In this post we will try to understand the **concept of equivalent ratios** with examples.

The concept requires basic understanding of ratios and fraction manipulation so ensure you have completed the previous chapters.

In this post we will first learn to identify the equivalent ratios and then will learn to create equivalent ratios from given ratio.

**What are Equivalent ratios?**

Two or more **ratios having the same lowest terms** are** equivalent ratios**.

**Ratios can be simplified into lowest terms** using fraction manipulations.

So the ratios, which after simplification have similar values are called equivalent ratios.

**How to find if ratios are equivalent?**

To check if the ratios are equivalent or not, do the following steps:

(a) **Convert all the ratios into fractions**.

(b) For a fraction, find the common factor present in numerator and denominator.

This can be done by finding **HCF of numerator and denominator**.

(c) **Divide the numerator and denominator by HCF **number.

(d) Now the fraction has been reduced to lowest terms.**Compare all the fractions** and check if the values are equal or not.

**Example 01**

Are ratios 1 : 2 and 2 : 4 equivalent ?

**Solution**

Follow the below steps:**(a) Convert the ratios into fraction**

1 : 2 ⟹ 1 / 2

2 : 4 ⟹ 2 / 4**(b) Reduce the fraction to lowest terms****Fraction 1/2**

1 / 2 is already in lowest form **Fraction 2/4**

(i) Take HCF of numerator and denominator

HCF ( 2, 4 ) = 2

(ii) Divide numerator and denominator by 2

\mathtt{\frac{2\ \div \ 2}{4\ \div \ 2} \ \Longrightarrow \frac{1}{2}}

Fraction 2/4 is reduced to 1/2

**(c) Compare the fractions**

Both the fractions reduced to value 1/2.

Hence the given ratio is equivalent ratio.

**Example 02**

Check if the below ratios are equivalent or not?

4 : 16 and 12 : 48

**Solution**

Follow the below steps:**(a) Convert the given ratios into fractions**

4 : 16 ⟹ 4 / 16

12 : 48 ⟹ 12 / 48

**(b) Reduce all the fractions to its lowest term**s**Fraction 4 : 16**

(i) Find HCF of numerator and denominator

HCF ( 4, 16 ) = 4

(ii) Divide numerator and denominator by HCF = 4

\mathtt{\frac{4\ \div \ 4}{16\ \div \ 4} \ \Longrightarrow \frac{1}{4} \ }

**Fraction 12 : 48**

(i) Find HCF of numerator and denominator

HCF ( 12, 48 ) = 12

(ii) Divide numerator and denominator by HCF = 12

\mathtt{\frac{12\ \div \ 12}{48\ \div \ 12} \ \Longrightarrow \frac{1}{4} \ }

**(c) Compare the fractions**

Here both the ratios have been reduced to value 1/4.

Hence the given ratios are equivalent ratios.

**Example 03**

Check if the given ratios are equivalent or not

15 : 25 and 30 : 50

**Solution**

Follow the below steps:

**(a) Convert all the ratios into fractions**

15 : 25 ⟹ 15 / 25

30 : 50 ⟹ 30 / 50

**(b) Reduce the fraction to its lowest terms**

**Fraction 15/25**

(i) Find HCF of numerator and denominator

HCF ( 15, 25 ) = 5

(ii) Divide numerator and denominator by HCF = 5

\mathtt{\frac{15\ \div \ 5}{25\ \div \ 5} \ \Longrightarrow \frac{3}{5} \ }

**Fraction 30 / 50**

(i) Find HCF of numerator and denominator

HCF (30, 50) = 10

(ii) Divide numerator and denominator by 10

\mathtt{\frac{30\ \div \ 10}{50\ \div \ 10} \ \Longrightarrow \frac{3}{5} \ }

**(c) Compare the fractions**

Both the ratios are reduced to fraction 3/5.

Hence, the given ratios are equivalent ratios.

