Question

A man borrowed a sum of money and agrees to pay it off by paying Rs. 43,800 at the end of the first year and Rs. 34,992 at the end of the second year. If the rate of compound interest is 8% per annum, find the sum borrowed.

Answer 1

Hint: Compound interest is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. The formula for compound interest is-
${\text{A}} = {\text{P}}{\left( {1 + \dfrac{{\text{r}}}{{100}}} \right)^{\text{t}}}$
Here A is the final amount, P is the initial principal amount, r is the rate of interest, and t is the time in years.

Complete step-by-step solution -
We have been given the amount paid at the end of the first year at an interest of 8% per annum. Also, we have been given the amount that has been paid at the end of the second year. It has been given that the debt is paid at the end of two years, so the total amount for the two years is-
$Rs. 43800 + Rs. 34992 = Rs. 78792$
We need to find the sum that was borrowed, which is the principal amount. So, we will apply the formula for compound interest at-
$A = Rs. 78792, r = 8% and t = 2 years$
$\begin{align}
  &{\text{A}} = {\text{P}}{\left( {1 + \dfrac{{\text{r}}}{{100}}} \right)^{\text{t}}} \\
  &78792 = {\text{P}}{\left( {1 + \dfrac{8}{{100}}} \right)^2} \\
  &78792 = {\text{P}}{\left( {\dfrac{{108}}{{100}}} \right)^2} \\
  &{\text{P}} = 78792{\left( {\dfrac{{100}}{{108}}} \right)^2} = Rs.\;67551.44 \\
\end{align} $
This is the sum of money that was borrowed and the required answer.

Note: One may find it confusing initially, because students apply the formula for compound interest at the end of the first year itself, substituting $A = 43800, r = 8, \text{and}\; \ t =1$. But this is incorrect. It is not necessary that the amount paid at the end of the first year will be equal to the amount calculated using the formula. But it has been given that the debt is cleared at the end of two years, which means that the total amount that has been paid is the one obtained by the formula and can be applied to find the value of P.