Last updated on July 4th, 2025
Ratios that simplify to the same value are known as equivalent ratios. These ratios have the same proportional relationship. In this article, we will explore equivalent ratios in detail.
Ratios are represented in the form a:b or a/b. Here, the first number (a) is known as the antecedent, and the second number (b) is known as the consequent. A ratio compares how much of one quantity is related to another.
For example, if the ratio of pens to books in a store is 3:4 (or 3/4), it means that for every 3 pens, there are 4 books. Thus, ratios are useful to compare numbers and quantities in various situations. Now, let us learn about equivalent ratios.
Equivalent ratios and equivalent fractions are similar concepts. Equivalent ratios have the same relationship or comparison between numbers. The numbers in the ratios may be different, but when we multiply or divide these terms by the same value, we get an equivalent ratio. We define the equality of two ratios as proportions. Let’s take an example to understand this concept better.
We can find whether 2:3 and 4:6 are equivalent by simplifying both ratios to their simplest form. To simplify the ratio 4:6, we find its highest common factor, which is 2. Here, the ratio 2:3 is in its simplest form, so now we can divide both terms of 4:6 by 2.
(4 ÷ 2) : (6 ÷ 2) = 2:3
Hence, both ratios are equivalent.
Multiplying or dividing both the antecedent and the consequent by the same number helps to find the equivalent ratios. If one ratio is a multiple of another, then both ratios are equivalent. Let us consider an example to find the equivalent ratios of 1:2.
To find the equivalent ratios, we multiply or divide both terms of the ratio by the same whole number.
Equivalent ratios of 1:2 are 2:4, 3:6, 4:8, 5:10, and so on.
We use the cross multiplication method and the greatest common factor to identify the equivalent ratios. Now, let’s discuss them in detail:
We use the cross multiplication method when the ratios have small numbers. It is easy to determine whether 6:9 and 8:12 are equivalent using this method. To find it, follow the steps given below:
Step 1: Convert the given ratio into fractional form.
6:9 = 6/9
8:12 = 8/12
Step 2: Cross-multiply the fractions.
6 × 12 = 72 and 9 × 8 = 72
We can see that the products are equal for equivalent ratios.
→ 6 × 12 = 9 × 8 = 72
Hence, we can say that 6:9 and 8:12 are equivalent ratios.
To identify whether the given ratios 6:9 and 8:12 are equivalent or not using the GCF method, follow these steps:
Step 1: Find the greatest common factor of the antecedent and consequent of the given ratios.
GCF of 6 and 9:
Factors of 6: 1, 2, 3, 6
Factors of 9: 1, 3, 9
Therefore, the GCF between 6 and 9 is 3.
Now, let’s find the GCF of 8 and 12:
Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
Thus, the GCF of 8 and 12 is 4.
Step 2: Divide the antecedent and consequent by their GCF.
(6 ÷ 3) : (9 ÷ 3) = 2:3
(8 ÷ 4) : (12 ÷ 4) = 2:3
Step 3: If both ratios reduce to the same simplest form, they are equivalent.
Here, 6:9 is equivalent to 8:12
We can represent equivalent ratios visually to get a better understanding of the concept. In the given representation, the shaded area to the unshaded area is the same for each ratio. For example, the ratios such as 2:6 and 4:12 are equivalent to 1:3.
And so on. Look at the given image to understand the visual representation more easily.
Equivalent ratios are used to show proportional relationships between two quantities, like in cooking or making dosage for medicines. Below are some real-world applications of the equivalent ratios:
Understanding the concept of equivalent ratios helps students compare and solve mathematical problems related to proportions. However, mistakes can happen when they are solving problems that include equivalent ratios. Here are some common errors and helpful solutions to avoid these mistakes.
In a class, the ratio of girls to boys is 2:3; find an equivalent ratio where the numbers are doubled.
4:6
The given ratio of girls to boys is 2:3. Since we need to double the numbers, multiply the antecedent and consequent by 2.
Multiply 2 and 3 by 2:
2 × 2 = 4
3 × 2 = 6
Hence, the equivalent ratio is 4:6.
Doubling the numbers in the ratio 2:3 gives an equivalent ratio of 4:6
The ratio of apples to oranges in a basket is 15:10. What is the equivalent ratio if you divide both numbers by 5?
3:2
The given ratio is 15:10.
To find the equivalent ratio, find the GCF of 15 and 10.
The factors of 15 are 1, 3, 5, 15
The factors of 10 are 1, 2, 5, 10
Hence, the GCF is 5.
Now, divide both numbers by 5:
15/5 : 10/5 = 3:2
Therefore, the equivalent ratio is 3:2.
Find two equivalent ratios of 5:8.
10:16 and 15:24
The given ratio is 5:8.
To find the first equivalent ratio, multiply both terms by 2:
5 × 2 : 8 × 2
= 10: 16
Therefore, the first equivalent ratio is 10:16.
Now, let us multiply both terms by 3:
5 × 3 : 8 × 3
= 15:24
Hence, the second equivalent ratio is 15:24.
Thus, the two equivalent ratios of 5:8 are 10:16 and 15:24.
Find three equivalent ratios of 7:2.
14:4, 21:6, 28:8.
To find the three equivalent ratios of 7:2, we need to multiply both terms by the same factor.
First, multiply the antecedents and consequents by 2:
(7 × 2) : (2 × 2)
= 14: 4
Hence, the first equivalent ratio is 14:4
Now, multiply the terms by 3:
(7 × 3) : (2 × 3)
= 21:6
Therefore, the second equivalent ratio is 21:6
Next, multiply the numbers by 4:
(7 × 4) : (2 × 4)
= 28:8
Thus, the third equivalent ratio is 28:8
The three equivalent ratios of 7:2 are 14:4, 21:6, and 28:8.
If the ratio 3:8 is equivalent to 15:x, find the value of x.
40
To find the value of x, we use the cross-multiplication method.
First, write the given ratio in fraction form:
3:8 = 3/8
15:x = 15/x
Now, we can cross-multiply the terms:
3 × x = 8 × 15
3x = 120
Next, solve for x:
Divide both sides of the expression by 3 to get the value of x.
x = 120/3
x = 40
Hence, the value of x is 40.
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.