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?

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Solution

## 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,$\begin{array}{rcl}A& =& 1600{\left(1+\frac{5}{200}\right)}^{2}+1600{\left(1+\frac{5}{200}\right)}^{1}\\ & =& 1600{\left(1+\frac{1}{40}\right)}^{2}+1600{\left(1+\frac{1}{40}\right)}^{1}\\ & =& 1600×{\left(\frac{41}{40}\right)}^{2}+1600×{\left(\frac{41}{40}\right)}^{1}\\ & =& 1600×\frac{41}{40}×\frac{41}{40}+1600×\frac{41}{40}\\ & =& 1681+1640\\ & =& 3321\end{array}$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}$$\mathrm{=3321-3200}\phantom{\rule{0ex}{0ex}}=₹121$Hence, the amount he would have gained by way of interest is $₹121$.

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