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





10th Grade Math

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