# Proportion

If a ratio a:b is equal to another ratio x:y, then a, b, x and y are said to be in proportion. Here, a and y are called the extremes, while b and x are called the means of the proportion.

When two ratios a:b and x:y are in proportion, it is written as a:b::x:y (Read as “a is to b as x is to y”.)

Example: Let Rahul’s height be 150 cm and his weight be 50 kg. Let Sachin’s height be 120 cm and his weight be 40 kg.

The ratio of the heights is Sachin’s height : Rahul’s height = 120:150 = 4:5

The ratio of the weights is Sachin’s weight : Rahul’s weight = 40:50 = 4:5

We can see that Sachin’s height : Rahul’s height = Sachin’s weight : Rahul’s weight.

Hence, we can say that the height and weight of the two boys are in proportion.

Note: In a ratio x:y, both quantities must be of the same kind, while in a proportion a:b = x:y, all four quantities need not be of the same kind. The first two quantities must be of the same kind and the last two quantities must be of the same kind.

As we saw in the above example, the first two quantities are heights of the two boys and the last two are weights.

If a:b = x:y, then bx = ay, i.e., product of means is equal to product of extremes.

Example
Find the fourth proportion of the numbers 6, 8 and 15.
Solution
If x be the fourth proportion, then 6:8 = 15:x
Hence,
If a:b = b:c, then a, b, c are said to be in continued proportion, where b is called the mean proportion between a and c. a is called the first proportion and c is called the third proportion.
When a, b, c are in continued proportion, b 2 = ac (i.e., product of means is equal to product of extremes).

Example
Find the mean proportion between 3 and 75.
Solution
If x is the mean proportion, then we have 3:x :: x:75
⇒ x 2 = 3 × 75 = 225 ⇒ x = 15