Basic Mathematics for College Students

To simplify a fraction,we write it in simplest form by removing a factor equal to 1.

For example, to simplify we note that the greatest factor common to the numerator

and denominator is 5 and proceed as follows:

1

Factor 10 and 15. Note the form of 1 highlighted in red.

Use the rule for multiplying fractions in reverse:

write as the product of two fractions, and

A number divided by itself is equal to 1:.

Use the multiplication property of 1: the product

of any fraction and 1 is that fraction.

We have found that the simplified form of is To simplify we removed a

factor equal to 1in the form of The result, is equivalent to

To streamline the simplifying process, we can replace pairs of factors common to

the numerator and denominator with the equivalent fraction^11.

10

15.

2

3 ,

5

5.

10

15 ,

2

3.

10

15



2

3

5

 5  1

2

3

 1

5

5.

2

3

2  5

3  5



2

3



5

5

10

15



2  5

3  5

10

15 ,

214 Chapter 3 Fractions and Mixed Numbers

Self Check 7

Simplify each fraction:

a.

b.

Now TryProblems 65 and 69

3

9

10

25

EXAMPLE 7

Simplify each fraction: a. b.

StrategyWe will factor the numerator and denominator. Then we will look for

any factors common to the numerator and denominator and remove them.

WHYWe need to make sure that the numerator and denominator have no

common factors other than 1. If that is the case, then the fraction is in simplest form.

Solution

a.

1

To prepare to simplify, factor 6 and 10. Note the form of 1 highlighted in red.

Simplify by removing the common factor of 2 from the numerator and

denominator. A slash / and the 1’s are used to show that is replaced by

the equivalent fraction. A factor equal to 1 in the form of was removed.

Multiply the remaining factors in the numerator: 1  3 3. Multiply the

remaining factors in the denominator: 1  5 5.

Since 3 and 5 have no common factors (other than 1), is in simplest form.

b. To prepare to simplify, factor 21.

Simplify by removing the common factor of 7 from the numerator and

denominator.

Multiply the remaining factors in the denominator: 1 3 = 3.

Caution! Don't forget to write the 1’s when removing common factors of

the numerator and the denominator. Failure to do so can lead to the

common mistake shown below.

We can easily identify common factors of the numerator and the denominator

of a fraction if we write them in prime-factored form.

7

21



7

3  7



0

3



1

3



7

1

3  7

1

7

21



7

3  7

3

5



3

5

2

2

1

1

2

 2

2

1

 3

2

1

 5

6

10



2  3

2  5

7

21

6

10