Algebra Demystified 2nd Ed

Chapter 1 FraCtions 7

Simplifying Fractions

When working with fractions, we are usually asked to “reduce the fraction to

lowest terms” or to “write the fraction in lowest terms” or to “simplify the fraction.”

These phrases mean that the numerator and denominator have no common

factors (other than 1). For example,^23 is in written in lowest terms but^46 is not

because 2 is a factor of both 4 and 6. Simplifying fractions is like fraction mul-

tiplication in reverse. For now, we will use the most basic approach to simplify-

ing fractions. In the next section, we will learn a quicker method.

First write the numerator and denominator as a product of prime numbers.

(Refer to the Appendix if you need to review finding the prime factorization of

a number.) Next collect the prime numbers common to both the numerator and

denominator (if any) at beginning of each fraction. Split each fraction into two

fractions, the first with the common prime numbers. This puts the fraction in

the form of “1” times another fraction. This might seem like unnecessary work

(actually, it is), but it will drive home the point that the factors that are common

in the numerator and denominator form the number 1. Thinking of simplifying

fractions in this way can help you avoid common fraction errors later in algebra.

EXAMPLE

Simplify the fraction with the method outlined above.

6

18

SOLUTION

We begin by factoring 6 and 18.

6

18

23

233

231

233

= ⋅

⋅⋅

= ⋅⋅

⋅⋅

()

()

We now write the common factors as a separate fraction.

6

18

23

233

231

233

23

23

1

3

6

6

1

3

= ⋅

⋅⋅

= ⋅⋅

⋅⋅

= ⋅

⋅

() ⋅=⋅

()

Because^66 is 1, we see that 186 simplifies to^13.

6

18

23

233

231

233

23

23

1

3

6

6

1

3

1

3

= ⋅

⋅⋅

= ⋅⋅

⋅⋅

= ⋅

⋅

() ⋅=⋅=

()

42

49

723

77

7

7

23

7

16

7

6

7

= ⋅⋅

⋅

=⋅⋅ =⋅ =

SOLUTION

We begin by factoring 6 and 18.

✔

EXAMPLE

Simplify the fraction with the method outlined above.