In how much time can a debt of Rs. 7920, due at a certain future date, be cleared by an immediate cash down payment of Rs. 3600, if money is worth 0.5% per month simple interest?

Introduction / Context:
This is a classic true discount application where an immediate cash payment is used to settle a larger debt due after some time. The question gives the cash down payment, the full amount of the debt, and the monthly simple interest rate. We need to find the time at which the debt would otherwise be due so that both arrangements are financially equivalent in present worth terms.

Given Data / Assumptions:

• Debt amount (sum due in the future), S = Rs. 7920.
• Immediate cash down payment (present worth), PW = Rs. 3600.
• Rate of simple interest r = 0.5 percent per month.
• Time t is in months, to be converted into years if needed.
• Simple interest and true discount concepts apply.

Concept / Approach:
If S is due after t months and money earns r percent per month simple interest, then present worth PW and amount S are related by: S = PW * (1 + r * t / 100) or equivalently: PW = S * 100 / (100 + r * t) We know S, PW, and r, so we can solve for t. Then we convert the number of months into years by dividing by 12.

Step-by-Step Solution:
Step 1: Use PW formula: PW = S * 100 / (100 + r * t). Step 2: Substitute PW = 3600, S = 7920, r = 0.5. 3600 = 7920 * 100 / (100 + 0.5 * t). Step 3: Rearrange to solve for the denominator. 100 + 0.5 * t = 7920 * 100 / 3600. Step 4: Compute 7920 * 100 / 3600 = 220. Step 5: Thus 100 + 0.5 * t = 220, so 0.5 * t = 120. Step 6: t = 120 / 0.5 = 240 months. Step 7: Convert to years: 240 months / 12 = 20 years.

Verification / Alternative check:
Check using the amount formula S = PW * (1 + r * t / 100). With PW = 3600, r = 0.5, t = 240: r * t / 100 = 0.5 * 240 / 100 = 120 / 100 = 1.2. S = 3600 * (1 + 1.2) = 3600 * 2.2 = 7920. This matches the given future debt amount exactly, confirming that 20 years is correct.

Why Other Options Are Wrong:
Values such as 10, 30, or 40 years would produce very different multipliers when used in the simple interest formula. They would not convert Rs. 3600 into Rs. 7920 at a rate of 0.5 percent per month. Only 20 years leads to the correct future value.

Common Pitfalls:
A typical mistake is to treat the 0.5 percent as an annual rate or to forget that the rate is per month. Another error is not converting months to years properly or mismanaging the algebra when isolating t in the PW equation. Always keep the time unit consistent with the rate and verify the answer by reconstructing the original future value from the computed time and present worth.

Final Answer:
The debt is due after 20 years.