Simplifying Ratios

Here we will learn about simplifying ratios, including how to simplify a ratio, how to write equivalent ratios and using simplifying ratio in real life problems.

There are also ratio worksheets and word problems based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is simplifying ratios?

Simplifying ratios is a way of using common factors to divide all the numbers in a ratio until they cannot be divided further.

A simplified ratio is equivalent to an non simplified ratio.

E.g.

Let’s look at katex is not defined red counters and katex is not defined blue counters.

If we organised them into a the two colours, we can get something that looks like this:

Each of the four rows contains katex is not defined red counters and katex is not defined blue counters. This allows us to express the ratio katex is not defined as the ratio in the simplest form katex is not defined.

We can simplify the ratio by finding the highest common factor or HCF (otherwise known as the greatest common factor or GCF) of each of the numbers in the ratio.

For the example above, the highest common factor would be katex is not defined as we can divide katex is not defined and katex is not defined by katex is not defined.

Bar modelling

We can use bar modelling to help us simplify ratios.

E.g.

Here are some circles.

Rearranging them into coloured bars, we can show the number of orange circles and the number of purple circles.

The number of orange circles is katex is not defined, and the number of purple circles is katex is not defined. This can be written as the ratio katex is not defined

We can also rearrange these circles into another pattern:

Here, we still have the ratio of katex is not defined but we have arranged them into two rows, each with the same proportion of circles.

Each row contains katex is not defined orange circles, and three purple circles. This means that we can divide each side of the ratio by katex is not defined, and still obtain the same ratio of orange to purple.

So by using bar modelling, we can simplify the ratio katex is not defined to get katex is not defined

How to simplify a ratio using the HCF

The fastest way to simplify a ratio is to use the highest common factor of the numbers in the ratio.

For example

If you are given a ratio of katex is not defined to simplify it: first find the highest common factor which is katex is not defined next divide each part of the ratio by the highest common factor katex is not defined which gives you the answer katex is not defined

Ratio Simplifier

We can summarise simplifying ratios using this formula

The simplified ratio katex is not defined

For example.

Simplify katex is not defined

katex is not defined of katex is not defined

katex is not defined

What is simplifying ratios?

How to simplify a ratio

In order to simplify a ratio:

1. Calculate the highest common factor of the parts of the ratio.
2. Divide each part of the ratio by the highest common factor.

How to simplify a ratio

Related lessons on ratio

Simplifying ratios is part of our series of lessons to support revision on ratio. You may find it helpful to start with the main ratio lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Ratio
• How to work out ratio 
• Ratio to fraction
• Dividing ratios
• Ratio problem solving
• Ratio scale
• Ratio to percentage

Simplifying ratios examples

Example 1: standard question

Simplify the ratio

1. Calculate the highest common factor of the parts of the ratio.

Listing the factors of katex is not defined and katex is not defined, we have:

2Divide each part of the ratio by the highest common factor.

Solution:

Example 2: three part ratio

Simplify the following ratio

Divide each part of the ratio by the highest common factor.

Show step

Solution:

katex is not defined

Example 3: ratio including decimals

Simplify the following ratio

Divide each part of the ratio by the highest common factor.

Show step

Solution:

katex is not defined

Example 4: fractions

Simplify the ratio

katex is not defined

Divide each part of the ratio by the highest common factor.

Show step

Solution:

katex is not defined

Example 5: mixed numbers

Simplify the ratio

Calculate the highest common factor of the parts of the ratio.

Show step

As we have two mixed numbers, we can convert them to improper fractions or decimals. We will look at converting the mixed numbers to decimals and then simplify the ratio.

Converting both of the mixed numbers to decimals we have katex is not defined s the same as katex is not defined

Multiplying both of these parts of the ratio by katex is not defined will quickly remove the decimal so we can then simplify the ratio.

katex is not defined

Calculating the highest common factor of katex is not defined and katex is not defined quickly by using prime factor trees, we get:

katex is not defined

katex is not defined

From this, we can show that the highest common factor of katex is not defined and katex is not defined is katex is not defined

katex is not defined

Divide each part of the ratio by the highest common factor.

Show step

Solution:

katex is not defined

Example 6: algebraic terms

Given that katex is not defined and katex is not defined are all positive integers, simplify the ratio

Divide each part of the ratio by the highest common factor.

Show step

Solution:

katex is not defined

Example 7: bar modelling

Use a bar model to simplify the ratio katex is not defined

Calculate the highest common factor of the parts of the ratio.

Show step

Representing the ratio katex is not defined as a bar model, we have

Where red represents katex is not defined shares, and purple represents katex is not defined shares.

katex is not defined

Divide each part of the ratio by the highest common factor.

Show step

As the highest common factor is katex is not defined, we divide the bar above into two bars containing the same amount of red shares, and purple shares.

Each bar is the ratio katex is not defined

Solution:

katex is not defined

Common misconceptions

• Not fully simplified

A common error is that the ratio is not cancelled to its simplest form due to not using the highest common factor,

E.g.

Simplify the ratio katex is not defined

We can simplify by dividing parts of the ratio by dividing both numbers by katex is not defined, leaving the ratio katex is not defined.

This is not the final answer as it can be simplified further by dividing by katex is not defined to get the ratio katex is not defined

• Solution not written in the correct order

A common error is to change the order of the ratio. It is important that the order of the numbers in the ratio remains the same while simplifying.

• Not simplifying all parts of the ratio

When simplifying a ratio with more than two parts, a common misconception is to not simplify all of the parts of the ratio. 

E.g.

“Simplify the ratio katex is not defined” could be incorrectly written as katex is not defined as the middle part is not cancelled down.

• Incorrect division

Some numbers are more tricky to divide than others, especially when calculating it mentally.

Common incorrect calculations include katex is not defined and katex is not defined

• Addition and subtraction?

We cannot use addition and subtraction to simplify a ratio, we can only use multiplication and division.

Let’s look at why using a visual representation:

Below are katex is not defined rectangles katex is not defined and katex is not defined stars katex is not defined.

If we were to express the number of stars and rectangles as a ratio, we would write the ratio katex is not defined. Remember that the order is important and so we have written the number of stars first and the number of rectangles second katex is not defined.

Dividing the number of rectangles by the number of stars we get katex is not defined.

Here the number katex is not defined is known as the constant of proportionality (Step-by-step guide: Direct proportion).

Now if we were to subtract katex is not defined from each of the stars and the rectangles above and divide the quantities again the answer should be katex is not defined if the ratio was equivalent.

Subtracting katex is not defined from each we now have katex is not defined stars katex is not defined and katex is not defined rectangles katex is not defined. I katex is not defined.

As this is not katex is not defined the ratios are not equivalent.

• Ratios and decimals

Simplified ratios should not include decimals.

E.g.

If we want to simplify the ratio katex is not defined, we need to divide each quantity in the ratio by a common factor. 

Here we have simplified by a common factor of katex is not defined, and then simplified again by dividing these new numbers by katex is not defined.

When simplifying a ratio, your solution must contain whole numbers (integers)

So the correct simplified ratio is:

The next lessons are

• Proportion
• Compound measures
• Best buy maths

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