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 xy = ab ........................... (ii)

From (ii), xa = yb = k (say).

Then , x = ak, y = bk

Substituting in (i), ak + bk = M

⟹ (a + b)k = M

⟹ k = Ma+b

Therefore, x = ak = aa+b M and y = bk = ba+b M

Two parts of M in the ratio a : b are aMa+b and bMa+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 22+3 × 60 and 32+3 × 60

i.e., 25 × 60 and 35 × 60

i.e., 24 and 36



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

Solution:

The two parts are 88+7 × 75 and 78+7 × 75

i.e., 815 × 75 and 715 × 75

i.e., 40 and 35

● Ratio and proportion





10th Grade Math

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