# 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

1. | Determine the HCF of the numerator and the denominator. |

2. | Divide both the numerator and the denominator by the HCF. The resulting fraction is in its simplest form. |

##### 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

1. | Identify the multiplier needed to give both fractions a common denominator. |

2. | Multiply the numerator and denominator by the answer found in Step 1. |

##### 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 $\underline{\frac{3}{4}<\frac{7}{8}}$*