Question

John deposited 2500 rupees on the first of January in a bank while interest is compounded half yearly at 6% annual rate. On the first July, he deposits 2500 rupees more. How much would he have in the account at the end of the year?

Answer 1

Hint:
 For solving this question, we will first find the compound amount for 6 months using the given rate of interest and time period as six months. Then we will add this amount to the new amount added. Since interest will remain the same, so we will find a new compound amount for the next six months. As interest is compounded half-yearly and interest is given per annum, so we will divide interest by half. The formula of the compound amount is given by $A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}$ where P is the principal amount, r is the rate of interest annually and n is the number of times amount compounds.

Complete step by step answer:
Here we are given the principal amount as Rs.2500 so P = Rs.2500. The rate of interest is given as 6% per annum. As we are given that, rate of interest is compounded half-yearly, so our interest for six months will reduce to $\dfrac{6}{2}\%=3\%$. Therefore r = 3%.
Since interest is compounded semiannually and we are given a time period between January to June so we have six months. So, the number of times the number of compounds (t) will be 1. Compound amount is given by $A=P{{\left( 1+\dfrac{r}{100} \right)}^{t}}$.
So compounded amount become equal to,
\[\begin{align}
  & A=2500{{\left( 1+\dfrac{3}{100} \right)}^{1}} \\
 & \Rightarrow A=2500\left( 1+0.03 \right) \\
 & \Rightarrow A=2500\left( 1.03 \right) \\
 & \Rightarrow A=2575 \\
\end{align}\]
Hence amount in July will be equal to Rs.2575
John deposits Rs.2500 more, so amount in bank becomes equal to Rs.2575+2500 = Rs.5075 in July. Principal amount becomes Rs.5075.
Now, new compounded amount after 6 more months will be,
\[\begin{align}
  & A=5075{{\left( 1+\dfrac{3}{100} \right)}^{1}} \\
 & \Rightarrow A=5075\left( 1+0.3 \right) \\
 & \Rightarrow A=5075\left( 1.03 \right) \\
 & \Rightarrow A=5227.25 \\
\end{align}\]
Hence amount compounded at the end of 1 year will be Rs.5227.25

Note:
 Students always forget to convert the rate of interest for compounding semi-annually. In the formula, t denotes the number of times our amount gets compounded. Take care of the signs while solving this question. Take care while solving decimal numbers.