Question

Yadaish for his family needs to borrow Rs.$5120\,at\,12\dfrac{1}{2}\% $per annum compound annually. How much does he have to pay to clear the debt at the end of two year nine months? Also find the total how much interest he has Paid.

Answer 1

Hint: We will convert to year. And then by using the formula of amount , we will calculate and then compound interest .
Amount$ = \dfrac{{P \times R \times T}}{{100}}$.n
Amount $ = P{\left( {1 + \dfrac{R}{{100}}} \right)^x}$
C.I $ = Amount - Principle$
S.I $ = \dfrac{{P \times R \times T}}{{100}}$
Here, $P = Principle\,\,sum$
$R = $Rate of interest
$T = $Time


Complete step by step solution:
P\[ = \]₹$5120$
$R = 12\dfrac{1}{2}\% $
$R = \dfrac{{25}}{2}\% $
$T = 2years$9 months
$T = 2\dfrac{9}{{12}}$
\[T = 2\dfrac{3}{4}\]years
So, we will break time into two parts
First in $2$years and second $T = \dfrac{3}{4}$years
Now, we will find the amount , we have
$Amount = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$
Amount $ = 5120{\left( {1 + \dfrac{{\dfrac{{25}}{2}}}{{100}}} \right)^2}$
Amount $ = 5120{\left( {1 + \dfrac{{25}}{{2 \times 100}}} \right)^2}$
Amount $ = 5120{\left( {\dfrac{1}{1} + \dfrac{1}{8}} \right)^2}$
Amount $ = 5120{\left( {\dfrac{{1 + 8}}{8}} \right)^2}$
Amount $ = 5120 \times {\left( {\dfrac{9}{8}} \right)^2}$
Amount $ = 5120 \times \dfrac{9}{8} \times \dfrac{9}{8}$
Amount $ = 640 \times 9 \times \dfrac{9}{8}$
Amount $ = 80 \times 9 \times 9$
Amount $ = 80 \times 81$
Amount $ = 6480$
Now, we will take $\dfrac{3}{4}$years and find the value of simple interest, here $ = 6480$
Simple interest $ = \dfrac{{P \times R \times T}}{{100}}$
Simple interest $ = \dfrac{{6480 \times 25 \times \dfrac{3}{4}}}{{2 \times 100}}$
Simple interest $ = \dfrac{{6480 \times 25 \times 3}}{{2 \times 4 \times 100}}$
Simple interest $ = \dfrac{{648 \times 25 \times 3}}{{2 \times 4 \times 10}}$
Simple interest $ = \dfrac{{648 \times 5 \times 3}}{{2 \times 4 \times 2}}$
Simple interest $ = \dfrac{{324 \times 5 \times 3}}{{4 \times 2}}$
Simple interest $ = \dfrac{{324 \times 5 \times 3}}{{4 \times 2}}$
Simple interest $ = \dfrac{{81 \times 5 \times 3}}{2}$
Simple interest $ = \dfrac{{81 \times 15}}{2}$
Simple interest $ = \dfrac{{1215}}{2}$
Simple interest $ = $₹$607.50$
So, the amount at the end of $2$year’s $9$months,
$
   = 6480 + 607.50 \\
   = 7087.50 \\
 $
Therefore, the amount $ = $₹$1967.50$
Hence, the compound interest is ₹$1967.50$


Note: Students can convert the time into the years because some time it is given in years, some time in months or some time in years and months.