Question

Find the compound interest on Rs. 2,50,000 at the rate of $8\%$ per annum for 1.5 years when the interest is compounded half yearly.

Answer 1

Hint: To solve this question, we need to know the method for calculating the compound interest (C I) at the end of a certain specified period is equal to the difference between the amount at the end of the period and the original amount, that is C I = Amount - Principal. If we take P as the principal and $R\%$ as the rate of interest per annum and if the interest is compounded half yearly, then the amount A and C I at the and of n years is given by the formulas, $A=P{{\left( 1+\dfrac{R}{100} \right)}^{2n}}$ and $CI=P\left[ {{\left( 1+\dfrac{R}{100} \right)}^{2n}}-1 \right]$ respectively. We will use these to find the desired answer.

Complete step-by-step answer:
We have been asked to find the compound interest of Rs. 2,50,000 at the rate of $8\%$ per annum for 1.5 years when the interest is compounded half yearly. Here, in the question we have been given the principal amount, $P=Rs.2,50,000$ and the rate of interest, $R=8\%$ per annum and the time period as $n=1.5=\dfrac{3}{2}$ years. To find the compound interest, we have to first find the Amount, A at the end of the time period. So, we will use the formula, $A=P{{\left( 1+\dfrac{R}{100} \right)}^{2n}}$ to calculate the same. So, we get,
$\begin{align}
  & A=250000{{\left[ 1+\dfrac{8}{100} \right]}^{2\times \dfrac{3}{2}}} \\
 & \Rightarrow A=250000{{\left[ 1+0.08 \right]}^{3}} \\
 & \Rightarrow A=250000\times 1.259712 \\
 & \Rightarrow A=Rs.3,14,928 \\
\end{align}$
So, now that we have obtained the Amount, we can find the C I, using the formula,
$\begin{align}
  & CI=A-P \\
 & \Rightarrow CI=314928-250000 \\
 & \Rightarrow CI=Rs.64,928 \\
\end{align}$
Therefore, Rs. 64,928 is the compound interest earned on Rs. 2,50,000 at the rate of $8 \%$ per annum for 1.5 years, when interest is compounded half yearly.

Note: We have to note here that we have used the formula for calculating the amount as, $A=P{{\left( 1+\dfrac{R}{100} \right)}^{2n}}$ since the interest was compounded half yearly. If the interest was to be calculated on a yearly basis, then the formula for the amount and C I will become, $A=P{{\left( 1+\dfrac{R}{100} \right)}^{n}}$ and $CI=P\left[ {{\left( 1+\dfrac{R}{100} \right)}^{n}}-1 \right]$. And if the compound interest has to be calculated for a quarterly basis, then the formula becomes, $A=P{{\left( 1+\dfrac{R}{100} \right)}^{4n}}$ and $CI=P\left[ {{\left( 1+\dfrac{R}{100} \right)}^{4n}}-1 \right]$. So, the students should be careful with the application of the formulas according to the questions.