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Rs. $12,500$ lent at compound interest for two years at $10\% $ per annum fetches Rs. _______ more, if the interest was payable half yearly than if it was payable annually-
A.Zero
B.${\text{Rs}}{\text{. 10}}{\text{.48}}$
C.${\text{Rs}}{\text{. 38}}{\text{.50}}$
D.${\text{Rs}}{\text{. 68}}{\text{.82}}$

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Last updated date: 21st Jun 2024
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Answer
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Hint: Interest is the amount of money paid for using someone else’s money. There are two types of interest. $1)$ Simple Interest and $2)$ Compound interest. Interest can be calculated on the basis of various factors. Here we will calculate interest annually and semi-annually and then its difference.

Complete step-by-step answer:
Principal Amount,$P = {\text{Rs}}{\text{. }}12,500$
Rate of interest, $R = 10\% $
Term period, $T = 2{\text{ years}}$
Interest is paid annually –
$\Rightarrow A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$
Place the known values in the above equations –
$\eqalign{
\Rightarrow & A = 12500{\left( {1 + \dfrac{{10}}{{100}}} \right)^2} \cr
\Rightarrow & A = 12500{\left( {\dfrac{{110}}{{100}}} \right)^2} \cr
\Rightarrow & A = 15125{\text{ Rs}}{\text{.}} \cr} $
Therefore, Interest = Amount – Principal
$\eqalign{
 & \therefore I = A - P \cr
 & \therefore I = 15125 - 12500 \cr
 & \therefore I = Rs.{\text{ 2625 }}......{\text{(a)}} \cr} $
When interest is paid semi-annually –
$\Rightarrow A = P{\left( {1 + \dfrac{{R/2}}{{100}}} \right)^{2T}}$
Place values in the above equations –
$\eqalign{
\Rightarrow & A = 12500{\left( {1 + \dfrac{{10/2}}{{100}}} \right)^{2 \times 2}} \cr
\Rightarrow & A = 12500{\left( {1 + \dfrac{{10}}{{200}}} \right)^4} \cr} $
Simplify the above equation using the basic mathematical operations –
$\Rightarrow A = 15193.8{\text{ Rs}}{\text{.}}$
Interest = Amount – Principal
$\eqalign{
& \therefore I = A - P \cr
 & \therefore I = 15193.8 - 12500 \cr
 & \therefore I = 2693.8{\text{ Rs}}{\text{. }}......{\text{(b)}} \cr} $
By using the equations (a) and (b)
Difference between the interests calculated annually and semi-annually
$\eqalign{
\Rightarrow & I = 2693.8 - 2625 \cr
\Rightarrow & I = 68.8{\text{ Rs}}{\text{.}} \cr} $
Therefore, the required answer - if Rs. $12,500$ lent at compound interest for two years at $10\% $ per annum fetches Rs. $Rs.{\text{ }}68.8$ more, if the interest was payable half yearly than if it was payable annually.

So, the correct answer is “Option D”.

Note: In other words present value shows that the amount received in the future is not as worth as an equal amount received today. Always remember the relation among the present value and the principal amount. Always convert the percentage rate of interest in the form of fraction or the decimals and then substitute further for the required solutions.