Question

What amount is to be repaid on a loan of Rs. 12000 for $ 1\dfrac{1}{2} $ year at 10% per annum if interest is compounded half-yearly.

Answer 1

Hint: To calculate compound interest, we have given formula:
 $ A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}} $
Where, A = final amount
P = initial amount
r = interest rate
t = number of time periods
Hence, compound interest is the difference of final amount and initial amount.
 $ \Rightarrow CI=A-P $
Substitute the values in the formula to calculate compound interest.

Complete step-by-step answer:
As given in the question, the interest is compounded half-yearly, therefore the rate of interest is reduced half times.
That means, interest rate = 10% per annum, so, for compounding half-yearly, the interest rate = 5%.
So, r = 5 %
In the given question, the time period is $ 1\dfrac{1}{2} $ , i.e. three times a six-months interval.
So, t = 3
We have principal = Rs 12000
Using values of P, r and t, we get amount (A) as:
 $ \begin{align}
  & {{\left( SI \right)}_{3}}=\dfrac{{{A}_{2}}\times r\times t}{100} \\
 & =\dfrac{13230\times 5\times 1}{100\times 2} \\
 & =Rs.661.50
\end{align} $
Hence, Amount = Rs 13891.50

So, compound interest is:
 $ \begin{align}
  & CI=A-P \\
 & =13891.50-12000 \\
 & =1891.50
\end{align} $
Hence, compound interest = Rs 1891.50

Note: The other way to find compound interest compounded half-yearly is applying simple interest for every 6 months for the same interest rate and adding the interest in the initial value to calculate for another 6 months until for the total time period:
As it is given:
P = Rs 12000
r = 5%
t = $ 1\dfrac{1}{2} $ years = 3 $ \times $ 6 months
So, Simple interest for first 6 months is:
 $ \begin{align}
  & {{\left( SI \right)}_{1}}=\dfrac{P\times r\times t}{100} \\
 & =\dfrac{12000\times 5\times 1}{100\times 2} \\
 & =Rs.600
\end{align} $
Amount after first 6 months is:
\[\begin{align}
  & CI={{\left( SI \right)}_{1}}+{{\left( SI \right)}_{2}}+{{\left( SI \right)}_{3}} \\
 & =600+630+661.50 \\
 & =Rs.1891.50
\end{align}\]

Now, consider $ {{A}_{1}} $ as principal for another 6 months. So simple interest for another 6 months is:
 $ \begin{align}
  & {{\left( SI \right)}_{2}}=\dfrac{{{A}_{1}}\times r\times t}{100} \\
 & =\dfrac{12600\times 5\times 1}{100\times 2} \\
 & =Rs.630
\end{align} $
Amount after another 6 months is:
 $ \begin{align}
  & {{A}_{2}}={{A}_{1}}+{{\left( SI \right)}_{2}} \\
 & =12600+630 \\
 & =Rs.13230
\end{align} $

Now, consider $ {{A}_{2}} $ as principal for another 6 months. So simple interest for another 6 months is:
 $ \begin{align}
  & {{\left( SI \right)}_{3}}=\dfrac{{{A}_{2}}\times r\times t}{100} \\
 & =\dfrac{13230\times 5\times 1}{100\times 2} \\
 & =Rs.661.50
\end{align} $
Final amount after another 6 months is:
 $ \begin{align}
  & {{A}_{3}}={{A}_{2}}+{{\left( SI \right)}_{3}} \\
 & =13230+661.50 \\
 & =Rs.13891.50
\end{align} $

Hence, total interest is
\[\begin{align}
  & CI={{\left( SI \right)}_{1}}+{{\left( SI \right)}_{2}}+{{\left( SI \right)}_{3}} \\
 & =600+630+661.50 \\
 & =Rs.1891.50
\end{align}\]