Question

Nihal took a 1,00,000 loan at 15% interest per year. If he repays in two equal yearly instalments, the amount per instalment is x. If he repays in three equal yearly instalments, the amount per instalment is y. What is the approximate difference between x and y?

Hint:

For loan installment calculations, remember that the present value of all payments must equal the loan amount. While the formula is useful, understanding the concept helps. Each installment pays off some interest accrued during the period and some principal. A longer loan period means more total interest paid, but smaller individual installments.

Answer 1

Step 1: Understanding the Concept:
This problem deals with loan amortization and calculating equal periodic installments (annuities). The present value of all future installments must equal the principal loan amount.
Step 2: Key Formula or Approach:
The formula for the present value (P) of an ordinary annuity (a series of equal payments E) is: \[ P = E \left[ \frac{1 - (1+r)^{-n}}{r} \right] \] where \(r\) is the interest rate per period and \(n\) is the number of periods. We can rearrange this to find the installment amount E: \[ E = P \left[ \frac{r}{1 - (1+r)^{-n}} \right] \] Step 3: Detailed Explanation:
Given: Loan Principal \(P = 1,00,000\), Annual interest rate \(r = 15% = 0.15\).
Case 1: Two yearly instalments (\(n=2\))
The instalment amount is \(x\). \[ x = 1,00,000 \left[ \frac{0.15}{1 - (1+0.15)^{-2}} \right] = 1,00,000 \left[ \frac{0.15}{1 - (1.15)^{-2}} \right] \] \[ 1.15^2 = 1.3225 \] \[ x = 1,00,000 \left[ \frac{0.15}{1 - \frac{1}{1.3225}} \right] = 1,00,000 \left[ \frac{0.15}{\frac{1.3225 - 1}{1.3225}} \right] \] \[ x = 1,00,000 \left[ \frac{0.15 \times 1.3225}{0.3225} \right] = 1,00,000 \left[ \frac{0.198375}{0.3225} \right] \] \[ x \approx 1,00,000 \times 0.615116 \approx 61511.6 \] Case 2: Three yearly instalments (\(n=3\))
The instalment amount is \(y\). \[ y = 1,00,000 \left[ \frac{0.15}{1 - (1+0.15)^{-3}} \right] = 1,00,000 \left[ \frac{0.15}{1 - (1.15)^{-3}} \right] \] \[ 1.15^3 = 1.15 \times 1.3225 = 1.520875 \] \[ y = 1,00,000 \left[ \frac{0.15}{1 - \frac{1}{1.520875}} \right] = 1,00,000 \left[ \frac{0.15}{\frac{1.520875 - 1}{1.520875}} \right] \] \[ y = 1,00,000 \left[ \frac{0.15 \times 1.520875}{0.520875} \right] = 1,00,000 \left[ \frac{0.22813125}{0.520875} \right] \] \[ y \approx 1,00,000 \times 0.437972 \approx 43797.2 \] Step 4: Final Answer:
The approximate difference between \(x\) and \(y\) is: \[ x - y \approx 61511.6 - 43797.2 = 17714.4 \] The closest option is 17714.