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# Solving ratio problems

## In a nutshell

Ratios are quantitative relationships between amounts. They can be simplified or scaled up in order to help solve problems.

## Ratio basics

The ratio of one quantity to another is written as two or more figures separated by a colon. Ratios are useful when solving problems comparing one amount to another.

*Tip:** The colon is read as 'to' so *$3:2$* is read as 'three to two'.*

##### Example 1

*Here, the ratio of shaded to non-shaded parts of the circle is *$3:2$* - three parts are shaded for every two that are not.*
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### Simplifying ratios

Ratios are simplified in the same way as fractions: by doing the same thing to one side as to the other. To simplify, simply divide each figure by the same amount.

##### Example 2

*Simplify the ratio $15:10$* .
*Fifteen and ten can both be divided by five so divide each side.*
$\begin {aligned} &&15&:10 \\\div5 &&&& \div 5 \\&& \underline 3&\underline{:2}\end {aligned}$
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### Scaling up ratios

Ratios can also be scaled up: meaning to multiply each side by the same amount. This is useful when you are given a ratio in its simplest form and need to work out an amount.

##### Example 3

*For every sunny day in England there are three rainy days. If it was sunny for six days this month, how many days did it rain?*
*Write down the ratio of sunny to rainy days.* $1:3$
*Scale up by multiplying by six.*
$\begin {aligned} &&1&:3\\ \times 6 &&&&\times 6 \\&&6&:18\end {aligned}$
*For every six days of sun there are *$18$* days of rain so this month it rained for $\underline{18}$ days.*
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