Question

Zaved got a loan of Rs $8000$ against his fixed deposits to purchase a scooter. If the rate of interest is ten percent per annum compounded half-yearly, find the amount that he pays after a year and a half.

Answer 1

Hint: For solving this problem, we need to have a clear understanding of what compound interest means. Due to the fact that interest is compounded half-yearly, the compounding frequency will be two. Hence, using the standard formula, the amount that Zaved pays after a year and a half is Rs $9261$ .

Complete step-by-step answer:
Compound interest is the addition of interest to the principal sum of a loam or deposit, or in other words, interest on interest. It is the result of reinvesting interest rather than paying it out, so that interest in the next period is then earned on the principal sum plus the previously accumulated interest. The rate at which compound interest accrues depends on the frequency of compounding, such that the higher the number of compounding periods, the greater the compound interest.
In the given problem, the principal amount is the loan Zaved gets against his fixed deposits which is equal to Rs $8000$ . The rate of interest is 10% per annum compounded half yearly. The total accumulated value, including the principal sum, P plus the compounded interest I is given by the formula,
$P'=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}$
Where, P is the original principal sum which is Rs $8000$ , P’ is the new principal sum or the amount that Zaved has to pay, r is the nominal annual interest rate which is ten percent, n is the compounding frequency which is two due to the fact that the interest is compounded half-yearly. t is the overall length of the time the interest is applied which is one and half years. Now, substituting the values in the equation we get that,
\[\begin{align}
  & P'=8000{{\left( 1+\dfrac{10}{200} \right)}^{2\times \dfrac{3}{2}}} \\
 & \Rightarrow P'=8000{{\left( 1+\dfrac{1}{20} \right)}^{3}} \\
 & \Rightarrow P'=8000{{\left( \dfrac{21}{20} \right)}^{3}} \\
 & \Rightarrow P'=9261 \\
\end{align}\]
Thus, the amount that Zaved pays after a year and a half is Rs $9261$ .

Note: These problems might seem to be pretty easy to solve, but slight miscalculations can lead to a totally different answer. We need to carefully understand what the compounding frequency is before solving the problem. We also should not get confused between the compound interest and the new principal sum before stating the final answer.