On what sum of money will the difference between the compound interest and simple interest for 2 years be equal to ₹25 if the rate of interest charged for both is 5% p.a.?

Given,

Let Sum (P) = ₹x,

Rate (R) = 5% p.a.

Period (n) = 2 years.

We know that,

$\text{S.I.} = \dfrac{\text{PRT}}{100}$

Substituting values we get,

$\text{S.I.} = \dfrac{x \times 5 \times 2}{100} \\[1em] = ₹\dfrac{x}{10}.$

When interest is compounded annually,

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

Substituting values,

$A = x\Big(1 + \dfrac{5}{100}\Big)^2 \\[1em] = x\Big(1 + \dfrac{1}{20}\Big)^2 \\[1em] = x \times \Big(\dfrac{21}{20}\Big)^2 \\[1em] = x \times \dfrac{441}{400} \\[1em] = ₹\dfrac{441x}{400}.$

C.I. = A - P = $₹\dfrac{441x}{400} - ₹x = ₹\dfrac{41x}{400}.$

$\text{C.I.} - \text{S.I.} = ₹\dfrac{41x}{400} - ₹\dfrac{x}{10} \\[1em] = ₹\dfrac{41x - 40x}{400} \\[1em] = ₹\dfrac{x}{400}$

Given,

Difference = ₹25.

$\therefore \dfrac{x}{400} = 25 \\[1em] \Rightarrow x = 25 \times 400 \\[1em] \Rightarrow x = ₹10000.$

Hence, sum of money = ₹10000.