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## Comparing Fractions

#### Introduction

Sometimes if fractions have a different denominator (the number below the line), it can be difficult to work out which is larger.

For example, which of the following three fractions is largest:

$rac{2}{3}$ or $rac{4}{5}$ or $rac{6}{7}$?

To work out the correct order, you'll need to do some calculations to make them easier to compare.

## Lesson

If you are comparing two fractions with the same denominator, you can just look at which numerator is larger.

For example, if you are comparing the following two fractions:

$rac{2}{5}$ and $rac{3}{5}$

Both have the same denominator (5) and so you can see that $rac{3}{5}$ is larger than $rac{2}{5}$.

If you have two numbers with different denominators, you'll need to make them the same in order to compare.

In order to make the denominators the same, you need to find the lowest common denominator of the two numbers.

If we want to compare $rac{3}{5}$ and $rac{4}{6}$, we need to find the lowest common denominator of 5 and 6. To do this, we can write out the multiples of 5 and 6.

Multiples of 5Multiples of 6
56
1012
1518
2024
2530
3036
3542

30 is the lowest common denominator of 5 and 6. Therefore, we need to convert both $rac{3}{5}$ and $rac{4}{6}$ to have 30 as a denominator.

To convert $rac{3}{5}$ to have a denominator of 30, we need to multiply 5 by 6. However, we need to multiply both the numerator and the denominator by the same value or we will change the fraction's value. Therefore, we can multiply it by $rac{6}{6}$.

$rac{3}{5}$ x $rac{6}{6}$ = $rac{18}{30}$

Then we repeat the process with $rac{4}{6}$. To make the denominator 30, we need to multiply by $rac{5}{5}$.

$rac{4}{6}$ x $rac{5}{5}$ = $rac{20}{30}$

Now we have both numbers with a common denominator, we can easily compare them.

$rac{20}{30}$ is larger than $rac{18}{30}$

Therefore $rac{4}{6}$ is larger than $rac{3}{5}$.

## Examples

$\dfrac{5}{7}$ ? $\dfrac{4}{7}$
$\dfrac{5}{7}$ ? $\dfrac{4}{7}$
$\dfrac{5}{7} > \dfrac{4}{7}$
$\dfrac{5}{6}$ ? $\dfrac{3}{5}$
$\dfrac{5*5}{6*5}$ ? $\dfrac{3*6}{5*6}$
$\dfrac{25}{30}$ ? $\dfrac{18}{30}$
$\dfrac{25}{30} > \dfrac{18}{30}$
$1\dfrac{5}{7}$ ? $2\dfrac{1}{7}$
$\dfrac{12}{7}$ ? $\dfrac{15}{7}$
$\dfrac{12}{7} < \dfrac{15}{7}$