The difference between simple interest and compound interest on a certain sum is ₹ 54.40 for 2 years at 8 percent per annum. Find the sum.

Given,

n = 2 years

r = 8%

Let sum of money be ₹ P.

C.I. = A - P

$= P\Big(1 + \dfrac{r}{100}\Big)^n - P \\[1em] = P\Big(1 + \dfrac{8}{100}\Big)^2 - P \\[1em] = P \times \Big(\dfrac{108}{100}\Big)^2 - P \\[1em] = P \times \Big(\dfrac{27}{25}\Big)^2 - P\\[1em] = P \times \dfrac{729}{625} - P \\[1em] = \dfrac{729P}{625} - P \\[1em] = \dfrac{729P - 625P}{625} \\[1em] = \dfrac{104P}{625}.$

By formula,

$S.I. = \dfrac{P \times R \times T}{100} \\[1em] = \dfrac{P \times 8 \times 2}{100} \\[1em] = \dfrac{4P}{25}.$

Given,

Difference between S.I. and C.I. = ₹ 54.40

$\Rightarrow \dfrac{104P}{625} - \dfrac{4P}{25} = 54.40 \\[1em] \Rightarrow \dfrac{104P - 100P}{625} = 54.40 \\[1em] \Rightarrow \dfrac{4P}{625} = 54.40 \\[1em] \Rightarrow P = \dfrac{54.40 \times 625}{4} \\[1em] \Rightarrow P = ₹ 8500.$

Hence, sum = ₹ 8500.