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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.

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

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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.

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. ​$$\begin {aligned} &&15&:10 \\\div5 &&&& \div 5 \\&& \underline 3&\underline{:2}\end {aligned}$$​​

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.​