Question

What sum will become Rs. 9826 in 18 months if the rate of interest is \[2\dfrac{1}{2}\% \] per annum and the interest is compounded half-yearly?
A. Rs. 9466.54
B. Rs. 9646.54
C. Rs. 9566.54
D. Rs. 9456.54

Answer 1

Hint: Half a year consists of 6 months, \[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\] where A is the final amount, P is the initial principal balance r is the interest rate, number of times interest applied per time period and t is the number of time period elapsed.

Complete step-by-step answer:
In the question we are given the total amount as 9826 and interest per annum is \[2\dfrac{1}{2}\% \] and also n will be one as 3 as half year means 6 months and there are 3, 6 months in total of 18 months also it is given that interest per annum is \[2\dfrac{1}{2}\% \] but the interest half yearly will be half of it
\[\therefore r = \dfrac{5}{{2 \times 2}} = \dfrac{5}{4}\% \]
Now let us put all of this in \[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\] and find the value of P.
\[\begin{array}{l}
\therefore A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\\
 \Rightarrow 9826 = P{\left( {1 + \dfrac{{\dfrac{5}{4}}}{{100}}} \right)^3}\\
 \Rightarrow 9826 = P{\left( {1 + \dfrac{5}{{400}}} \right)^3}\\
 \Rightarrow 9826 = P{\left( {\dfrac{{405}}{{400}}} \right)^3}\\
 \Rightarrow 9826 = P{\left( {1.0125} \right)^3}\\
 \Rightarrow P = \dfrac{{9826}}{{{{\left( {1.0125} \right)}^3}}}\\
 \Rightarrow P = \dfrac{{9826}}{{1.03797}}\\
 \Rightarrow P = 9466.54
\end{array}\]
So from here it is clear that option A is the correct option.


Note: Finding the value of n is one of the most important key steps in this question as the other things depend upon that only. Also beawar while doing calculation. There are a lot of chances of making a mistake there and by doing everything correct also students get penalised due to that as finding a cube of 1.0125 and then dividing by 9826 needs a lot of precision.