Lesson
5 Simplifying Fractions
Monitoring Progress:
Quiz 1
Simplifying Fractions
Vocabulary
What are factors?
Let’s review factors . We can use arrays to show the factors of
a number. Here are different arrays that show the factors of the
number 12.
1 × 12
2×6
3×4
The numbers 1, 2, 3, 4, 6, and 12 are all factors of 12.
What are common factors?
Let’s review common factors . When two different numbers have the
same factors, those factors are called common factors.
Factors for 12: 1, 2, 3, 4, 6, and 12
Factors for 16:
1, 2, 4, 8, and 16
The common factors for 12 and 16 are 1, 2, and 4.
104 Unit 2 • Lesson 5
factors
common factors
greatest common
factor
Lesson 5
How do common factors help us simplify
fractions?
We can easily find or “pull out” factors of whole numbers. For example,
we can “pull” 10 out of 40 this way:
40 = 4 · 10
The equation 4 · 10 is just another way of writing the number 40.
Pulling out numbers is also important when we work with fractions. By
pulling out the same number from the numerator and the denominator,
we can simplify a fraction.
Let’s simplify 4
6 by pulling out a common factor.
The numerator and denominator have a common factor of 2.
We find the simplified fraction by thinking about our basic facts.
Pulling out the 2 gives us
a simpler fraction.
2 · ? = 4 2 · ? = 6
T T
2 · 2 = 4 2 · 3 = 6
Basic facts help us make a connection between the original numerator
and denominator and the common factor. When we know our basic
facts, we can complete our problem by simplifying.
·
·
4 2 2
4 2 2
6 = 2 3 is the same as 6 = 2 · 3
4
2
Because 2
2 = 1, the problem can be written as: 6 = 1 · 3 .
Remember, a number
multiplied by 1 will always
equal itself.
2
4 2
Because 1 · 2
3 = 3 , we can rewrite the problem as 6 = 3 .
2
We can check to see if 4
6 = 3 by using fraction bars.
4
6 2
3
Unit 2 • Lesson 5 105
Lesson 5
How do we simplify fractions by using the
greatest common factor?
It is easy to pull out the same number from the numerator and
denominator when the numerator and denominator are small numbers.
It’s a little harder when they are bigger numbers. Let’s look at how to
simplify fractions with bigger numbers. We can use what we’ve learned
about greatest common factors.
We know the greatest common factor (GCF) is the biggest factor for
two or more numbers.
Example 1
Find the GCF of 8 and 12. Use the GCF to simplify 182 .
Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
The GCF of 8 and 12 is 4.
8
So, to simplify 12
, we “pull out” the GCF of 4 from both the numerator
and denominator. We do this by thinking about the basic facts.
4 · ? = 8
4 · ? = 12
T T
4 · 2 = 8
4 · 3 = 12
We find the missing factor when we think about basic facts. Then we
can simplify.
8
4 2
8
2
12 = 4 · 3 is the same as 12 = 1 · 3
We also know from basic multiplication facts that multiplying any
number by 1 will equal itself. So now we have our simplified fraction.
The GCF is 182 = 2
3.
106 Unit 2 • Lesson 5
Lesson 5
We call this process simplifying fractions. Let’s look at another example.
Example 2
Simplify 162 using the GCF.
Factors of 6: 1, 2, 3, 6
Factors of 12: 1, 2, 3, 4, 6, 12
The GCF of 6 and 12 is 6.
Now we “pull out” the GCF of 6 from both the numerator and
denominator.
6
6 1
12 = 6 · 2
6
We have the fraction 6
6, which is equal to 1. We replace 6 with the
number 1.
6
1
12 = 1 · 2
We know from our basic multiplication facts that anything times
1 is itself.
1
1· 1
2=2
6
1
12 = 2
How can we be sure these two fractions are equivalent?
Let’s look at fraction bars.
6
12 1
2
They are the same size.
Apply Skills
Turn to Interactive Text,
page 59.
Monitoring Progress
Quiz 1
Reinforce Understanding
Use the mBook Study Guide
to review lesson concepts.
Unit 2 • Lesson 5 107
Lesson 5
Homework
Activity 1
Solve the problems by multiplying across. You don’t need to simplify your
answers.
1.
1
3
4.
6
8
4
· 45 15
42
56
2 4 8
5 ·7
35
7
· 39 18
72
6
2
3
6
· 34 12
2. 8 · 7
3.
5.
6. 8 · 10
3
6
18
80
Activity 2
Simplify by pulling out the GCF from the numerator and the denominator.
1.
6
8
2 3
2 4
3
4
3
2. 9
3 1
3 3
1
3
3.
5
10
3.
4
5
5 1
5 2
1
2
Activity 3
Multiply the two fractions using an area model.
1.
2
3
· 34
1
6
2. 2 · 8
· 14
Activity 4 • Distributed Practice
Solve.
7
1.
3
7
+ 47 7
5.
6
8
+ 24
10
8
7
1
4
5
2. 8 − 8
6.
2
8
1,976
 897
1,079
108 Unit 2 • Lesson 5
1
4
+ 13
11
9
3.
8
9
7.
536
5
2,680
2
9
4.
8.
5
12
− 14
2
12
1
6
325
+ 137
462
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