Question & Answer
QUESTION

Mishael borrowed Rs.16000 from a finance company at 10% per annum compounded half-yearly. What amount of money will charge his debt after $1\dfrac{1}{2}$ years.
(a). 17652
(b). 18000
(c). 19000
(d). 18522

ANSWER Verified Verified
Hint: Here, it is given that the interest is compounded half-yearly. Therefore the rate of interest will be halved from $r$ to $\dfrac{r}{2}$ and the time is doubled from $n$ to $2n$. Then we have to calculate the amount of interest for the principal amount P, by the formula:
$A=P{{\left( 1+\dfrac{\dfrac{r}{2}}{100} \right)}^{2n}}$

Complete step-by-step answer:
We are given that Mishael borrowed Rs.16000 from a finance company at 10% per annum compounded half-yearly. We have to calculate the amount of debt after $1\dfrac{1}{2}$ years.
Here we have:
Principal amount, $P=16000$
Rate of interest, \[r\text{ }=10%\] per annum
Time, $n=1\dfrac{1}{2}=\dfrac{3}{2}$ years
 When the interest is compounded annually, we have the formula for amount, A. i.e.
$A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}$
 But, here the interest is compounded half-yearly. Therefore the number of years, $n$ is doubled and the rate of annual interest $r$ is halved. Hence we can say that:
$\begin{align}
  & n=2n \\
 & r=\dfrac{r}{2} \\
\end{align}$
 Then the amount A, is calculated using the formula:
$A=P{{\left( 1+\dfrac{\dfrac{r}{2}}{100} \right)}^{2n}}$
 Now, by substituting all the values we get:
$A=16000{{\left( 1+\dfrac{\dfrac{10}{2}}{100} \right)}^{2\times \dfrac{3}{2}}}$
Now, by cancellation we obtain:
$A=16000{{\left( 1+\dfrac{5}{100} \right)}^{3}}$
 Next, by taking the LCM we get:
$\begin{align}
  & A=16000{{\left( \dfrac{100+5}{100} \right)}^{3}} \\
 & A=16000{{\left( \dfrac{105}{100} \right)}^{3}} \\
\end{align}$
Hence by cancellation we obtain:
$\begin{align}
  & A=16000\times {{\left( \dfrac{21}{20} \right)}^{3}} \\
 & A=16000\times \dfrac{21\times 21\times 21}{20\times 20\times 20} \\
\end{align}$
 Now, again by cancellation we get:
$\begin{align}
  & A=2\times 21\times 21\times 21 \\
 & A=18522 \\
\end{align}$
 Therefore, the amount of debt after $1\dfrac{1}{2}$ years is Rs.18,522.
Hence, the correct answer for this question is option (d).

Note: Here, you have to calculate the amount for 6 months not for one year. When the interest is compounded half yearly, the principal amount will be the same, only rate of interest and time changes.