The difference between compound interest for a year payable half-yearly and 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 of money lent out be ₹ x.

Calculating C.I. payable half-yearly :

P = ₹ x

Rate of interest = 10%

Time = 1 year

C.I. = A - P

$C.I. = P\Big(1 + \dfrac{r}{2 \times 100}\Big)^{n \times 2} - P \\[1em] = x \times \Big(1 + \dfrac{10}{200}\Big)^{1 \times 2} - x \\[1em] = x \times \Big(\dfrac{210}{200}\Big)^2 - x \\[1em] = \dfrac{441x}{400} - x \\[1em] = \dfrac{441x - 400x}{400} \\[1em] = ₹ \dfrac{41x}{400}.$

Calculating S.I. :

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

Given,

Difference between compound interest for a year payable half-yearly and simple interest on ₹ x lent out at 10% for a year is ₹ 15.

$\therefore \dfrac{41x}{400} - \dfrac{x}{10} = 15 \\[1em] \Rightarrow \dfrac{41x - 40x}{400} = 15 \\[1em] \Rightarrow \dfrac{x}{400} = 15 \\[1em] \Rightarrow x = 15 \times 400 = ₹ 6000.$

Hence, sum of money lent out = ₹ 6000.