Question

A bank offers $5\% $ compound interest calculated on a half-yearly basis. A customer deposits Rs.$16000$ 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 the way of interest is-
A.$Rs.1200$
B.$Rs.1210$
C.$Rs.1220$
D.$Rs.1230$

Answer 1

Hint: We will use the given formula to calculate compound interest calculated half-yearly-
$ \Rightarrow $ I=${\text{P}}{\left( {{\text{1 + }}\dfrac{{\text{r}}}{{{\text{100n}}}}} \right)^{{\text{nt}}}} - {\text{P}}$
Where n is the number of times the interest is compounded in a year and t is the time in years, P is the principal.
Here n= $2$ and t=$1$ for the interest generated for principal deposited on ${\text{1st}}$January and n=$2$ and t=$\dfrac{1}{2}$ for the interest generated on ${\text{1st}}$ July. Put the given values in the formula and solve. Then add both the interests calculated to find the amount gained by the way of interest.

Complete step-by-step answer:
Given, the rate offered by bank =$5\% $
The compound interest is calculated on half yearly basis. So in a year, the interest is compounded two times.
Principal= Rs.$16000$
We have to find the amount.
We will use the formula of interest-
$ \Rightarrow $ I=${\text{P}}{\left( {{\text{1 + }}\dfrac{{\text{r}}}{{{\text{100n}}}}} \right)^{{\text{nt}}}} - {\text{P}}$
Where n is the number of times the interest is compounded in a year and t is the time in years, P is the principal.
Here n= $2$ and t=$1$
So the interest generated for principal deposited on ${\text{1st}}$January is-
$ \Rightarrow I = 16000{\left( {1 + \dfrac{5}{{100 \times 2}}} \right)^{2 \times 1}} - 16000$
On solving, we get-
$ \Rightarrow I = 16000{\left( {1 + \dfrac{1}{{40}}} \right)^1} - 16000$
On taking LCM, we get-
$ \Rightarrow I = 16000{\left( {\dfrac{{41}}{{40}}} \right)^2} - 16000$
On taking $16000$ common and solving, we get-
$ \Rightarrow I = 16000\left( {\dfrac{{1681 - 1600}}{{1600}}} \right)$
On solving, we get-
$ \Rightarrow I = 16000 \times \dfrac{{81}}{{1600}} = 810$
Now n=$2$ and t=$\dfrac{1}{2}$ for the interest generated on ${\text{1st}}$ July
Now the interest generated for principal deposited on ${\text{1st}}$ July is-
$ \Rightarrow I = 16000{\left( {1 + \dfrac{5}{{100 \times 2}}} \right)^{2 \times \dfrac{1}{2}}} - 16000$
On solving, we get-
$ \Rightarrow I = 16000\left( {\dfrac{{41}}{{40}}} \right) - 16000$
On solving, further, we get-
$ \Rightarrow I = 16000 \times \dfrac{{41 - 40}}{{40}} = \dfrac{{16000}}{{40}}$
On solving, we get-
$ \Rightarrow I = 400$
Then total interest=$810 + 400 = 1210$Rs
The correct answer is option B.

Note: Here the student may go wrong if he/she takes t=$1$ for both compound interest as then the interest will be the same and the amount will be zero. For the first CI the time will be one year but for the second CI the time will be only one half of the year because the money is deposited in July.