# Proportion

We will discuss about the proportion of four quantities.

Four quantities w, x, y, and z are in proportion if w : x :: y : z or, $$\frac{w}{x}$$ = $$\frac{y}{z}$$.

Definition: If four quantities w, x, y and z are such that the ratio w : x is equal to the ratio y : z then we say w, x, y and z are in proportion or, w, x, y and z are proportional. We express it by writing w : x :: y : z.

w : x :: y : z if and only if $$\frac{w}{x}$$ = $$\frac{y}{z}$$ i.e., wz = xy.

In w : x : : y : z, w, x, y and z are the first, second, third and fourth terms respectively.

Also, w and z are called the extreme terms while x and y are called the middle terms or mean terms.

That is in a proportion the first and fourth terms are called extremes, while the second and third terms are called means.



Product of extremes = product of means

or, wz = xy.

If w, x, y and z are in proportion then the last term z is also called the fourth proportional.

For example, 2 : 7 :: 8 : 28 because $$\frac{2}{7}$$ = $$\frac{8}{28}$$.

Solved examples on Proportion:

Check whether the following numbers form a proportion or not.

(i) 2.5, 4.5, 5.5, 9.9

(ii) 1$$\frac{1}{4}$$, 1$$\frac{3}{4}$$, 1.5, 1.4

Solution:

(i) 2.5 : 4.5 = $$\frac{2.5}{4.5}$$ = $$\frac{25}{45}$$ = $$\frac{5}{9}$$

5.5 : 9.9 = $$\frac{5.5}{9.9}$$ = $$\frac{55}{99}$$ = $$\frac{5}{9}$$

Therefore, $$\frac{2.5}{4.5}$$ = $$\frac{5.5}{9.9}$$

Hence, 2.5, 4.5, 5.5 and 9.9 are in proportion.

(ii) 1$$\frac{1}{4}$$ : 1$$\frac{3}{4}$$ = $$\frac{5}{4}$$ : $$\frac{7}{4}$$ = $$\frac{5}{4}$$ × 4: $$\frac{7}{4}$$ × 4 = 5 : 7 = $$\frac{5}{7}$$

1.5 : 1.4 = $$\frac{1.5}{1.4}$$ = $$\frac{15}{14}$$

Since, $$\frac{5}{7}$$  ≠ $$\frac{15}{14}$$

Hence, 1$$\frac{1}{4}$$, 1$$\frac{3}{4}$$, 1.5 and 1.4 are not in proportion.

● Ratio and proportion