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# The compound interest on Rs.20,480 at $6\dfrac{1}{4}\%$ per annum for 2 years 73 days isA.Rs.2929B.Rs.3000C.Rs.3131D.Rs.3636

Last updated date: 14th Aug 2024
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Hint: We will use the formula to calculate the amount A given by $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}$ where P is the principal amount, R is rate of interest, n is the number of years. We will convert n = 2 years 73 days in years only using the formula: $x{\text{ years y days = }}x \cdot \dfrac{y}{{365}}{\text{ years}}$ and rate of interest R as a whole fraction. Then, compound interest (CI) will be given by: $CI = A - P$.

We are required to calculate the Compound Interest (CI) on Rs.20480 at rate of interest $6\dfrac{1}{4}\%$ per annum for a time period of 2 years 73 days.
We know that CI is given by $CI = A - P$ where A is the amount and P is the principal.
Here, P is Rs.20480 and amount A can be calculated by the formula: $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}$, where R is the rate of interest and n is the time period.
Now, R = rate of interest = $6\dfrac{1}{4}\%$. We can write this equation, by using the formula: $x\dfrac{y}{z} = \dfrac{{(xz) + y}}{z}$as: $6\dfrac{1}{4}\% = \dfrac{{25}}{4}\%$
And, n = time period = 2 years 73 years. We can write the time period using the formula: $x{\text{ years y days = }}x \cdot \dfrac{y}{{365}}{\text{ years}}$ as $2{\text{ years 73 years = }}2\dfrac{{73}}{{365}} = 2\dfrac{1}{5}{\text{ years}}$
Substituting the values of P, R and n in the equation of A, we get
$\Rightarrow A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n} = 20480{\left( {1 + \dfrac{{25}}{{4\left( {100} \right)}}} \right)^2}\left( {1 + \dfrac{{\dfrac{{25}}{4} \cdot \dfrac{1}{5}}}{{100}}} \right)$ (since n is a fraction here so, we have first used for time period of 2 years and then for 73 days or $\dfrac{1}{5}{\text{ years}}$)
$\Rightarrow A = 20480{\left( {1 + \dfrac{1}{{16}}} \right)^2}\left( {1 + \dfrac{1}{{80}}} \right) = 20480{\left( {\dfrac{{17}}{{16}}} \right)^2}\left( {\dfrac{{81}}{{80}}} \right) \\ \Rightarrow A = 20480\left( {\dfrac{{23409}}{{20480}}} \right) = Rs.23409 \\$
Hence, the value of the amount is Rs.23409 and principal P is Rs.20480.
Therefore, the value of the compound interest: $CI = A - P$ will be:
$\Rightarrow CI = Rs.23409 - Rs.20480 \\ \Rightarrow CI = Rs.2929 \\$
Hence, the compound interest on Rs.20,480 at $6\dfrac{1}{4}\%$ per annum for 2 years 73 days is Rs.2929.
Therefore, option (A) is correct.

Note: In this question, you may get confused in the step when we calculated the value of the amount A by putting $n = 2\dfrac{1}{5}years$ and then using it individually for 2 years and then $\dfrac{1}{5}years$. You can also solve this question by calculating the compound interest for 2 years and then simple interest for 73 days given by $\dfrac{{PRT}}{{100}}$ and then by adding them, we will get the value of total compound interest by adding them.