# Equivalent Ratios

Equivalent ratios are ratios having the same value when they are simplified. The equality of two ratios is termed proportion; thus, equivalent ratios are always in proportion. In other words, two ratios are equivalent if one can be expressed as the multiple of the other.

Thus, in equivalent ratios, the corresponding numbers of the two ratios are not the same. However, after simplification, we get the same value. It is similar to finding equivalent fractions.

For example, 1:3 and 3:9 and 2:4 and 12 are equivalent ratios.

## How to Find Equivalent Ratios

To identify if the given ratios are equivalent, we should reduce them to their simplest form. If the simplified forms are the same, the ratios are equivalent.

However, for finding equivalent ratios, there are two methods. Let us determine whether ratios 8:5 and 24:15 are equivalent or not.

### Cross-Multiplication Method

Step 1: Writing both the ratios in fractional form, numerator over denominator

${\dfrac{8}{5},\dfrac{24}{15}}$

Step 2: Doing cross multiplication

8 Ã— 15 = 5 Ã— 24

Step 3: Checking if the products are the same. If same, the ratios are equivalent.

As

8 Ã— 15 = 5 Ã— 24 = 120

Hence, 8:5 and 24:15 are equivalent ratios.

### HCF Method

Step 1: Finding the HCF of the antecedent and consequent of the ratios

Here, the HCF of 8 and 24 is 8, and the HCF of 5 and 15 is

Step 2: Dividing the terms in both ratios by their corresponding HCF

Thus, (8 Ã· 8): (24 Ã· 8) and (5 Ã· 5): (15 Ã· 5)

Step 3: If the reduced forms of both ratios are equal, it means they are equivalent

(8 Ã· 8): (24 Ã· 8) = 1:3 and (5 Ã· 5): (15 Ã· 5) = 1:3

1:3 = 1:3

Hence, 8:5 and 24:15 are equivalent ratios.

## Solved Examples

What ratio is equivalent to 3:4

Solution:

Let us first write the given ratio as a fraction.
3:4 â‡’ ${\dfrac{3}{4}}$
Now, to find an equivalent ratio to the given ratio, we will multiply each term by a natural number starting from 2.
3:4 = ${\dfrac{3\times 2}{4\times 2}=\dfrac{6}{8}}$
Thus an equivalent ratio to 3:4 is 6:8

Find two equivalent ratios of 4:9

Solution:

Let us first write the given ratio as a fraction.
4:9 â‡’ ${\dfrac{4}{9}}$
Now, to find equivalent ratios to the given ratio, we will multiply the terms by any natural numbers starting from 2.
${\dfrac{4}{9}}$ = ${\dfrac{\left( 4\times 2\right) }{\left( 9\times 2\right) }=\dfrac{8}{18}}$
${\dfrac{4}{9}}$ = ${\dfrac{\left( 4\times 3\right) }{\left( 9\times 3\right) }=\dfrac{12}{27}}$
Hence, two equivalent ratios of 4:9 are 8:18 and 12:27

Compare the given ratios and find whether they are equivalent.

Solution:

8:12, 4:6, 16:24
Let us first write the given ratios as a fraction.
8:12 = ${\dfrac{8}{12}}$
4:6 = ${\dfrac{4}{6}}$
16:24 = ${\dfrac{16}{24}}$
Dividing the terms in all the ratios by their respective HCF
HCF of 8 and 12 is 4
Thus, ${\dfrac{\left( 8\div 4\right) }{\left( 12\div 4\right) }=\dfrac{2}{3}}$
HCF of 4 and 6 are 2
Thus, ${\dfrac{\left( 4\div 2\right) }{\left( 6\div 2\right) }=\dfrac{2}{3}}$
HCF of 16 and 24 is 8
Â Thus, ${\dfrac{\left( 16\div 8\right) }{\left( 24\div 8\right) }=\dfrac{2}{3}}$
Hence, all the above ratios are equivalent as their simplified forms are the same.

Find the value of x if 4:5 is equivalent to 16:x.

Solution:

Given,
4:5 = 16:x
Here, we need to multiply 4 and 5 by the same natural number such that 4 becomes 16 and 5 becomes x
Since 4 Ã— 4 = 16, we need to multiply 5 with 4 to make the ratios equivalent.
Thus, x = 5 Ã— 4 = 20 and ratios 4:5 and 16:20 are equivalent

## Equivalent Ratio Tables

A ratio table is a list containing the equivalent ratios of any given ratio in an ordered form.

For example, below is an equivalent ratio table of the ratio 1:2.

Similarly, we can prepare an equivalent ratio table of any ratio.

## FAQs

Q.1. How are unit rates and equivalent ratios related to each other?

Ans. Unit rates and equivalent ratios are both used to represent ratios and proportions. A unit rate is a kind of ratio in which the second number, or the denominator, is equal to one. In contrast, equivalent ratios are different ratios that have the same value.