The ages of Pramod and Rohit are 16 years and 18 years respectively. In what ratio must they invest money at 5% p.a. compounded yearly so that both get the same sum on attaining the age of 25 years ?

Let Pramod invest ₹ x and Rohit invest ₹ y.

By formula,

A = $P\Big(1 + \dfrac{r}{100}\Big)^n$

For Pramod :

P = ₹ x

r = 5%

n = 9 years (As it will take 9 years for Pramod to reach 25 years of age)

Substituting values we get :

$\text{Amount received by Pramod} = x\Big(1 + \dfrac{5}{100}\Big)^9 \\[1em] = x \times \Big(\dfrac{105}{100}\Big)^9 \\[1em] = x \times \Big(\dfrac{21}{20}\Big)^9.$

For Rohit :

P = ₹ y

r = 5%

n = 7 years (As it will take 7 years for Pramod to reach 25 years of age)

Substituting values we get :

$\text{Amount received by Rohit} = y\Big(1 + \dfrac{5}{100}\Big)^7 \\[1em] = y \times \Big(\dfrac{105}{100}\Big)^7 \\[1em] = y \times \Big(\dfrac{21}{20}\Big)^7.$

Since,

Amount received by both are equal.

$\Rightarrow x \times \Big(\dfrac{21}{20}\Big)^9 = y \times \Big(\dfrac{21}{20}\Big)^7 \\[1em] \Rightarrow \dfrac{x}{y} = \dfrac{\Big(\dfrac{21}{20}\Big)^7}{\Big(\dfrac{21}{20}\Big)^9} \\[1em] \Rightarrow \dfrac{x}{y} = \dfrac{1}{\Big(\dfrac{21}{20}\Big)^2} \\[1em] \Rightarrow \dfrac{x}{y} = \dfrac{20^2}{21^2} \\[1em] \Rightarrow \dfrac{x}{y} = \dfrac{400}{441}.$

Hence, ratio in which sum must be invested = 400 : 441.