Equivalent Ratios

Equivalent ratios are the ratios that are the same when we compare them. Two or more ratios can be compared with each other to check whether they are equivalent or not. For example, 1:2 and 2:4 are equivalent ratios.

In other words, we can say, two ratios are equivalent to each other if one of them can be expressed as the multiple of the other. Hence, to get the equivalent ratio of another ratio, we have to multiply the two quantities (antecedent and consequent) by the same number. This method is similar to the method of finding equivalent fractions.

Let us learn in this article how to find the equivalent ratios with examples. But before we proceed, first we need to understand about ratios and their quantities.

What is Ratio?

In Mathematics, a ratio compares two quantities named as antecedent and consequent, by the means of division. For example, when we cook food, then each ingredient has to be added in a ratio. Thus, we can say, a ratio is used to express one quantity as a fraction of another quantity.

A ratio is usually expressed with the symbol ‘:’. The comparison or simplified form of two quantities of the same kind is referred to as ratio.

Numerator and Denominator

We can also express the ratio as a fraction. If a:b, is a ratio, then a/b is its fraction form. Thus, we can easily compare two or more equivalent ratios in the form of equivalent fractions.

Standard Form of Ratio

The standard form of the ratio is given below:

How to Find Equivalent Ratios?

As we know, two or more ratios are equivalent if their simplified forms are the same. Thus, to find a ratio equivalent to another we have to multiply the two quantities, by the same number.

Another way to find equivalent ratios is to convert the given ratio into fraction form and then multiply the numerator and denominator by the same number to get equivalent fractions. Then again we can write the resulting fraction as an equivalent ratio.

Also, if we have to compare any two equivalent ratios, then we can divide the two quantities by the highest common factor and get the simplest form of ratio. Hence, we can compare them.

The examples of equivalent ratios are:

• 2 : 4 :: 4 : 8
• 10 : 20 :: 20 :40
• 1 : 2 :: 2 : 4
• 0.5 : 1 :: 2:4

Solved Examples

Q.1: Find the equivalent ratios of 8 : 18.

Solution: Let us first write the given ratio as a fraction.

8:18 ⇒ 8/18

Now multiply the numerator and denominator by 2

= (8 × 2)/(18 × 2)

= 16/36

Or we can write, the above fraction as a ratio;

= 16 : 36

So, 16 : 36 is an equivalent ratio of 8 : 18.

Q.2. Find any two equivalent ratios of 4 : 5.

Solution: Let us first write the given ratio as a fraction.

4:5 ⇒ 4/5

Now multiply the numerator and denominator by 2, to get the first equivalent fraction.

= 4/5

= (4 × 2)/(5 × 2)

= 8/10

Or

4:5 = 8:10

Again, multiply and divide ⅘ by another natural number, such as 3, as given below:

= 4/5

= (4 × 3)/(5 × 3)

= 12/15

Or

4:5 = 12:15

Hence, the two equivalent ratios of 4 : 5 are 8 : 10 and 12 : 15.

Q.3. Compare the given ratios if they are equivalent or not.

14:21, 2:3, 1:1.5, 6:9

Solution: Let us write the given ratios as fractions.

14:21 ⇒ 14/21

2:3 ⇒ ⅔

1:1.5 ⇒ 1/1.5

6:9 ⇒ 6/9

Now, we have to find the common factors that divide the numerator and denominator evenly and hence we get the simplified form of fractions.

14/21 = ⅔ (HCF = 7)

⅔ = ⅔ (Already simplified form)

1/1.5 = 10/15 = ⅔ (HCF = 5)

6/9 = ⅔ (HCF = 3)

Thus, we can see all the above fractions are equivalent since their simplified forms are the same. Therefore, the given ratios are also equivalent to each other.

Q.4: A bag contains 4 red balls and 9 white balls. What is the ratio of red balls to the white balls?

Solution: Number of red balls = 4

Number of white balls = 9

Therefore, the ratio of red balls to the white balls is 4:9.