Question

Find the amount which Ram will get on$Rs{\text{ }}4096$ , if he gave it for $18$ month at $12\dfrac{1}{2}\%$ per annum, interest being compounded half-yearly.
A.$Rs.4913$
B.$Rs.4425$
C.$Rs.4814$
D.$Rs.4124$

Answer 1

Before proceeding further we need the formula to calculate the amount when the interest is compounded half-yearly.
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \ldots \left( 1 \right)$.
Where,
$A = {\text{Amount}} \\\ P = {\text{Principal}} \\\ R = {\text{Rate}} \\\ t = {\text{time}} \\\$

Complete step by step solution
When we are calculating compound interest yearly there is no impact but in case of the half-yearly, there is impact on time as well as rate.
In this question we have,
$A = {\text{?}} \\\ P = {\text{4096}} \\\ R = 12\dfrac{1}{2} = \dfrac{{12 \times 2 + 1}}{2} = \dfrac{{25}}{2} \\\ t = {\text{18 month}} \\\$
Now we need to calculate time in a year.
$12\,{\text{month}} = 1\,{\text{year}} \\\ 1\,{\text{month}} = \dfrac{1}{{12}} \times {\text{year}} \\\ 18\,{\text{month}} = \dfrac{{18}}{{12}}{\text{year}} \\\ {\text{ = }}\dfrac{3}{2}{\text{year}} \\\$
In half-yearly, time is double and rate is half then we can rewrite the rate and time as,
$t = \dfrac{3}{2} \times 2 = 3\,{\text{half yearly}} \\\ R = \dfrac{{25}}{2} \times \dfrac{1}{2} = \dfrac{{25}}{4}\% \\\$
Now we have all the values except A which we need to calculate substitute all the value in the equation $\left( 1 \right),$

$$A = 4096{\left( {1 + \dfrac{{25}}{4} \times \dfrac{1}{{100}}} \right)^3} \\\ = 4096{\left( {1 + \dfrac{1}{4} \times \dfrac{1}{4}} \right)^3} \\\ = 4096{\left( {1 + \dfrac{1}{{16}}} \right)^3} \\\ = 4096{\left( {\dfrac{{17}}{{16}}} \right)^3} \\\ = 4096 \times \left( {\dfrac{{4913}}{{4096}}} \right) \\\ = 4913 \\\$$

So by the discussion, we see that the amount is$Rs{\text{ }}4913$ .
Thus, the amount is $Rs{\text{ }}4913$.

Note: Here the main point of concern is while calculating time and rate in case of half year.
We must note that if time is $1$ year and rate is $1\%$ then in case of half-year time will be $2$ year and the rate will be $0.5\% .$
While calculating the time year from month we must follow a unitary method of calculation.