Question

A sum of 10000 is invested at the rate of 8% per year for 12 months. What is the compound interest if interest is compounded half yearly?

Answer 1

Hint:
 In this question, we are given the principal amount, rate of interest compounded half-yearly, and time period. We will first change the rate of interest per annum and then change the time period as per the number of times the amount increases. After that, we will use the formula of the compound amount and compound interest to find out the required answer. Formula for compound amount is given by $A=P{(1+{\displaystyle \frac{r}{100}})}^n$ where P is the principal amount, r is the numerical value of rate of interest and n is the time period.
The formula for compound interest is given by,
Compound interest = Compounds amount - principal amount.

Complete step by step answer:
Here we are given the principal amount as Rs.10000. Therefore, P = 10000.
Now we are given the rate of interest as 8% compounded half yearly but rate of interest is in per annum. So we will change it according to the half year that is interest becomes half so that it can be compounded after every half year. Hence, the rate of interest becomes ${\displaystyle \frac{8}{2}}=4\mathrm{\%}$ . Therefore, r = 4%.
Interest is compounded half-yearly so time period should also be half years. As the number of months are 12 so it means we have 2 half years. Therefore, n = 12.
Now we know compound amount is given by $A=P{(1+{\displaystyle \frac{r}{100}})}^n$ where A is compound amount, P is the principal amount, r is rate of interest and n is time period. Putting in all the values we get,
$$A=10000{(1+{\displaystyle \frac{4}{100}})}^2$$
Taking LCM as 100 we get,
$$\begin{array}{rl}& A=10000{\left({\displaystyle \frac{100+4}{100}}\right)}^2\\ & \Rightarrow A=10000{\left({\displaystyle \frac{104}{100}}\right)}^2\\ & \Rightarrow A=10000\times {\displaystyle \frac{104}{100}}\times {\displaystyle \frac{104}{100}}\end{array}$$
Cancelling $100\times 100$ with 10000 we get,
$$A=104\times 104=10816$$
Hence the amount after 12 months becomes Rs.10816. Now we know that compound interest can be calculated using the formula,
Compound interest = compound amount - principal amount.
Putting in the values we get,
Compound interest = Rs.10816 - Rs.10000 = Rs.816.
Hence the required compound interest is Rs.816

Note:
 Students should not forget to convert the rate of interest and time as for half-yearly. Students should note that in the formula for calculating the compound amount, we fill the numerical value of r only. For percentage, 100 is already divided.