The difference between the compound interest for a year payable half-yearly and the simple interest on a certain sum of money lent out at 10% for a year is ₹15. Find the sum of money lent out.

Let Sum (P) = ₹x.

Given,

Rate = 10% p.a. or 5% half-yearly.

Period = 1 year or 2 half-years.

We know that,

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

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 = $₹\dfrac{441x}{400} - ₹x = ₹\dfrac{41x}{400}.$

We know that,

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

Substituting values we get,

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

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

Given,

Difference = ₹15,

$\therefore \dfrac{x}{400} = 15 \\[1em] \Rightarrow x = 15 \times 400 = ₹6000.$

Hence, sum of money = ₹6000.