Equivalent 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.

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. 

• 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.

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: 

• 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