Simplifying fractions

In a nutshell

To simplify a fraction, you reduce the fraction to its lowest form.  The lowest form of a fraction occurs when the numerator (top number) and denominator (bottom number) have no common factors except $$1$$​. 

Example 1

Below are some examples of fractions in their lowest form.

$$\frac{1}{3}, \frac{2}{9}, \frac{13}{20}$$​​

Simplifying fractions

The quickest way to simplify a fraction is to divide by the highest common factor (HCF). The highest common factor is the largest factor that the numerator and denominator have in common.

Procedure

Example 2

Give the fraction $$\frac{12}{20}$$​ in its simplest form.

List the factors of the numerator and the denominator:

Numerator: $$1, 2, 3, 4, 6, 12$$​​

Denominator: $$1, 2, 4, 5, 10, 20$$​​

Identify the largest number that appears on both lists:

$$4$$​

Divide the numerator and denominator by the HCF:

$$\frac{12\div4}{20\div4}= \underline{\frac{3}{5}}$$​​

Tip: If the numerator and denominator are even, you can always divide by $$2$$.

Expanding fractions

To add, subtract or compare fractions the denominators need to be the same. 

procedure

Example 3

Which fraction is greater, $$\frac{3}{4}$$​ or $$\frac{7}{8}$$​?

Identify the multiplier needed to give each fraction a common denominator:

$$4\times2 = 8$$​​

Multiply the numerator and denominator by the multiplier:

$$\frac{3\times2}{4\times2} = \frac{6}{8}$$​​

Compare the two fractions:

$$\frac{6}{8}<\frac{7}{8}$$​ therefore $$\underline{\frac{3}{4}<\frac{7}{8}}$$​