Dividing into a Given Ratio

We will learn how to divide a number into two parts in a given ratio (i.e., dividing into a given ratio).

Let the number be M. It is to be divided into two parts in the ratio a : b.

The two parts are x and y if x + y = M ........................... (i)

and $$\frac{x}{y}$$ = $$\frac{a}{b}$$ ........................... (ii)

From (ii), $$\frac{x}{a}$$ = $$\frac{y}{b}$$ = k (say).

Then , x = ak, y = bk

Substituting in (i), ak + bk = M

⟹ (a + b)k = M

⟹ k = $$\frac{M}{a + b}$$

Therefore, x = ak = $$\frac{a}{a + b}$$ M and y = bk = $$\frac{b}{a + b}$$ M

Two parts of M in the ratio a : b are $$\frac{aM}{a + b}$$ and $$\frac{bM}{a + b}$$

Solved examples on dividing a number into a given ratio:

1. Divide 60 into two parts in the ratio 2 : 3.

Solution:

The two parts are $$\frac{2}{2 + 3}$$ × 60 and $$\frac{3}{2 + 3}$$ × 60

i.e., $$\frac{2}{5}$$ × 60 and $$\frac{3}{5}$$ × 60

i.e., 24 and 36

2. Divide 75 into two parts in the ratio 8 : 7

Solution:

The two parts are $$\frac{8}{8 + 7}$$ × 75 and $$\frac{7}{8 + 7}$$ × 75

i.e., $$\frac{8}{15}$$ × 75 and $$\frac{7}{15}$$ × 75

i.e., 40 and 35

● Ratio and proportion