**Example 04**

Check if the below ratios are equivalent or not

5 : 4 and 16 : 12

**Solution**

Follow the below steps:**(a) Convert the ratios into fraction**

5 : 4 ⟹ 5 / 4

16 : 12 ⟹ 16 / 12

**(b) Reduce the fraction to its lowest terms**

**Fraction 5/4**

The fraction is already in reduced form.

**Fraction 16/12**

(i) Find HCF of numerator and denominator

HCF (16, 12 ) = 4

(ii) Divide numerator and denominator by 4

\mathtt{\frac{16\ \div \ 4}{12\ \div \ 4} \ \Longrightarrow \frac{4}{3} \ }

**(c) Compare the fractions**

Both the fractions have been reduced to its lowest forms.

5 : 4 ⟹ 5 / 4

16 : 12 ⟹ 4 / 3

Both the values are different.**Hence the given ratios are not equivalent.**

**Example 05**

Check if the ratios are equivalent or not

11 : 13 and 33 : 39

**Solution**

Follow the below steps:**(a) Convert the ratio into fraction**

11 : 13 ⟹ 11 / 13

33 : 39 ⟹ 33 / 39

**(b) Reduce the fraction into lowest terms****Fraction 11/13**

The fraction is already reduced to lowest terms.**Fraction 33/39**

(i) Find HCF of numerator and denominator

HCF ( 33, 39 ) = 3

(ii) Divide numerator and denominator by 3

\mathtt{\frac{33\ \div \ 3}{39\ \div \ 3} \ \Longrightarrow \frac{11}{13}}

**(c) Compare the fractions**

We have converted the ratios into lowest terms.

11 : 13 = 11/13

33 : 39 = 11/13

Both the fractions are same.

Hence, the given ratios are equivalent.

**How to create equivalent ratios?**

From the given ratio, you can create equivalent ratio by **multiplying/dividing the ratio with any possible number**.

All the ratios developed by multiplication/division have the same lowest terms, that’s why they are also equivalent ratio.

Given below are some examples for your understanding.

**Example 01**

Find two equivalent ratio of 2 : 3

**Solution**

First convert the ration into fraction.

2 : 3 ⟹ 2 / 3

To get equivalent ratio, just multiply the fraction with any possible number.

**(i) Multiply the numerator and denominator by 2**

\mathtt{\frac{2\ \times \ 2}{3\ \times \ 2} \ \Longrightarrow \frac{4}{6} \ }

Here we get the ratio 4 : 6

**(ii) Now multiply numerator and denominator by 3.**

\mathtt{\frac{2\ \times \ 3}{3\ \times \ 3} \ \Longrightarrow \frac{6}{9} \ }

Here we get the ratio 6 : 9.

Both the above ratios 4 : 6 and 6 : 9 are equivalent ratios as they are derived from same base ratio 2 : 3.

Also, if desired, both the ratios can be reduced to its same lowest form 2 : 3.

**Example 02**Find three equivalent ration of 5 : 3

**Solution**

First convert the ration into fraction form

5 : 3 ⟹ 5 / 3

**(i) Multiply numerator and denominator by 4**

\mathtt{\frac{5\ \times \ 4}{3\ \times \ 4} \ \Longrightarrow \frac{20}{12}}

Here we get the ratio 20 : 12.

**(ii) Multiply numerator and denominator by 6**

\mathtt{\frac{5\ \times \ 6}{3\ \times \ 6} \ \Longrightarrow \frac{30}{18}}

Here we get the ratio 30 : 18.

**(iii) Multiply numerator and denominator by 7.**

\mathtt{\frac{5\ \times \ 7}{3\ \times \ 7} \ \Longrightarrow \frac{35}{21}}

Here all the ratios 20 : 21, 30 : 18 and 35 : 21 are equivalent ratios since they are derived from the same base ratio 5 : 3.

Also, if desired, all the ratios can be converted back into the base ratio 5 : 3.