Proportions | Definition and Examples

Proportions are an important concept in Mathematics that is frequently used for the comparison of two ratios or fractions. It is closely related to another concept called ratios. When two things are in proportion, it means they change in a way that keeps their relationship the same. For example, if 2 apples cost 4 pounds, 4 apples will cost 8 pounds, keeping the same ratio

Example :

• Cooking: If a recipe calls for 2 cups of flour for 4 servings, for 8 servings, you'd need 4 cups of flour. The amount of flour is in proportion to the number of servings.
• Speed and Distance: If a car travels 60 miles in 1 hour, in 3 hours, it would travel 180 miles. The distance is in proportion to the time.

Proportions Symbol

Proportions between two fractions or ratios can be represented by using an '=' or '::' sign.

a/b = c/d
a : b :: c : d

Here the '=' or '::' sign shows that the ratio between a and b is equivalent to the ratio between c and d.

Types of Proportions

There are different types of proportions depending on the nature of relationships between two or more quantities. The different categories of proportion are as follows:

• Direct Proportion
• Inverse Proportion

• Direct proportion: It describes the relationship between two quantities in which the increase in value of one quantity results in an increase in the value of the other quantity, and vice versa. In this case, the quantities are said to be in direct proportion to each other.

For example, we find that the height of an infant increases with age. So, we can say that the height of a child is directly proportional to their age.

• Inverse Proportion: This type of proportion implies that if the value of one quantity is increased, the value of another quantity will decrease, and vice versa. In this case, the quantities are said to be inversely proportional.

For example, if the speed of a vehicle increases, the time taken to cover a fixed distance will decrease, and vice versa. So, the time and speed vary in inverse proportion.

Fourth, Third and Mean Proportional

If two ratios a : b and m : n is in proportion, then the two terms ‘b’ and 'm’ are called the 'means’ or 'mean term’. The terms ‘a’ and ‘n’ are known as ‘extremes’ or ‘extreme terms '. Here, 'm’ is called the third proportional of ‘a’ and 'b', and ‘n’ is known as the fourth proportional of ‘a’ and ‘b’.

Example : If p : q = r : s, then:

• s is called the fourth proportional to p, q, r.
• r is called the third proportional to p and q.

The mean proportional between p and q is √(pq).

Proportions Formula

The concept of proportion involves expressing the equality of two ratios. The general formula for proportion and its related forms can be outlined as follows:

a/b = c/d or a:b::c:d

Important Properties of Proportions

The important properties of proportion are mentioned below.

• Addendo: When m:n = q:r, then (m + q):(n + r) = m:n = q:r

• Subtrahendo: When p:q = m:n, then (m - q):(n - r) = p:m = q:n

• Dividendo: When a:b = c:d, then (a - b):b = (c - d):d

• Componendo: When a:b = c:d, then (a + b):b = (c + d):d

• Alternendo: When x:y = a:b, then x:a = y:b

• Invertendo: If x:y = m:n, then y:x = n:m

• Componendo and Dividend: When a: b = c: d, then (a + b):(a - b) = (c + d):(c - d)

Difference Between Ratio and Proportion

Proportion and ratios are two important mathematical concepts that are closely related to each other. Proportion indicates equality in two or more ratios. To get a better understanding of ratios and proportions, it is important to know the differences between the two, which are given below.

Solved Examples on Proportions

Example 1. There are 21 boys and 14 girls in a class. In another class, the number of boys and girls is 27 and 18. Find out whether the ratio of boys and girls in two classes is in proportion.
Solution:

Ratio of boys and girls in one class is 21/14 = 3/2.

Ratio of boys and girls in another class is 27/18 = 3/2

Value of the two ratios is equal; therefore, the ratio of the number of boys and girls in two classes is in proportion

Example 2. Find out whether the two ratios 8/36 and 12/40 are in proportion.
Solution:

First ratio is 8/36 = 2/9, and the second ratio is 12/40 = 3/10

Given ratios are unequal

Therefore, they are not in proportion

Example 3. Check whether the following statements are true or false.
(i) 9 : 15 = 20 : 35
(ii) 16 kg: 36 kg = 28 apples: 63 apples
Solution:

(i)

9/15 = 3/5

20/35 = 4/7

They are unequal, so the statement is false

(ii)

16 kg/36 kg = 4/9

28 apples/63 apples = 4/9

These ratios are equal, so the statement is true

Articles Related to Proportions:

• Ratio and Proportions
• Quiz about ratio and proportion
• Use of ratio and proportion in real-life