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**

Follow the below steps:**(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

Follow the below steps:**(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.