The Basics of Ratio and Proportion

In Mathematics, we use the ratio to compare two quantities, whereas proportion compares two ratios. 

• A ratio can be expressed in many ways, such as p:q or p/y, and is often understood as p is to q (where p and q are two terms). For example, 5:7 can be written as 5/7 (here, 5 is the antecedent and 7 is the consequent).

• In contrast, proportion states that two ratios are equal. It can be expressed as x/y=y/z=z/w and can be read as x to y to z to w. (Here, the ratio x/y is equivalent to z/w  ≠ 0.) If two ratios are in proportion, we express it as p : q :: r : s → p/q = r/s (Here, q and r are mean terms; and the outermost terms p and s are extreme terms). The product of extremes will always be equal to the product of means.

Ratios come in different forms. Learning the types helps students understand which type to use in different situations. We will now learn about the various types:

Equivalent Ratios: Two ratios that are the same when simplified.

For example, 5:10 and 1:2


Part-to-Part Ratio: When one part is compared to another in a whole, it is said to be in a part-to-part ratio.

For example, the ratio of blue ribbons to white ribbons in a shop is 3:5.

Compound Ratio: When we compare two or more ratios by taking their products.

For example: (5:6) × (3:8) = 15: 48.

Part-to-Whole Ratio: When a part is compared to the whole or the total amount, the ratio is part to whole.

For example, if there are 5 blue ribbons and 3 red ribbons, the total number of ribbons is 8. Now, the part-to-whole ratio of blue ribbons to total ribbons is 5:8.

If two ratios are identical, then they are proportional to each other. Proportions are of two types based on which they compare two ratios:

Direct Proportion

When two ratios have a direct relationship, they are directly proportional to each other. If a change in a quantity occurs, the other quantity also changes proportionally. We often use the symbol ‘∝’ to denote the proportionality. For example: The number of chocolates you buy increases, and the amount you have to pay also increases.

Inverse Proportion

If two quantities have an inverse relationship that is, one quantity increases the other decreases, and vice versa, they are inversely proportional to each other. 
It can be written as a ∝ 1/b.

To calculate the ratio and proportion, you can use the following formulas:

• The formula for the ratio can be represented as x:y = x/y

• If two ratios are equal, they are proportional and can be written as: p/q = r/s 
  i.e., (p × q = r × s)

Ratio and Proportion enable children to solve many real-life problems. To grasp it easily, let’s look at a few tips and tricks:

• If a ratio is expressed as x: y: z, the total can be calculated by simply adding x +y +z.

• Do not forget to express the ratio in its fractional form. For example: 4:6 = 4/6.

• You can multiply the denominator and numerator of a fraction with the same number to get the same ratios. For example, 2: 5 is equivalent to 4:10 (when multiplied by 2).

• Cross-multiplication can be used to identify the missing values. For example: If y:3 = 6: 9 then:
  y × 9 = 3 ×6 = 9y =18
  y = 18/ 9 = 2.

• Ratios can be expressed in any of these three forms: 5 to 6, 5:6, or 5/6.

Ratios and proportions are two fundamental concepts that have multiple real-life applications. Let’s look at a few:

• Understanding real-world situations helps children adjust the ingredient proportions when cooking their favorite dish.

• They will be able to comprehend the conversion of currency, which are ratios of various currencies.

• To determine the discounts, we apply the ratios to understand the amount that can be saved.

• We use proportion to evaluate the population growth. For example, demographers conclude that population growth is directly proportional to the consumption of resources.

• Using the ratios, we can calculate the investment returns or the growth rate.