 10
is 6 and 2/3.
1/2 of 7 apples is 31/2 apples.
Example: What is 2/3 of 10?
2
• 10
3
2 10
= •
3 1
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 1
•
3 2
1• 1 1
=
=
3•2 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.
© 2010 Cheryl Wilcox
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Lesson 12  page 3
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
1 2
•
2 3
1• 2
2•3
2
6
1
1
3
1 2
•
2 3
3
1
2
•
2 3
1
3
1
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?
1
5 2
•
6 3
5•2
6•3
5• 2
5•1
3•3
6•3
5
9
3
5/6 of 2/3 is 5/9.
Example: What is 28/15 of 20/21?
28 20
•
15 21
4
4
28
20
15
•
3
21
16
9
3
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 • 20 28 • 20 560 35 16 .
15 21
© 2010 Cheryl Wilcox
15 • 21
315 35
9
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Lesson 12  page 4
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 . First let’s just think about it with pictures.
2
3
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
• 18
2
18
2
9
1 2
•
2 3
1• 2
2 •3
1
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
18
2
3
3 • 18 2
3
56
3
Step 2
Multiply.
1 56
•
2 3
Step 3
Convert the result to a mixed number.
1• 56
2•3
56
6
56 6
9r 2
56
2
9
6
6
9
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 56
•
5 3
1• 56
5•3
56
15
I’ve inherited 3 and 11/15 acres.
© 2010 Cheryl Wilcox
56 15 3r 11
56
11
3
15
15
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Lesson 12  page 5
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
3
times 1 cup rice is how much rice?
2
4
Re-write both mixed numbers as improper fractions before you begin multiplying:
2
5 7
•
2 4
5•7
2•4
35
8
1
2
5
2
1
3
4
7
4
(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
8
4
3
.
8
I should add 4 3/8 cups rice.

© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 12  page 6
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
• 12 .
4
3. Draw a sketch to illustrate each multiplication.
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?
4. Write out the steps for each multiplication.
a.
2
• 15
5
a.
2
• 15
5
b.
2
• 16
5
b.
2
• 16
5
5. Sketch a picture to show that 1/3 of 5/7 is 5/21.
© 2010 Cheryl Wilcox
6. Sketch a picture to show that 1/2 of 4/5 is 2/5.
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7. Mulltiply the fractions. Use cross-cancelling if possible. Do
not convert answers to mixed numbers.
a.
2 7
•
7 3
b.
2 14
•
7 3
Lesson 12  page 7
8. Convert the mixed numbers to improper fractions before
multiplying. Write the answer as a mixed number.
a.
2
1
•2
7
3
b. 3
1
4
d. 1
7 5
•
10 21
9. You are making 2 1/2 times a recipe with the following
ingredients:
1/
1
27
c. 11 • 1
6 14
c. •
7 3
d.
1
1
•3
8
5
2
cup butter; 4 eggs; 3 1/2 cups flour
How much of each will you use?
© 2010 Cheryl Wilcox
1 7
•
4 20
10. When you multiply a whole number with a fraction, (as in
the problem 1 • 20 ), is the answer greater than or less than
4
the whole number? Why?
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Lesson 12 ! page 8
Lesson 12: Multiplying Fractions
Homework 12A
(
Name
)
2. Draw a line 2 5/8 inches long.
Convert 2 5/8 to an improper fraction. Show the eighths on
the ruler.
3
1. Simplify 6 ! 4 3 + 2 • 5 .
1
3. List all the factor pairs of the numbers, then find the GCF.
68
2
4. Find the prime factorization of the numbers.
102
126
231
Use the prime factorizations to simplify the fractions.
Use the GCF to simplify the fractions:
68
102
102
68
5. Write 5 fractions equivalent to
11
.
6
126
231
231
126
6. Simplify
8
to lowest terms.
10
Could the mixed number 5
terms?
© 2010 Cheryl Wilcox
8
be simplified to lowest
10
3
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Lesson 12 ! page 9
7. Change the mixed numbers to improper fractions.
a. 4
3
8
8. Change the improper fractions to mixed numbers.
a.
72
12
b. 12
4
5
b.
72
13
c. 23
17
100
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?
10. Multiply the fractions. Use cross-cancelling if possible.
a.
