# Simplifying ratios || How to reduce ratios to lowest terms

Most of the times the ratios contain big numbers which causes difficulty in further calculation.

We know that ratios can be written in the form of fractions. So using the basic fractions rule, it is possible to simplify ratios to small number using basic division technique.

In this post we will learn methods of simplifying ratios to its lowest terms.

To understand the chapter fully, you should have basic understanding of ratios, fractions and Greatest Common Factors (GCF)

## Methods of Simplifying ratios

In order to simplify the ratios we have to find the common factors present between the given numbers.

The common factors can be calculated by finding HCF of the given numbers.

### Ratio Simplification using HCF

Given below are the steps to simplify ratios:

(a) Convert the ratio in the form of fraction.

(b) Find the common factors between numerator and denominator by finding HCF.

(c) Divide both numerator and denominator by the calculated HCF.

I hope the above steps are clear to you.
Given below are some examples for further explanation.

Example 01
Simplify the ratio 28 : 12

Solution

(a) Write the ratio in the form of fractions
28 : 12 ⟹ 28/12

(b) Use GCF to find common factors present in numerator and denominator

HCF (28, 12) = 4

Hence, 4 is highest factor of both 12 and 28.

With factor 4, the fraction can be expressed as:

\mathtt{\frac{28}{12} \Longrightarrow \ \frac{4\ \times \ 7}{4\ \times \ 3} \ }

(c) Divide numerator and denominator by 4

\mathtt{\frac{28}{12} \Longrightarrow \ \frac{( 4\ \times \ 7) \div 4}{( 4\ \times \ 3) \div 4} \ \Longrightarrow \frac{7}{3} \ }

After division we get 7/3 as fraction.
Hence, the ratio 28 : 12 is simplified to ration 7 : 3.

Example 02
Simplify the ratio 18 : 40

(a) Convert ratio into fractions
18 : 40 ⟹ 18/40

(b) Find common factor between numbers using HCF

HCF (18, 40) = 2

Hence, number 2 is the highest common factor present in both numerator and denominator.

(c) Divide numerator and denominator by 2

\mathtt{\frac{18\ \div \ 2}{40\ \div \ 2} \ \Longrightarrow \frac{9}{20}}

Hence, ratio 18 : 40 is reduced to 9 : 20.

Example 03
Simplify the ratio 12 : 96

(a) Convert the ratio into fraction

12 : 96 ⟹ 12 / 96

(b) Find the common factors using HCF

HCF (12, 96) = 12

Hence, 12 is the highest factor present in both the given numbers.

(c) Divide numerator and denominator by 12

\mathtt{\frac{12\ \div \ 12}{96\ \div \ 12} \ \Longrightarrow \frac{1}{8} \ }

Hence the ratio 12 : 96 is simplified to 1 : 8

Example 04
Simplify the ratio 2 : 4 : 16

Solution
Here ratio of three numbers are given.
There is no need to convert the numbers into fraction.
Just find the common factor and divide the given numbers with it.

(a) Find the HCF of all numbers.

HCF (2, 4, 16) = 2

So, 2 is the highest factor present in all the numbers.

(b) Divide all the numbers in ratio by 2

\mathtt{\Longrightarrow \ \frac{2}{2} \ :\ \frac{4}{2} \ :\ \frac{16}{2}}\\\ \\ \mathtt{\Longrightarrow \ 1\ :\ 2:\ 8}

Hence, the ratio 2: 4: 16 is reduced to 1 : 2 : 8

Example 05
Simplify the given ratio; 5 : 25 : 55

Solution
(a) Find the common factors of numbers using HCF

HCF ( 5, 25, 55) = 5

(b) Divide all the number in ratios by 5

\mathtt{\Longrightarrow \ \frac{5}{5} \ :\ \frac{25}{5} \ :\ \frac{55}{5}}\\\ \\ \mathtt{\Longrightarrow \ 1\ :\ 5:\ 11}

Hence, the ratio 5 : 25 : 55 is reduced to 1 : 5 : 11

### Can we simplify ratios without using HCF?

Yes!!
If you are good in division calculations, you can divide the fractions directly.

The trick is to divide the given numbers with prime numbers starting with 2, 3, 5, 7 . . . etc. till no common factors are left.

For Example
Simplify the ratio 8 : 20

Solution
First convert the ratio in form of fraction.
8 : 20 ⟹ 8 / 20

(a) Both numerator and denominator is divisible by 2

8 ÷ 2 = 4
20

8 ÷ 2 = 4
20 ÷ 2 = 10

(b) Again numerator and denominator are divisible by 2
4 ÷ 2 = 2
10 ÷ 2 = 5

Now there is no common factor between 2 & 5.

Hence ratio 8 : 20 is reduced to 2 : 5.

Example 02
Simplify the ratio 15 : 75

Solution
Write the ratio in the form of fraction.
15 : 75 ⟹ 15 / 75

(a) Both numerator and denominator are divisible by 3
15 ÷ 3 = 5
75 ÷3 = 25

(b) Both numerator and denominator is divisible by 5
5 ÷ 5 = 1
25 ÷ 5= 5

Now the fraction 1/5 cannot be divided further.

Hence ratio 15 : 75 has been simplified to 1 : 5.

Example 03
Simplify the ratio 20 : 140

Solution
Write the ratio in form of fraction
20 : 140 ⟹ 20/140

(a) Both numerator and denominator are divisible by 2

20 ÷ 2 = 10
140 ÷ 2 = 70

(b) Again numerator and denominator is divisible by 2

10 ÷ 2 = 5
70 ÷ 2 = 35

(c) The numerator and denominator is divisible by 5

5 ÷ 5 = 1
35 ÷ 5 = 7

Hence, the ratio 20 : 140 is simplified to 1 : 7.

Example 04
Simplify the ratio 4 : 16 : 100

(a) All the numbers are divisible by 2

4 ÷ 2 = 2
16 ÷ 2 = 8
100 ÷ 2 = 50

(b) Again all the numbers are divisible by 2

2 ÷ 2 = 1
8 ÷ 2 = 4
50 ÷ 2 = 25

Now there is no common factor left between 1 : 4 : 25.

Hence ratio 4 : 16 : 100 is reduced to 1 : 4 : 25.

Example 05
Simplify the ratio 30 : 45 : 90

(a) Check if all numbers are divisible by 2
30 & 90 is divisible by 2.
But 45 is not divisible.

Move on to check the divisibility by 3

(b) All numbers are divisible by 3

30 ÷ 3 = 10
45 ÷ 3 = 9
90 ÷ 3 = 30

Now there is no common factor left between numbers.

Hence, ratio 30 : 45 : 90 is simplified to 10 : 9 : 30

Example 06
Simplify ratio 11 : 10

Solution
The ratio cannot be further simplified as there is no common factor between numbers.