**Ratios **

**Ratios** are a way of expressing one thing compared to another, if x:y is in the ratio of 1:2 this means y is twice the size of x. There are **8** skills involving ratios you need to learn.

Make sure you are happy with the following before continuing:

**Skill 1:** Writing** Ratios as Fractions**

Ratios can be written as a fraction in a couple of ways.

**Example:** Red to blue counters are in a bag in the ratio (\textcolor{red}{3} :\textcolor{blue}{2} ).

There are \dfrac{\textcolor{red}{3}}{\textcolor{blue}{2}} as many red counters as blue counters.

There are \dfrac{\textcolor{blue}{2}}{\textcolor{red}{3}} as many blue counters as red counters.

Alternatively, we can write either part as a fraction of the total i.e. \dfrac{\textcolor{blue}{2}}{\textcolor{black}{5}} of the counters in the bag are blue.

**Skill 2:** Simplifying

To **simplify** a ratio we **divide** all parts of the ratio by a common factor.

**Example:** Write the following ratio in its simplest form, \textcolor{red}{15}:\textcolor{blue}{30}:\textcolor{limegreen}{24}

All we do is divide each number by the highest common factor of all three numbers, which is 3.

We can not simplify this any more, therefore the ratio is in its **simplest form**.

**Skill 3:** Scaling** Ratios**

To **scale** a ratio we **multiply** by a common factor.

**Example:** Meringue is made by mixing cups of egg whites and cups of sugar in the ratio \textcolor{limegreen}{3}:\textcolor{blue}{7}. How many cups of sugar are needed if \textcolor{limegreen}{12} cups of egg whites are used in the mixture? We know that \textcolor{limegreen}{12} = \textcolor{limegreen}{3} \times \textcolor{black}{4}, so we need to multiply the ratio by \textcolor{black}{4}

So when \textcolor{limegreen}{12} egg whites are used, \textcolor{blue}{28} cups of sugar are needed.

**Skill 4:** Part** to Whole Ratios**

Sometimes you may see ratios x:y where y includes x. These are part : whole ratios.

**Example:** Adam has some **apples** and **oranges** in his bag. The ratio of **oranges** to **fruit** in his bag is \textcolor{orange}{2}:\textcolor{blue}{7}.

**a) Finding the fraction of a whole – **What fraction of Adam’s fruit are oranges?

For every \textcolor{blue}{7} pieces of fruit, \textcolor{orange}2 of these are oranges.

\large{\frac{\text{part}}{\text{whole}} = \frac{2}{7}}

**b) Finding the ratio of a part – **What is the ratio of oranges to apples?

\textcolor{orange}{2} out of every \textcolor{blue}{7} pieces of fruit are oranges, so \textcolor{blue}{7} - \textcolor{orange}{2} = \textcolor{limegreen}{5} out of every \textcolor{blue}{7} pieces of fruit are apples. For every \textcolor{orange}{2} oranges there are \textcolor{limegreen}{5} apples

\text{\textcolor{Orange}{oranges} : \textcolor{limegreen}{apples}} = \textcolor{orange}{2} : \textcolor{limegreen}{5}

**c) Finding the missing amount –** Adam has \textcolor{orange}{4} oranges, how many apples does he have?

We need to scale up the ratio \textcolor{orange}{2}:\textcolor{limegreen}{5}, so that the left is equal to \textcolor{orange}{4}. So we multiply the ratio by \textcolor{black}{2}.

So, Adam has \textcolor{limegreen}{10} apples.

**Skill 5:** Dividing** amounts into Ratio**

Being able to split a total amount into a ratio is a key skill needed.

**Example:** Aaron, Kim and Paul split \textcolor{purple}{£6000} in the ratio of \textcolor{red}{3}:\textcolor{limegreen}{4}:\textcolor{blue}{5}. How much money does Aaron receive?

