Question

A bank offers 5% C.I. calculated on a half-yearly basis. A customer deposits Rs. 1600 each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of interest is
(a) Rs. 120
(b) Rs. 121
(c) Rs. 122
(d) Rs. 123

Answer 1

Hint: Start by using the formula that $ I=P{{\left( 1+\dfrac{r}{100n} \right)}^{nt}}-P $ for the finding the amount when the interest is compounded half yearly. First, find the interest on the principal deposited on January 1, which is deposited for 1 year and then for the principal deposited on July 1, for which $ t=\dfrac{1}{2} $ . Add the two interests to get the answer.

Complete step-by-step answer:
Before starting with the question, let us know about interest.
Interest in the financial term is the amount that a borrower pays to the lender along with the repayment of the actual principal amount.
Broadly, there are two kinds of interest, first is the simple interest, and the other is the compound interest.
Now let us move to the case in which the interest is compounded half-yearly. The rate of interest is 5% per annum, and the formula that is to be used is $ I=P{{\left( 1+\dfrac{r}{100n} \right)}^{nt}}-P $ , where n is number of times the interest is compounded yearly, which is 2 in our case.
First, let us find the interest on the principal deposited on January 1, which is deposited for 1 year.
  $ I=1600{{\left( 1+\dfrac{5}{100\times 2} \right)}^{2\times 1}}-1600 $
 $ I=1600\left( {{\left( \dfrac{41}{40} \right)}^{2}}-1 \right)=1600\left( \dfrac{81}{1600} \right)=\text{Rs}\text{. }81 $
Now, let us calculate the interest for the principal deposited on July 1, for which $ t=\dfrac{1}{2} $ .
 $ I=1600{{\left( 1+\dfrac{5}{100\times 2} \right)}^{2\times \dfrac{1}{2}}}-1600 $
\[\Rightarrow I=1600\left( \dfrac{41}{40}-1 \right)=1600\times \dfrac{1}{40}=\text{Rs}\text{. }40\]
Now, to get the total interest, we will add the two results.
 $ \text{Total interest}=81+40=\text{Rs}\text{. 121} $
So, the correct answer is “Option B”.

Note: Be careful with the calculations and solving part as there is a possibility of making a mistake in the calculations. It is recommended to learn all the basic formulas related to simple as well as compound interests as they are very much useful in the problems related to money exchange. Also, don’t get confused and take the time period t for both the cases to be 1 years, make sure that you take $ t=\dfrac{1}{2} $ for the second case.