Question

A person borrowed some money and returned it in three equal quarterly installments of Rs 4630.50 each. What sum(approximately) did he borrow if the rate of interest was 20% per annum compounded quarterly?
A. Rs 12613.48
B. Rs 10613.48
C. Rs 11613.48
D. None of these

Answer 1

Hint:Compound interest is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. The formula for compound interest is-
${\text{A}} = {\text{P}}{\left( {1 + \dfrac{{\text{r}}}{{100}}} \right)^{\text{t}}}$
Here A is the final amount, P is the initial principal amount, r is the rate of interest, and t is the time in years. We have been given the amount A as Rs 4630.5. We should keep in mind that the payment is made quarterly, so the given rate of interest will be divided by 4, and we will calculate the principal amount separately for each quarter.

Complete step-by-step answer:
We have been given that the installments have been made quarterly, so that rate of interest for each quarter will be-
$\dfrac{{20}}{4}\% = 5\%$
At the end of the first quarter, using the formula for compound interest we get-
A = Rs 4630.5, t = 1 quarter, r = 5%
$\begin{aligned} &4630.5 = {{\text{P}}_1}{\left( {1 + \dfrac{5}{{100}}} \right)^1} \\\ &4630.5 = {{\text{P}}_1}{\left( {\dfrac{{105}}{{100}}} \right)^1} = {{\text{P}}_1}\left( {\dfrac{{21}}{{20}}} \right) \\\ &{{\text{P}}_1} = 4630.5\left( {\dfrac{{20}}{{21}}} \right)...\left( 1 \right) \\\ \end{aligned}$

Similarly for the second installment, we can apply the formula as-
A = Rs 4630.5, t = 2 quarter, r = 5%
$\begin{aligned} &4630.5 = {{\text{P}}_2}{\left( {1 + \dfrac{5}{{100}}} \right)^2} \\\ &4630.5 = {{\text{P}}_2}{\left( {\dfrac{{105}}{{100}}} \right)^2} = {{\text{P}}_2}{\left( {\dfrac{{21}}{{20}}} \right)^2} \\\ &{{\text{P}}_2} = 4630.5{\left( {\dfrac{{20}}{{21}}} \right)^2}...\left( 2 \right) \\\ \end{aligned}$

Similarly for the third installment, we can apply the formula as-
A = Rs 4630.5, t = 3 quarter, r = 5%
$\begin{aligned} &4630.5 = {{\text{P}}_3}{\left( {1 + \dfrac{5}{{100}}} \right)^3} \\\ &4630.5 = {{\text{P}}_3}{\left( {\dfrac{{105}}{{100}}} \right)^3} = {{\text{P}}_3}{\left( {\dfrac{{21}}{{20}}} \right)^3} \\\ &{{\text{P}}_3} = 4630.5{\left( {\dfrac{{20}}{{21}}} \right)^3}...\left( 3 \right) \\\ \end{aligned}$

To find the total sum borrowed, we can write that-
$\begin{aligned} &Total;\;sum = {{\text{P}}_1} + {{\text{P}}_2} + {{\text{P}}_3} \\\ &Total;\;sum = 4630.5\left( {\dfrac{{20}}{{21}}} \right) + 4630.5{\left( {\dfrac{{20}}{{21}}} \right)^2} + 4630.5{\left( {\dfrac{{20}}{{21}}} \right)^3} \\\ &Total;\;sum = 4630.5\left( {{{\left( {\dfrac{{20}}{{21}}} \right)}^1} + {{\left( {\dfrac{{20}}{{21}}} \right)}^2} + {{\left( {\dfrac{{20}}{{21}}} \right)}^3}} \right) \\\ &Total;\;sum = Rs\;12610 \approx Rs\;12613.48 \\\ \end{aligned}$

This is the total sum borrowed, the correct option is A.

Note: Here, a common mistake is that the students take the rate of interest as 20%, instead of 5%. When we look closely, we can see that the rate of interest is given as 20% per annum, but the installments are made quarterly. This means that we need to divide the rate of interest by 4 to get the quarterly rate of interest. There is an alternative shortcut method to solve such problems using the concept of present worth. The formula can be apply directly as-
$Total\;sum = \dfrac{{4630.5}}{{{{\left( {1 + \dfrac{5}{{100}}} \right)}^1}}} + \dfrac{{4630.5}}{{{{\left( {1 + \dfrac{5}{{100}}} \right)}^2}}} + \dfrac{{4630.5}}{{{{\left( {1 + \dfrac{5}{{100}}} \right)}^3}}}$