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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) $Rs.3579.71$

Last updated date: 10th Sep 2024
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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.