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Roma took a loan of Rs. \[16,000\] against her insurance policy at the rate of \[12\dfrac{1}{2}\% \] per annum. Calculate the total compound interest that will be paid by Roma after \[3\] years.
A) \[Rs.6781.25\]
B) \[Rs.6925.30\]
C) \[Rs.4296.82\]
D) \[Rs.3579.71\]

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Last updated date: 29th May 2024
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Answer
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Hint: In the given question, we have to find the compound interest where the principal amount is borrowed and the rate of annual interest is given. We can use the formula \[A = P{(1 + \dfrac{r}{n})^{n\,t}}\] to calculate the amount payable after three years and then deduct the principal amount to arrive at the compound interest i.e. \[CI = A - P\].

Complete step by step solution:
Compound interest (also known as compounding interest) is the interest on a loan or deposit that is measured using both the original principal and the interest accrued over time.
The total compounded amount payable after a specific period can be calculated with the following formula:
\[A = P{(1 + \dfrac{r}{n})^{n\,t}}\]
Where,
\[A = \]Final Amount
\[P = \]Principal Amount
\[r = \]Rate of interest
\[n = \]Number of times interest is to be compounded per year/annum
\[t = \]Time (in years)
For example, Rs.\[100\] compounded at \[10\% \] per annum for \[2\] year will result in:
\[ = 100{(1 + \dfrac{{10\% }}{1})^{(1)(2)}}\]
\[ = 100{(1 + 0.1)^2}\]
On simplifying in further we get
\[ = 100{(1.1)^2}\]
\[ = 100(1.21)\]
\[ = 121\]
Here, if the interest was to be compounded half-yearly, then \[n = 4\]as interest will be compounded four times over the two years half-yearly. Similarly, if it was quarterly, then \[n = 8\].
Now we can solve the sum as follows:
The loan taken will become our principal amount and rate of interest is given to be compounded annually. Hence-
\[P = 16,000\]
\[n = 1\] (annually)
\[t = 3\] (years)
\[r = 12\dfrac{1}{2}\% = 0.125\]
Applying the formula, we get,
\[A = P{(1 + \dfrac{r}{n})^{n\,t}}\]
\[A = 16000{(1 + \dfrac{{0.125}}{1})^{(1)(3)}}\]
\[A = 16000{(1 + 0.125)^3}\]
On simplifying in further we get
\[A = 16000{(1.125)^3}\]
\[A = 16000(1.4238)\]
\[A = 22781.25\]
Now Compound Interest (CI) can be found out as follows:
\[CI = A - P\]
\[CI = 22781.25 - 16000\]
\[CI = 6781.25\]
Hence compounded Interest will be \[Rs.6781.25\] over the period of three years. Therefore, Option (A) is correct.

Note:
We can also apply the following formula to solve the sum directly:
\[CI = P[{(1 + \dfrac{r}{{100}})^n} - 1]\] where \[n = \]number of years
\[CI = 16000[{(1 + \dfrac{{12.5}}{{100}})^3} - 1]\]
\[CI = 16000[{(1 + 0.125)^3} - 1]\]
On simplifying in further we get
\[CI = 16000[{(1.125)^3} - 1]\]
\[CI = 16000[1.4238 - 1]\]
\[CI = 16000[0.4238]\]
\[CI = 6780.8\]
Difference of \[6781.25 - 6780.8 = 0.45\] in the answer is due to decimal points.