Question

Sam invested Rs15000 at 10% per annum for one year. If the interest is compounded half-yearly, then the amount received by Sam at the end of year will be
A. Rs16500
B. Rs16525.50
C. Rs16537.50
D. Rs18150

Answer 1

Hint: Here we use the formula of compound interest to calculate the value of the amount Sam receives. We calculate the rate of interest for half the year and calculate the time period in terms of half year. Substitute values in the formula to find the value of the amount received.
* Half-yearly means \[\dfrac{1}{2}\]part of a year.
* If amount from compound interest is denoted by A, Principal amount by P, Rate of interest by R and time period by T, then we have the formula\[A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}\].

Complete step by step answer:
We know that the principal amount given is Rs15000
So, \[P = 15000\]
We are given the rate of interest for one year is \[10\% \]
When we have compound interest per annum, then we write rate of interest same as given to us, but here we convert the rate of interest from per annum to half-yearly by multiplying the rate of interest by \[\dfrac{1}{2}\].
Rate of interest for 1 year is \[10\% \]
So rate of interest for \[\dfrac{1}{2}\]of a year is given by \[\dfrac{1}{2} \times 10\% \]
Calculate the value of Rate of interest.
Therefore, \[R = 5\% \]
Also, the time period is 1 year
We write time period with reference to half-yearly time period
Using unitary method, we can write
\[ \Rightarrow \]1 month is \[\dfrac{1}{{12}}\]of a year.
\[ \Rightarrow \]6 months will be \[\dfrac{1}{{12}} \times 6\] of the year.
\[ \Rightarrow \]6 months is \[\dfrac{1}{2}\]year.
We know 12 months give us 1 year
\[\Rightarrow \]1 year has 2 half-yearly time periods in it
So, time period becomes \[T = 2\]
We know \[A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}\]
Now we substitute the values of \[P = 15000,R = 5,T = 2\]in the formula for compound interest.
\[ \Rightarrow A = 15000{\left( {1 + \dfrac{5}{{100}}} \right)^2}\]
Take LCM inside the bracket
\[ \Rightarrow A = 15000{\left( {\dfrac{{100 + 5}}{{100}}} \right)^2}\]
\[ \Rightarrow A = 15000{\left( {\dfrac{{105}}{{100}}} \right)^2}\]
Now square the terms inside the bracket.
\[ \Rightarrow A = 15000\dfrac{{105 \times 105}}{{100 \times 100}}\]
Cancel the same terms from numerator and denominator.
\[ \Rightarrow A = 15000\dfrac{{21 \times 21}}{{20 \times 20}}\]
Cancel the same terms from numerator and denominator.
\[ \Rightarrow A = \dfrac{{66150}}{4}\]
\[ \Rightarrow A = 16,537.50\]

Thus, the amount received by Sam at the end of year is Rs16537.50. Hence, option (C) is correct.

Note:
Students are likely to make mistakes in the solution because they don’t convert the values of time period and rate of interest given in terms of years to half-yearly. Keep in mind to always convert the terms according to the need of the question and then substitute values in the formula.