Question

The compound interest on Rs. 8000 for 1 year at 5 %p.a. payable half-yearly is:
\[
  \left( a \right)Rs.800 \\
  \left( b \right)Rs.810 \\
  \left( c \right)Rs.400 \\
  \left( d \right)Rs.405 \\
\]

Answer 1

Hint-In this question, we use the concept of compound interest. We use formula of Compound interest = Amount - Principal and ${\text{Amount}} = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}$ where, P is principal, R is rate and t is time.

Complete step-by-step solution -
Given, Principal (P) =Rs.8000
Rate (R) = 5% payable half yearly
Time (t) = 1 year
Now, we have to find the amount for half yearly so the rate becomes half and time becomes double for half yearly.
 $
   \Rightarrow {\text{Amount}} = P{\left( {1 + \dfrac{R}{{100}}} \right)^t} \\
   \Rightarrow {\text{Amount}} = 8000{\left( {1 + \dfrac{5}{{2 \times 100}}} \right)^2} \\
   \Rightarrow {\text{Amount}} = 8000{\left( {1 + \dfrac{1}{{40}}} \right)^2} \\
   \Rightarrow {\text{Amount}} = 8000{\left( {\dfrac{{41}}{{40}}} \right)^2} \\
   \Rightarrow {\text{Amount}} = 8000 \times \dfrac{{41}}{{40}} \times \dfrac{{41}}{{40}} \\
   \Rightarrow {\text{Amount}} = 5 \times 41 \times 41 \\
   \Rightarrow {\text{Amount}} = Rs.8405 \\
 $
Now, we use the formula of compound interest.
$
  {\text{Compound interest}} = {\text{Amount}} - {\text{Principal}} \\
   \Rightarrow {\text{Compound interest}} = 8405 - 8000 \\
   \Rightarrow {\text{Compound interest}} = Rs.405 \\
$
So, the correct option is (d).

Note-In such types of problems first we find the value of the amount for half yearly and always remember one thing for half yearly rate becomes half and time becomes double of actual value and then uses compound interest formula. So, we will get the required answer.