What is the compound interest on Rupees 1600 at 20% per annum for one and a half years compounded half yearly?

Answer

Verified

Hint: Find the half yearly interest rate on the amount. Convert $1\dfrac{1}{2}$ years into half years and use the fact that if an amount P is compounded half-yearly with half yearly interest rate to be r for n half years, then the amount after n half years is given by $A=P{{\left( 1+r \right)}^{n}}$. Use the fact that the interest I is given by $I=A-P$. Hence find the interest on the given amount.

Complete step by step answer:
In the question the interest is levied on Rs 16000. Hence, we have P = 16,000
Also, we have rate of interest $r=10\%p.a$
Hence, we have
$r=\dfrac{10}{2}\%=5\%$ half-yearly
Also, we have $1\dfrac{1}{2}\text{ years = }\dfrac{3}{2}\text{ years = }\dfrac{3}{2}\times 2\text{ half years = 3 half years}$
Hence, we have $n=3\text{ half years}$
We know that if an amount P is compounded half-yearly with half yearly interest rate to be r for n half years, then the amount after n half years is given by $A=P{{\left( 1+r \right)}^{n}}$.
Hence, we have
$\begin{align}
  & A=P{{\left( 1+r \right)}^{n}}=16000{{\left( 1+\dfrac{5}{100} \right)}^{3}} \\
 & =16000\times {{\left( 1.05 \right)}^{3}}=18522 \\
\end{align}$
We know that if A is the total amount and P is the principle, then the interest is given by $I=A-P$
Hence, we have
$I=18522-16000=2522$
Hence the interest on the given amount after $1\dfrac{1}{2}$ years at 10% per annum compounded half yearly is Rs 2522.
Hence option [b] is correct.

Note: Alternative Solution: We calculate the amount at the end of each half year and hence calculate the total interest.
For the first half year, we have
$P=16000,r=10\%,t=\dfrac{1}{2}$
We know that $I=\dfrac{P\times r\times t}{100}$
Hence, we have
$I=\dfrac{16000\times 10\times 1}{2\times 100}=800$
For the second half year, we have
$P=16000+800=16800,$r = 10% and $t=\dfrac{1}{2}$
Hence, we have
$I=\dfrac{16800\times 10\times 1}{2\times 100}=840$
For the third half year, we have
$P=16800+840=17640$, r = 10% and $t=\dfrac{1}{2}$
Hence, we have
$I=\dfrac{17640\times 10\times 1}{2\times 100}=882$
Hence the total interest is $=800+840+882=2522$, which is the same as obtained above.
Hence option [b] is correct.

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