Question

Find the compound interest on Rs. 10000 for 12 months at 10% per annum. If the interest is compounded half yearly.

Answer 1

Hint: We know that $Amount=P{{\left( 1+\dfrac{R}{100} \right)}^{T}}$ when the interest is compounded annually. But in this question, we have interest compounded half-yearly, thus, time = 2 $\times $ (years) = 2, rate $=\dfrac{R}{2}=\dfrac{10}{2}=5%$. It means that 5% interest after every six months. Thus, our formula to find interest will be $A=P{{\left( 1+\dfrac{\dfrac{R}{2}}{100} \right)}^{T\times 2}}$.

Complete step-by-step solution -
It is given in the question that the principal amount is Rs. 10,000, rate of interest is 10% per annum and the time is 12 months and also it is given that the type of interest is half-yearly. Then we have to find the compound interest of this case. We know that $Amount=P{{\left( 1+\dfrac{R}{100} \right)}^{T}}$ where, P is the principal amount, R is the rate of interest and T is the time period. But it is given in the question that the type of interest is half-yearly. So, the rate will be $=\dfrac{R}{2}=\dfrac{10}{2}=5%$ and time $=\dfrac{12months}{6months}=2$. It means that the man will get an interest of 5% after every six months.
Total amount $=P{{\left( 1+\dfrac{R}{100} \right)}^{T}}$
We know that, P = 10000, R = 5% and T = 2. So, on substituting these values in the formula of the total interest, we will get,
$\begin{align}
  & =10000\times {{\left( 1+\dfrac{5}{100} \right)}^{2}} \\
 & =10000\times {{\left( \dfrac{105}{100} \right)}^{2}} \\
 & =\dfrac{10000\times 105\times 105}{100\times 100} \\
 & =105\times 105 \\
 & =11025 \\
\end{align}$
Thus, the total amount = Rs. 11025.
Now, compound interest = (total amount) – (principal amount)
= Rs. 11025 – Rs. 10000
= Rs. 1025
Thus, the compound interest = Rs. 1025.

Note: In these types of questions, the students usually get confused between per annum and compounded half-yearly. Compounded annually means the interest is increased once a year whereas compounded half-yearly means the interest is increased after every six months.