Question

A bank offers $5\%$ compound interest calculated on half-yearly basis. A customer deposits $₹1600$ each on $\text{1st}$ January and $\text{1st}$ July of a year. At the end of the year, the amount he would have gained by way of interest is?

Answer 1

Step $1:$ Finding the total amount at the end of the year:

As, the interest is compounded half yearly,

Therefore, Two times interest will be added on the amount which deposited on $\text{1st}$ January and One time nterest will be added on the amount which deposited on $\text{1st}$ July.

Now, $\text{A=P(1+}\frac{R}{200}{\text{)}}^n$, where A is the amount at the end of n half years, P is principal amount, R is rate of interest and n is number of half years.

Therefore, total amount at the end of one year on the money deposited on $\text{1st}$ January and $\text{1st}$ July is,

$\begin{array}{rcl}A& =& 1600{\left(1+\frac{5}{200}\right)}^2+1600{\left(1+\frac{5}{200}\right)}^1\\ & =& 1600{\left(1+\frac{1}{40}\right)}^2+1600{\left(1+\frac{1}{40}\right)}^1\\ & =& 1600\times {\left(\frac{41}{40}\right)}^2+1600\times {\left(\frac{41}{40}\right)}^1\\ & =& 1600\times \frac{41}{40}\times \frac{41}{40}+1600\times \frac{41}{40}\\ & =& 1681+1640\\ & =& 3321\end{array}$

Step $2:$ Finding the interest gained:

Total principal amount deposited in one year $=1600+1600=3200$

Total amount gained at the end of one year $=3321$

Therefore, Interest gained $=\mathrm{Amount}-\mathrm{Principal}$

$$\begin{gathered} \mathrm{=3321-3200} \\ =₹121 \end{gathered}$$

Hence, the amount he would have gained by way of interest is $₹121$.