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# Ratio

## What is a ratio?

A ratio is a relationship between two or more quantities. It expresses the relative proportions of various quantities. A ratio is written with a colon ($:$), as in $3:4$.

A café makes biscuits using $\Tblue{2}$ kilos of flour and $\Tred{1}$ kilo of sugar. The ratio of flour to sugar is $\Tblue{2}:\Tred{1}$.

I make biscuits with $\Tblue{8}$ ounces of flour and $\Tred{4}$ ounces of sugar. The ratio flour to sugar is the same $\Tblue{8}:\Tred{4} = \Tblue{2}:\Tred{1}$, because I need twice as much flour as I need sugar.

Ratios can compare more than two quantities.

My recipe uses as much butter as it does sugar and twice as much flour. The ratio of flour to sugar to butter is $\Tblue{2}:\Tred{1}:\Tviolet{1}$.

Recipes use ingredients in particular ratios.

## Simplifying ratios

A ratio is written in reduced form if the numbers in the ratio have no common factor.

$\Tblue{4:3}$ is in reduced form

$\Tgreen{20:25}$ is not reduced because $\Tred{5}$ is a common factor.

To reduce a ratio, find the highest common factor (HCF) of the numbers and divide the numbers by it.

The reduced form of $\Tgreen{20:35}$ is $\Tblue{4:7}$ because $\HCF(\Tgreen{20},\Tgreen{35}) = \Tred{5},\quad \Tgreen{20} = \Tblue{4}\times \Tred{5},\quad \Tgreen{35} = \Tblue{7}\times \Tred{5}.$

A TV screen has a width of $\Tblue{48}$ cm and a height of $\Tgreen{27}$ cm. Its width to height ratio is $\Tblue{48}:\Tgreen{27}= \Tblue{16}\times\Tred{3}:\Tgreen{9}\times\Tred{3} = \Tblue{16}:\Tgreen{9}.$

This television has an aspect ratio of 16:9. This refers to the lengths of its sides. The ratio 16:9 is in its simplest form.

## Comparing ratios

A ratio can be written in the form $1:n$ or $n:1$. The number $n$ can be any positive number, not just an integer. It is usually written in fractional or in decimal form.

The ratio $\Tblue{3}:\Tgreen{7}$ can be written as $\Tblue{1}:\frac{\Tgreen{7}}{\Tblue{3}}$ or in decimal form $\Tblue{1}:\Tgreen{2.33}$.

The form $1:n$ is mainly used to compare ratios.

An alloy has a ratio of $\Tblue{\text{copper}}$ to $\Tgreen{\text{tin}}$ of $\Tblue{36}:\Tgreen{17}$ and another has a ratio of $\Tblue{20}:\Tgreen{13}$. The second alloy has more $\Tgreen{\text{tin}}$ because $\Tblue{36}:\Tgreen{17} = \Tblue{1}:\frac{\Tgreen{17}}{\Tblue{36}} = \Tblue{1}:\Tgreen{0.47},\qquad \Tblue{20}:\Tgreen{13} = \Tblue{1}:\frac{\Tgreen{13}}{\Tblue{20}} = \Tblue{1}:\Tgreen{0.65}.$

Brass is an alloy of copper and zinc. Different types of brass use the metals in different ratios.

## Determining quantities from ratios

We show by examples how to determine quantities from ratios.

• Finding one quantity from the other. In a class with a ratio of boys to girls of $\Tblue{2}:\Tgreen{3}$, there are $\Tblue{12}$ boys. The number of girls is $\frac{\Tblue{12}}{\Tblue{2}}\times\Tgreen{3}= 6\times\Tgreen{3} = \Tgreen{18}.$
• Splitting quantities. There are $\Torange{45}$ pupils in the class. The proportion of boys is $\displaystyle\frac{\Tblue{2}}{\Tblue{2}+\Tgreen{3}} = \frac{\Tblue{2}}{\Torange{5}}$. The number of boys is $\frac{\Torange{45}}{\Torange{5}}\times\Tblue{2} = 9\times\Tblue{2} = \Tblue{18}.$
• Ratio with more than two numbers. A sum of $\Torange{20}$ euros is to be split between John, Mark and Paul in a ratio $\Tblue{2}:\Tgreen{7}:\Tviolet{1}$. They will receive in euros \begin{align*} &\frac{\Torange{20}}{\Tblue{2}+\Tgreen{7}+\Tviolet{1}}\times\Tblue{2} = \frac{\Torange{20}}{\Torange{10}}\times\Tblue{2} = 20\times\Tblue{2} =\Tblue{40},\\ &\quad\frac{\Torange{20}}{\Torange{10}}\times\Tgreen{7} = \Tgreen{140},\quad \frac{\Torange{20}}{\Torange{10}}\times\Tviolet{1} = \Tviolet{20}. \end{align*}

## Map scales

Maps are drawn by reducing real distances by a scale. The scale is represented as a ratio $1:n$ between the map and the original.

In a map, 2 cm corresponds to a distance of 10 km in reality.

To get the scale, we need to convert the lengths so that they have the same units.

As $10\ukm = 10,\!\Tred{000}\um = 1,\!0\Tred{000,\!0}\Tblue{00}\ucm$, the scale of the map is $\Tblue{2}:\Tgreen{1,\!000,\!000} = \Tblue{1} : \Tgreen{50,\!000}.$

The scale applies to distances. It needs to be squared to apply to areas. So if the length scaling is $1:n$, the area scaling is $1:n^\Tred{2}$.

The area scaling for the above map is $1 : 50,\!000^\Tred{2} = 1:(5\times 10^{4})^\Tred{2} = 1:25\times 10^{8}$

An example of a map scale bar showing both kilometres and miles.