The difference between the compound interest for 1 year, compounded half-yearly and the simple interest for 1 year on a certain sum of money at 10% per annum is ₹ 360. Find the sum.

Let sum of money lent out be ₹ x.

Calculating C.I. payable half-yearly :

P = ₹ x

r = 10%

n = 1 year

C.I. = A - P

When rate of interest is compounded half-yearly :

By formula,

A = $P\Big(1 + \dfrac{r}{2 \times 100}\Big)^{2n}$

C.I. = A - P

Substituting values we get :

$C.I. = P\Big(1 + \dfrac{r}{2 \times 100}\Big)^{n \times 2} - P \\[1em] = x \times \Big(1 + \dfrac{10}{200}\Big)^{1 \times 2} - x \\[1em] = x \times \Big(\dfrac{210}{200}\Big)^2 - x \\[1em] = x \times \Big(\dfrac{21}{20}\Big)^2 - x \\[1em] = \dfrac{441x}{400} - x \\[1em] = \dfrac{441x - 400x}{400} \\[1em] = ₹ \dfrac{41x}{400}.$

Calculating S.I. :

T = 1 year

R = 10%

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

Given,

Difference between compound interest for a year payable half-yearly and simple interest on ₹ x lent out at 10% for a year is ₹ 360.

$\therefore \dfrac{41x}{400} - \dfrac{x}{10} = 360 \\[1em] \Rightarrow \dfrac{41x - 40x}{400} = 360 \\[1em] \Rightarrow \dfrac{x}{400} = 360 \\[1em] \Rightarrow x = 360 \times 400 = ₹ 1,44,000.$

Hence, the sum = ₹ 1,44,000.