## Simplifying ratios

A ratio is a simple way of comparing two quantities, often showing amounts, and is written with a colon between two numbers (such as 1:2).

A ratio of 1:2 could show that for every 1 apple in a fruit basket there are 2 bananas.

When ratios are written with exact figures, they are often too large to understand and need simplifying. For example, comparing the population of London to Manchester would create a raw ratio of 10,000,000: 500,000.

A raw ratio of 10,000,000: 500,000 would be easier to understand if it was simplified to 20:1, showing that for every 20 people that live in London, there is 1 person who lives in Manchester.

## How to simplify a ratio

To simplify a ratio, you need to find the highest common factor (HCF) of the two numbers in the ratio.

A factor is any number that can be multiplied to equal your original number without leaving a remainder.

For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. The highest common factor is the largest factor which is the same for both numbers.

For example, 15 has the factors 1, 3, 5 and 15. Both 12 and 15 have 1 and 3 as factors, as 3 is the larger number it would be the highest common factor (HCF).

Once you have worked out the HCF, you need to divide both numbers by it.

If the original ratio that you were trying to simplify was 12:15, you would divide both numbers by 3 to create a new simplified ratio.

12 divided by 3 = 4.

15 divided by 3 = 5.

Therefore, the simplified ratio is 4:3.

### Example question 1: How do you simplify the ratio 700:500 to its simplest form?

To start with you need to find the HCF. The best way to do this is to write out all the factors of both 700 and 50.

The factors of 700 are: 1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350 and 700.

The factors of 500 are: 1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, and 500.

1,2,4,5,10, 25, 50 and 100 are factors of both numbers. As 100 is the largest number, it is the HCF.

700 divided by 100 is 7.

500 divided by 100 is 5.

The new simplified ratio is 7:5.

### Example question 2: Big Ben is 96 metres tall. Sam makes a scale model of Big Ben to a ratio of 96: 27. What is the simplest form of this ratio?

First, find the HCF of 96 and 27.

96 has the factors 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 and 96.

27 has the factors 1, 3, 9 and 27.

Both 1 and 3 is a factor of both, therefore 3 is the HCF.

96 divided by 3 is 32.

27 divided by 3 is 9.

The simplified ratio Is 32:9.

## Tips for simplifying ratios

We’ve put together five tips for simplifying ratios that we hope will help support your learning when simplifying ratios.

**Tip 1: Freshen up on your timetables and use that knowledge**

For example, if you know 21 and 14 are both in the 7 times table, you can divide by 7 easily. This would give a new fraction of 3:2 and then you can decide if you need to simplify it further.

If you have 88:44 and know they both can be divided by 11 you’ll save a lot of time - it is easier to list factors of 8 and 4 rather than 8 and 44.

**Tip 2: If both numbers are multiples of ten (end in a zero) always divide by the largest multiple of ten that fits into both numbers.**

For example, if you have 70:50 divide by 10 to make 7:5.

If you have 3000:6000 divided by 1000 to make 3:6 (this can then be simplified again to make 1:2 – don’t forget to check it really is the simplest form).

**Tip 3: Always write all your factors out to make sure you don’t miss any numbers out.**

For example, if you were simplifying 60:40 you might write all your factors out first like this:

60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60

40: 1, 2, 4, 5, 8, 10, 20 and 40.

It is then very clear which is the largest common factor – 10.

You can also underline, highlight, or circle all of the factors that appear in both numbers to guarantee you find the HCF.

**Tip 4: Keep an eye out for prime numbers (numbers that are only divisible by themselves and 1).**

If you have a prime number in your ratio, you will not be able to simplify it. The first 5 prime numbers are 2,3,5,7,11.

**Tip 5: Practice other forms of simplifying**

You might consider trying other forms of simplification such as simplifying ratios to fractions.