**Step 1:** Find the total number of parts in the ratio:

\textcolor{red}{3}+\textcolor{limegreen}{4}+\textcolor{blue}{5} = \textcolor{black}{12} parts.

**Step 2:** Divide the total amount by the total number of parts in ratio, this finds the value of **1 part**.

\textcolor{purple}{£6000} \div \textcolor{black}{12} = \textcolor{orange}{£500} = \, **1 part**

**Step 3:** Multiply the value of one part by the number of parts Aaron has:

\textcolor{orange}{£500} \times \textcolor{red}{3} = £1500

So Aaron gets £1500.

**Skill 6:** Difference** between Parts of a ratio**

You may sometimes be given the **difference** between two parts of the ratio, instead of the total amount.

**Example:** Josh, James and John share sweets in the ratio \textcolor{orange}{1}:\textcolor{blue}{2}:\textcolor{red}{4}. Josh has \textcolor{limegreen}{9} sweets less than John. How many sweets does James have?

**Step 1:** Firstly, work out how many parts of the ratio \textcolor{limegreen}{9} sweets makes up:

\textcolor{limegreen}{9} \, \text{sweets} = \text{John's sweets} - \text{Josh's sweets} = \textcolor{red}{4} \, \text{parts} - \textcolor{orange}{1} \, \text{part} = 3 \, \text{parts}.

**Step 2:** Then divide to find 1 part:

\textcolor{limegreen}{9} \, \text{sweets} \div 3 = \textcolor{purple}{3} \, \text{sweets}

\textcolor{purple}{3} \text{ sweets } = 1 \text{ part }

**Step 3:** Multiply by the number of parts James has to find how many sweets he has:

**Skill 7:** Changing** Ratios**

You need to be prepared for questions where the ratio **changes**.

**Example:** Billy and Claire share marbles in the ratio \textcolor{blue}{5}:\textcolor{limegreen}{3}. Billy gives \textcolor{orange}{4} marbles to Claire and the ratio is now 1:1. How many sweets did each have initially?

**Step 1:** At first, Billy had \textcolor{blue}{5}x marbles and Claire had \textcolor{limegreen}{3}x marbles.

**Step 2:** Billy gives \textcolor{orange}{4} marbles to Claire. Now, Billy has \textcolor{blue}{5}x-\textcolor{orange}{4} marbles and Claire has \textcolor{limegreen}{3}x+\textcolor{orange}{4}.

**Step 3:** The ratio \textcolor{blue}{5}x-\textcolor{orange}{4} : \textcolor{limegreen}{3}x+\textcolor{orange}{4} is 1:1.

\dfrac{\textcolor{blue}{5}x-\textcolor{orange}{4}}{\textcolor{limegreen}{3}x+\textcolor{orange}{4}} = \dfrac{1}{1}

Rearranging and solving for x,

\begin{aligned} (\textcolor{blue}{5}x-\textcolor{orange}{4})& = (\textcolor{limegreen}{3}x+\textcolor{orange}{4}) \\ \textcolor{blue}{5}x &= \textcolor{limegreen}{3}x+\textcolor{orange}{8} \\ 2x &= \textcolor{orange}{8} \\ x &= 4 \end{aligned}

Initially, Billy had \textcolor{blue}{5x = 20} marbles and Claire had \textcolor{limegreen}{3x = 12}.

## Skill 8: Reducing Ratios to the form 1 : n

To reduce a ratio to the form 1:n or n:1, all you have to do is divide the whole ratio by the smallest number.

## Example 1: **Scaling and Simplifying Decimal Ratios:**

Write the following ratio in its whole number simplest form.

**[2 marks]**

First, we need to multiply all parts of the ratio until there are only whole numbers left before simplifying.

**Example 2: Simplifying Ratios that have Different Units:**

Change the following ratio into the same unit ratio in its simplest form.

**[2 marks]**

If ratios have **different** units, we need to convert one of the units to the other, then simplify the ratio to its **simplest form**.