# Simplifying Fractions

*To simplify a fraction, divide the top and bottom by the highest number that
can divide into both numbers exactly.*

## Simplifying Fractions

Simplifying (or *reducing*) fractions means to make the fraction as simple as possible.

Why say four-eighths (\frac{4}{8}) when we really mean half (\frac{1}{2}) ?

\frac{4}{8} | = | \frac{2}{4} | = | \frac{1}{2} |

(Four-Eighths) | (Two-Quarters) | (One-Half) | ||

= | = |

## How do I Simplify a Fraction ?

There are two ways to simplify a fraction:

## Method 1

Try to **exactly divide** (only whole number answers) both the top and bottom of the fraction by 2, 3, 5, 7 ,... etc, until we can't go any further.

### Example: Simplify the fraction \frac{24}{108} :

÷ 2 | ÷ 2 | ÷ 3 | ||||

\frac{24}{108} = \frac{12}{54} = \frac{6}{27} = \frac{2}{9} | ||||||

÷ 2 | ÷ 2 | ÷ 3 |

That is as far as we can go. The fraction simplifies to \frac{2}{9}

### Example: Simplify the fraction \frac{10}{35} :

Dividing by **2** doesn't work because **35 can't be exactly divided by 2** (35/2 = 17½)

Likewise we can't divide exactly by **3** (10/3 = 3\frac{1}{3} and also 35/3=11\frac{2}{3})

No need to check **4** (we checked 2 already, and 4 is just 2×2).

But **5** does work!

÷ 5 | ||

\frac{10}{35} = \frac{2}{7} | ||

÷ 5 |

That is as far as we can go. The fraction simplifies to \frac{2}{7}

Notice that after checking 2 we didn't need to check 4 (4 is 2×2)?

We also don't need to check 6 when we have checked 2 and 3 (6 is 2x3).

In fact, when checking from smallest to largest we use prime numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...

## Method 2

Divide both the top and bottom of the fraction by the Greatest Common Factor (you have to work it out first!).

### Example: Simplify the fraction \frac{8}{12} :

The largest number that goes exactly into both 8 and 12 is 4, so *the Greatest Common Factor is 4*.

Divide both top and bottom by 4:

÷ 4 | ||

\frac{8}{12} = \frac{2}{3} | ||

÷ 4 |

That is as far as we can go. The fraction simplifies to \frac{2}{3}

## Simplifying Fractions Automatically

OK, there is a *third* method, use this tool:

## Fractions on the Number Line

Also ... see an animation of Fractions on the Number Line where you can see many common fractions and their simpler version.

We also have a chart of fractions with the simplest fraction highlighted.