Question

At what rate per cent per annum, will Rs.32000 yield a compound interest of Rs.5044 in 9 months interest being compounded quarterly?
A) 25
B) 23
C) 20
D) 18

Answer 1

 Hint: The compound interest is the interest that adds back to the principal sum, so that interest is earned during the next compounding period. The compound interest is also called as “interest on interest”.
$C.I. = P{\left( {1 + \dfrac{r}{{100}}} \right)^n} - P$
where, P is the principal amount
A is the total amount
r is the rate of interest
n is the time(in years)
The amount is the sum of the principal amount and interest.
Amount = Principal + Interest
Here, we will calculate the total amount then by applying the formula of amount, calculate the rate per annum. Students must keep in mind that for how much time the interest is being compounded.

Complete step-by-step answer:
According to the question,
Principal (P) = Rs.32000
Interest (I) = Rs.5044
As we know that,
Amount (A) = Principal + Interest
Amount = Rs.32000 + Rs.5044
Amount = Rs.37044
Rate = r% p.a. or $\dfrac{r}{4}\% $ per quarter
Time = 9 months = 3 quarters i.e., n=3
∴Applying the formula of compound interest $A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}$
$\begin{gathered}
  37044 = 32000{\left( {1 + \dfrac{r}{{400}}} \right)^3} \\
  \dfrac{{37044}}{{32000}} = {\left( {1 + \dfrac{r}{{400}}} \right)^3} \\
  \dfrac{{9261}}{{8000}} = {\left( {1 + \dfrac{r}{{400}}} \right)^3} \\
  {\left( {\dfrac{{21}}{{20}}} \right)^3} = {\left( {1 + \dfrac{r}{{400}}} \right)^3} \\
  1 + \dfrac{r}{{400}} = \dfrac{{21}}{{20}} \\
\end{gathered} $
$\begin{gathered}
  \dfrac{r}{{400}} = \dfrac{{21}}{{20}} - 1 \\
  \dfrac{r}{{400}} = \dfrac{1}{{20}} \\
  r = \dfrac{{400}}{{20}} \\
  r = 20\% p.a. \\
\end{gathered} $
Hence, the rate per annum is 20%

∴Option(C) is correct.

Note: The major mistakes students make while calculating the compound interest is applying the proper frequency of compounding. Some important formulas in compound interest are:
When rate of interest is compounded half-yearly, Amount =$P{\left( {1 + \dfrac{r}{{200}}} \right)^{2n}}$
When rate of interest is compounded quarterly, Amount = $P{\left( {1 + \dfrac{r}{{400}}} \right)^{4n}}$
When rate of interest is compounded ‘a’ times a year, Amount = $P{\left( {1 + \dfrac{r}{{a \times 100}}} \right)^{an}}$
When rate of interest is r1%, r2% and r3% for 1st year, 2nd year and 3rd year respectively,
Amount = \[P\left( {\left[ {1 + \dfrac{{{r_1}}}{{100}}} \right] \times \left[ {1 + \dfrac{{{r_2}}}{{100}}} \right] \times \left[ {1 + \dfrac{{{r_3}}}{{100}}} \right]} \right)\]