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$ 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

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}\]

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