# Ratio

## In a nutshell

Ratios are used to describe how something is divided between two or more categories. You need to know how to read, interpret and simplify ratios, as well as solve problems involving ratios.

### Definition

A **ratio*** *is a piece of information that compares quantities.

## Representing ratios

A ratio is written using colons (:) to separate and compare numbers. When reading ratios, read the colons as "to". For example, the ratio $2:3$ is read as "two to three".

## Interpreting ratios

Ratios are best understood through examples.

##### Examples

- If a recipe says to mix flour and water in the ratio $3:1$, it means for every $3$ cups of flour you add, you have to also add $1$ cup of water.
- If money between three friends Adam, Becky and Claire is divided in the ratio $1:2:3$, it means that for every $1$ pound that Adam gets, Becky gets $2$ pounds and Claire gets $3$ pounds.
- If a class has boys and girls in the ratio $4:3$, it means that for every $4$ boys in the class, there are also $3$ girls.
- If the ratio of chairs to tables in a room is $1:1$, it means that for every chair, you also have a table. In other words, there is an equal amount of tables and chairs.

## Simplifying ratios

It is possible to simplify ratios to their lowest form, much like fractions. However, it is first important to understand how two ratios can be equal.

### Equal ratios

If all the numbers in a ratio are multiplied or divided by the same number, then the value of the ratio doesn't change.

##### Examples

- $4:5=(4\times2):(5\times2)=8:10$

- $9:30=(9\div3):(30\div3)=3:10$
- $11:2:6=(11\times4):(2\times4):(6\times4)=44:8:24$
- $4:5=(4\div4):(5\div4)=1:1.25$

### Simplifying a ratio to its simplest form

#### Procedure

- Find a number that all numbers in the ratio can be divided by. This is called a
**common factor**.

- Divide all the numbers in the ratio by this common factor.

- Keep doing this until the ratio cannot be simplified any further. This is called the
**simplest form** of the ratio.

##### Example 1

*Simplify the ratio $10:6:16$ to its simplest form.*

*$10$, $6$, and $16$ are all even numbers. So, they are all divisible by $2$. Hence, $2$ is a common factor. *

*Divide all three numbers in the ratio by $2$:*

$10\div2=5$

$6\div2=3$

$16\div2=8$

*Therefore:*

$10:6:16=5:3:8$

*The numbers $5$, $3$ and $8$ have no common factors, so this ratio cannot be simplified further. Therefore:*

$\underline{10:6:16=5:3:8}$* in its simplest form.*

## Ratio problems - scaling up

Ratios can be used to calculate how much of a certain quantity is needed in relation to another given quantity.

##### Example 2

*The ratio of boys to girls in a classroom is $4:3$*. *If there are $20$ boys in this class, how many girls are there?*

*Call the number of girls in the class $x$. The $20$ boys and $x$ girls must be in the ratio $4:3$. Therefore:*

$4:3=20:x$

*Find what number is multiplied by $4$ that gives $20$:*

$20\div4=5$

*Multiply $3$ by $5$ to get the number of girls in the class:*

$4:3=(4\times5):(3\times5)=20:15$

$x=15$

*There are *$\underline{15}$ *girls in the class.*

## Ratio problems - sharing

Another common type of ratio question is to divide and share a quantity into a given ratio. With these problems, it's easiest to think of "parts".

##### Example 3

*$£150$ is split between two friends Carl and Daniel in the ratio $7:3$. How much money does Carl get?*

*Calculate the total number of parts by adding the numbers in the ratio:*

*Carl gets $7$ "parts" of the money, and Daniel will get $3$ "parts".*

$7$ *parts + $3$ parts = $10$ parts in total.*

*Find the value of $1$ part:*

*If $10$ parts $=£150$, then $1$ part = $£150\div10=£15$.*

*Find how much money Carl gets:*

*If $1$ part = $£15$, and Carl gets $7$ parts, then Carl gets $£15\times7=£105$.*

*Carl gets *$\underline{£105.}$* *