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Last updated on July 4th, 2025

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Equivalent Ratios

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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.

Equivalent Ratios for Vietnamese Students
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What are Ratios?

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.

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What are 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. 

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How to Find Equivalent Ratios?

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. 

 

  • Multiply 1:2 by 2 → (1 × 2) : (2 × 2) = 2:4 
     
  • Multiply 1:2 by 3 → (1 × 3) : (2 × 3) = 3:6
     
  • Multiply 1:2 by 4 → (1 × 4) : (2 × 4) = 4:8 
     
  • Multiply 1:2 by 5 → (1 × 5) : (2 × 5) = 5:10

 

Equivalent ratios of 1:2 are 2:4, 3:6, 4:8, 5:10, and so on.

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How to Identify Equivalent Ratios?

We use the cross multiplication method and the greatest common factor to identify the equivalent ratios. Now, let’s discuss them in detail:

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Cross Multiplication Method

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. 

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Greatest Common Factor (GCF) method

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

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What is the Visual Representation of Equivalent Ratios?

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.

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Real Life Applications of Equivalent Ratios

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: 

 

  • When cooking, we need to maintain a proper balance of ingredients. We use the equivalent ratios to adjust recipes proportionally. For example, if a cake needs 2 cups of flour for 3 cups of milk, doubling the recipe would result in:
    (2 × 2) : (3 × 2) = 4:6. 

 

  • Engineers apply equivalent ratios to ensure structures remain proportionally accurate during construction. For instance, if a bridge model is built at a 1:50 scale and the model is 2 meters high, then the actual bridge will be: 
    2 × 50 = 100 meters

 

  • In medical and health cases, researchers and doctors use the equivalent ratio to determine the correct dosage of medicine. For instance, if a 5-year-old kid needs 10 ml of medicine, then a 10-year-old may need:
    (10/5) × 10 = 20 ml
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Common Mistakes and How to Avoid Them in 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. 

Mistake 1

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Performing Cross-Multiplication Incorrectly

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Always use the cross multiplication method for ratios with small numbers. Incorrect cross-multiplication will lead to the wrong equivalent ratio.

 

For example, check whether 10:15 and 14:21 are equivalent  
In the cross multiplication method, convert the ratio into its fractional form first.

10/15 and 14/21 

Now cross-multiply the fractions:

10 × 21 = 210 
15 × 14 = 210
  
Since we get 210 as the same product after the cross multiplication, we can say that the ratios 10:15 and 14:21 are equivalent.

Mistake 2

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Not multiplying both terms in the ratio

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Always multiply the terms by the same whole number. If students multiply only one term, the ratio becomes incorrect.

 

For instance, the given ratio is 6:9, and to find the equivalent ratio, the students might multiply the ratio by 2 as: 
6 × 2 : 9 = 12:9, which is incorrect. 

The correct ratio is: 
6 × 2 : 9 × 2 = 12:18

Mistake 3

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Using Addition Instead of Multiplication

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Sometimes, students mistakenly add the ratios to find the answer. When multiplying the terms by a whole number, make sure that you are multiplying them instead of adding.

 

For instance, the given ratio is 2:3, and to find the equivalent ratio by multiplying by 3 gives: 
2 × 3 : 3 × 3 = 6:9

Mistake 4

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Improper Finding of Highest Common Factor

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When determining whether the given ratios are equivalent or not, students use the GCF method. While finding the GCF, keep in mind to list all the factors and find the greatest factor among them. Incorrect GCF leads to wrong equivalent ratios.

Mistake 5

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Ignoring the Order of Ratio

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Always maintain the correct order while writing the ratio. If the ratio 3:2 represents “apples to oranges,” write it as 3:2 
If they write the ratio incorrectly, it will lead to incorrect equivalent ratios.

 

For instance, writing 2:3 is incorrect.

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Solved Examples on Equivalent Ratios

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Problem 1

In a class, the ratio of girls to boys is 2:3; find an equivalent ratio where the numbers are doubled.

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4:6

Explanation

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

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Problem 2

The ratio of apples to oranges in a basket is 15:10. What is the equivalent ratio if you divide both numbers by 5?

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3:2

Explanation

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.

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Problem 3

Find two equivalent ratios of 5:8.

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 10:16 and 15:24

Explanation

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.

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Problem 4

Find three equivalent ratios of 7:2.

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14:4, 21:6, 28:8.

Explanation

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.

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Problem 5

If the ratio 3:8 is equivalent to 15:x, find the value of x.

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40

Explanation

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.

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FAQs on Equivalent Ratios

1.What is an equivalent ratio?

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2.How do you determine if two ratios are equivalent?

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3.Note the two methods for identifying equivalent ratios.

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4.Explain the GCF method for finding the equivalent ratio.

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5.Do all ratios have the same value when multiplied by the same number?

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6.How can children in Vietnam use numbers in everyday life to understand Equivalent Ratios?

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7.What are some fun ways kids in Vietnam can practice Equivalent Ratios with numbers?

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8.What role do numbers and Equivalent Ratios play in helping children in Vietnam develop problem-solving skills?

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9.How can families in Vietnam create number-rich environments to improve Equivalent Ratios skills?

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Hiralee Lalitkumar Makwana

About the Author

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.

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Fun Fact

: She loves to read number jokes and games.

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