Question

Ratio of the ages of Ramu and Rahim 4 years ago was 1:5 . After 4year it would be in the ratio 9:29 .Find their ages​

Answer 1

S O L U T I O N :

Let the present age of Ramu be x years & the present age of Rahim be y years respectively.

A/q

$\underbrace{\bf{4\:years\:ago\::}}$

The age of Ramu was = (x - 4) years.

The age of Rahim was = (y - 4) years.

So,

$\mapsto\tt{(x-4):(y-4) = 1:5}$

$\mapsto\tt{\dfrac{x-4}{y-4} = \dfrac{1}{5} }$

$\mapsto\tt{5(x-4) = 1(y-4) \:\:\underbrace{\sf{cross - multiplication\::}}}$

$\mapsto\tt{5x -20 = y - 4}$

$\mapsto\tt{5x - y = - 4 + 20}$

$\mapsto\tt{5x - y = 16}$

$\mapsto\tt{5x = 16 + y}$

$\mapsto\tt{x = 16 + y/5..............(1)}$

&

$\underbrace{\bf{After\: 4\:years\::}}$

The age of Ramu will = (x + 4) years.

The age of Rahim will = (y + 4) years.

So,

$\mapsto\tt{(x+4):(y+4) = 9:29}$

$\mapsto\tt{\dfrac{x+4}{y+4} = \dfrac{9}{29} }$

$\mapsto\tt{29(x+4) = 9(y+4) \:\:\underbrace{\sf{cross - multiplication\::}}}$

$\mapsto\tt{29x +116 = 9y +36}$

$\mapsto\tt{29x - 9y = 36-116}$

$\mapsto\tt{29x - 9y = -80}$

$\mapsto\tt{29\bigg(\dfrac{16+y}{5}\bigg) - 9y = -80\:\:[from(1)]}$

$\mapsto\tt{\dfrac{464+29y}{5} - 9y = -80}$

$\mapsto\tt{464 +29y - 45y = -400}$

$\mapsto\tt{464 -16y = -400}$

$\mapsto\tt{-16y = -400 - 464}$

$\mapsto\tt{-16y = -864}$

$\mapsto\tt{y = \cancel{-864/-16}}$

$\mapsto\bf{y = 54\:years}$

∴ Putting the value of y in equation (1),we get;

$\mapsto\tt{x = \dfrac{16+54}{5} }$

$\mapsto\tt{x = \dfrac{70}{5} }$

$\mapsto\tt{x = \cancel{\dfrac{70}{5} }}$

$\mapsto\bf{x = 14\:years}$

Thus,

The present age of Ramu will be 14 years & the present age of Rahim will be 54 years .