7
of 64
10
b.
7 80
•
10 9
c.
7
4
•8
10
7
d.
7
of 10
10
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?
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 _____.
© 2010 Cheryl Wilcox
c. The sum of 18 and 6 is _____.
d. The quotient of 20 and 5 is _____.
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Lesson 12  page 10
Lesson 12: Multiplying Fractions
Homework 12A Answers
3
1. Simplify 6
2. Draw a line 2 5/8 inches long.
Convert 2 5/8 to an improper fraction. Show the eighths on
the ruler.
4 3 2•5 .
216 4 3 2 • 5
2
216 4 3 10
216 4 13
5
8
21
8
216 52
164
3. List all the factor pairs of the numbers, then find the GCF.
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.
Use the GCF to simplify the fractions:
68
102
68 34
102 34
2
3
102
68
5. Write 5 fractions equivalent to
11
6
22
12
33
18
44
24
55
30
11
.
6
66
36
102 34
68 34
3
2
126
231
2• 3 •3• 7
231
126
11
6
3 • 7 • 11
6. Simplify
8
8
to lowest terms.
10
10
Could the mixed number 5
terms?
5
© 2010 Cheryl Wilcox
6
11
4
5
8
be simplified to lowest
10
8
10
5
4
5
Free Pre-Algebra
Lesson 12  page 11
7. Change the mixed numbers to improper fractions.
a. 4
3
8
32
8
b. 12
4
5
c. 23
17
100
3
8
60
5
35
8
4
5
64
5
2300
100
17
100
2317
100
9. Use the picture to answer the questions.
8. Change the improper fractions to mixed numbers.
a.
72
72 12
12
6r 0
72
12
6
b.
72
72 13
13
5r 7
72
13
5
7
13
c.
72
72 14
14
5r 2
72
14
5
2
14
5
10. Multiply the fractions. Use cross-cancelling if possible.
32
a.
7
of 64
10
7
10
•
64
1
224
5
44
4
5
5
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
8
7 80
b.
•
10 9
7
80
•
10 9
56
9
6
2
9
1
c.
7
4
•8
10
7
8
4
7
56
7
There are 12 inches in a foot.
4
7
1
7 60
•
10 7
f. What fraction of a foot is 6 inches? 1/2
g. What fraction of a foot is 3 inches? 1/4
7
10
d.
7
of 10
10
7
10
•
60
7
6
•
1
h. What fraction of a foot is 2 inches? 1/6
i. What fraction of a foot is 1 inch? 1/12
10
1
60
7
6
1
6
1
7
1
7
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
© 2010 Cheryl Wilcox
1
7
c. The sum of 18 and 6 is __24___.
18 6 24
24 6 18
6 18 24
24 18
d. The quotient of 20 and 5 is __4___.
20 5
4
4 • 5 20
20 4
5
5• 4
20
6
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Lesson 12 ! page 12
Lesson 12: Multiplying Fractions
Homework 12B
(
1. Simplify 4 6 ! 3 12 + 23
)
2
Name
2. Draw a line 1 3/4 inches long.
Convert 1 3/4 to an improper fraction. Show the fourths on
the ruler.
.
1
3. List all the factor pairs of the numbers, then find the GCF.
100
2
4. Find the prime factorization of the numbers.
175
165
110
Use the prime factorizations to simplify the fractions.
Use the GCF to simplify the fractions:
165
110
100
175
110
165
175
100
5. Write 5 fractions equivalent to
5
.
14
6.Change
6
to a mixed number.
5
The mixed number 3
6
is not in the correct format,
5
because 6/5 is an improper fraction. What would be the
correct way to write it?
© 2010 Cheryl Wilcox
3
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Lesson 12 ! page 13
7. Change the mixed numbers to improper fractions.
a. 5
4
9
b. 15
7
12
c. 35
1
3
9. Use the picture to answer the questions.
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
10. Multiply the fractions. Use cross-cancelling if possible.
a.
5
of 8
8
b.
5 80
•
8 7
c.
5
4
•8
8
5
d.
5
of 16
8
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?
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 _____.
© 2010 Cheryl Wilcox
c. The product of 8 and 4 is _____.
d. The sum of 20 and 5 is _____.