Question

Rs.20,000 is lent for 2 years at 5% compound interest. Calculate the compound interest and the amount after 2 years if the interest is compounded annually. What would have been the interest earned if the same amount was lent for simple interest at the same rate for the same duration?

Answer 1

Hint: Use the formula for calculating the amount received \[\text{A = P}{{\left( 1\text{ + r} \right)}^{\text{t}}}\], where P is the initial principal balance, r is the compound interest rate, t is the required time (in years) and A is the final amount received in hand. Now, after calculating the amount received, subtract the initial principal from it to calculate the compound interest balance. To find what would have been the interest if the same amount was lent for simple interest at the same rate for the same duration, use the formula ${A = P(1 + r \times t)}$, and get the final amount received and thereby subtract the principal from it.

Complete step-by-step answer:
Here, it is given that,
Initial principal balance lent = Rs. 20,000
Rate of compound interest $=\text{ 5 }\!\!%\!\!\text{ = }\frac{5}{100}\text{ = 0}\text{.05}$
Duration = 2 years
Thus, putting the values P = Rs. 20,000, r = 0.05 and t = 2 in the formula \[\text{A = P}{{\left( 1\text{ + r} \right)}^{\text{t}}}\], we get,
$\begin{align}
  & \text{A = 20000}{{\left( 1\text{ + 0}\text{.05} \right)}^{2}} \\
 & \text{ = 20000 x 1}\text{.05 x 1}\text{.05} \\
 & \text{ = 22050} \\
\end{align}$
Thus, the final amount received at hand is Rs. 22,050.
$\therefore $ The compound interest received = Amount received – Initial principle
$\begin{align}
  & =\text{ Rs}\text{.}\left( 22,050\text{ }-\text{ 20,000} \right) \\
 & =\text{ Rs}\text{. 2,050} \\
\end{align}$
Now, we intend to find the interest amount if the same money was lent at the same rate of simple interest for the same duration.
Then, putting the values P = Rs. 20,000, r = 0.05 and t = 2 in the formula $\text{A = P}\left( 1{ + r \times t} \right)$, we get,
$\begin{align}
  & \text{A = 20000 x }\left( \text{1 + 0}{.05 \times 2} \right) \\
 & \text{ = 20000 x 1}\text{.1} \\
 & \text{ = 22000} \\
\end{align}$
Thus, we receive Rs. 22,000 at hand if interest is calculated at the rate of simple interest.
$\therefore $ The simple interest received = Amount received – Initial principle
$\begin{align}
  & =\text{ Rs}\text{.}\left( 22,000\text{ }-\text{ 20,000} \right) \\
 & =\text{ Rs}\text{. 2,000} \\
\end{align}$
Answers: a) The compound interest received is Rs. 2,050
          b) The final amount at 5% compound interest rate is Rs. 22,050
         c) The simple interest received is Rs. 2,000.

Note: Compound interest means the amount received at the end of each year at a simple interest rate becomes the principal for the following year, and so on. For example, in the given problem, duration is 2 years. So, the amount received at 5% simple interest rate becomes the principal for the 2nd year, and again on it we receive the amount at the same 5% simple interest rate.