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.

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.

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}