A sum of ₹ 44200 is divided between John and Smith, 12 years and 14 years old respectively, in such a way that if their portions be invested at 10 percent per annum compound interest, they will receive equal amounts on reaching 16 years of age.

(i) What is the share of each out of ₹ 44200?

(ii) What will each receive, when 16 years old ?

(i) Let share of John = ₹ x and share of Smith = ₹ 44200 - ₹ x.

For John :

P = ₹ x

r = 10%

n = 4 years (John has 4 years to reach 16 years of age)

By formula,

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

Substituting values we get :

$\Rightarrow \text{Amount } = x\Big(1 + \dfrac{10}{100}\Big)^4 \\[1em] = x \times \Big(\dfrac{110}{100}\Big)^4\\[1em] = x \times \Big(\dfrac{11}{10}\Big)^4 ……..(1)$

For Smith :

P = ₹ (44200 - x)

r = 10%

n = 2 years (Smith has 2 years to reach 16 years of age)

By formula,

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

Substituting values we get :

$\Rightarrow \text{Amount } = (44200 - x) \times \Big(1 + \dfrac{10}{100}\Big)^2 \\[1em] = (44200 - x) \times \Big(\dfrac{110}{100}\Big)^2 \\[1em] = (44200 - x) \times \Big(\dfrac{11}{10}\Big)^2 ………..(2)$

Since, amount received by A and B are equal.

$\therefore x \times \Big(\dfrac{11}{10}\Big)^4 = (44200 - x) \times \Big(\dfrac{11}{10}\Big)^2 \\[1em] \Rightarrow x \times \Big(\dfrac{11}{10}\Big)^2 = (44200 - x) \\[1em] \Rightarrow \dfrac{121}{100}x = (44200 - x) \\[1em] \Rightarrow 121x = 100(44200 - x) \\[1em] \Rightarrow 121x = 4420000 - 100x \\[1em] \Rightarrow 121x + 100x = 4420000 \\[1em] \Rightarrow 221x = 4420000 \\[1em] \Rightarrow x = \dfrac{4420000}{221} \\[1em] \Rightarrow x = ₹ 20000.$

₹ (44200 - x) = ₹ (44200 - 20000) = ₹ 24200.

Hence, share of John and Smith are ₹ 20,000 and ₹ 24,200 respectively.

(ii) Substituting value of x in equation (1), we get :

$\Rightarrow 20000 \times \Big(\dfrac{11}{10}\Big)^4 \\[1em] \Rightarrow 20000 \times \dfrac{14641}{10000} \\[1em] \Rightarrow ₹ 29282.$

Since, both receive equal amount.

Hence, each receive ₹ 29282.