Question

Shyam borrowed a certain sum of money and pays it back in 2 years in two equal yearly instalments. The compound interest is charged at 5 % per annum and he pays back Rs. 441 every year, what was the amount that Shyam borrowed?

• A. INR 810
• B. INR 820
• C. INR 840
• D. INR 850

Hint:

Notice that the installment amount of 441 is a perfect square ($21^2$), which is deliberately chosen to cancel out with the denominator of the compound interest factor $\left(\frac{21}{20}\right)^2$. Keep an eye out for these mathematical simplifications.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To find the borrowed sum (present value) under a compound interest installment plan, we calculate the present value of each individual installment.
Key Formula or Approach:
The present value $P$ of two equal installments $x$ at an interest rate $r\%$ is: \[ P = \frac{x}{1 + \frac{r}{100}} + \frac{x}{\left(1 + \frac{r}{100}\right)^2} \]

Step 2: Detailed Explanation:
Given values:
- Installment amount ($x$) = Rs. 441
- Rate of interest ($r$) = $5\%$
Calculate the discount factor: \[ 1 + \frac{r}{100} = 1 + 0.05 = 1.05 = \frac{21}{20} \] Substitute these values into the present value formula: \[ P = \frac{441}{\frac{21}{20}} + \frac{441}{\left(\frac{21}{20}\right)^2} \] \[ P = \left(441 \times \frac{20}{21}\right) + \left(441 \times \frac{400}{441}\right) \] Since $441 = 21^2$: \[ P = (21 \times 20) + 400 \] \[ P = 420 + 400 = 820 \text{ INR} \] Thus, the borrowed sum was INR 820.

Step 3: Final Answer:
The borrowed amount is INR 820, matching Option (B).