We will learn about the properties of proportion. We know that a proportion is an expression which states that the two ratios are in equal.

In general, four numbers are said to be in proportion, if the ratio of the first two quantities is equal to the ratio of the last two. In general, the symbol for representing a proportion is “: :”

**(i)** The numbers a, b, c and d are in proportional if the ratio of the first two quantities is equal to the ratio of the last two quantities, i.e., a : b : : c : d and is read as ‘a is to b is as c is to d’. The symbol ‘ : : ‘ stands for ‘is as’.

**(ii)** Each quantity in a proportion is called its term or its proportional.

**(iii)** In a proportion; the first and the last terms are called the extremes; whereas the second and the third terms are called the means.

If four numbers a, b, c and d are in proportional (i.e., a : b : : c : d), then a and d are known as extreme terms and b and c are called middle terms.

**(v)** The fourth term of a proportion is called fourth proportional.

**(vi)** For every proportion, the product of
the extremes is always equal to the product of the means, i.e., a : b : : c : d
if and only if ad = bc.

For example; in proportion 3 : 4 : : 9 : 12;

Product of extremes = 3 × 12 = 36 and product of means = 4 × 9 = 36

**(vii)** From the terms of a given proportion,
we can make three more proportions.

**(viii)** If x : y = y : z, then x, y, z are
said to be continued proportion.

**(ix)** If x, y, z are in continued
proportion, (i.e., x : y : : y : z), then y is the mean proportional between x
and z.

**(x)** If x, y, z are in continued proportion,
(i.e., x : y : : y : z), then the third quantity is called the third
proportional to the first and second i.e., z is the third proportional to x and
y.

Properties of proportion will help us to solve different types of problems on ratio and proportion.

Solved Example:

Find the fourth proportional of 3, 4 and 18.

**Solution:**

Let the fourth proportional be x.

Therefore, 3 : 4 = 18 : x

⇒ 3 × x = 4 × 18; from the above property (vi) we know product of extremes = product of means

⇒ 3x = 72

⇒ x = 72/3

⇒ x = 24

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