Difference SI vs. true discount for multi-year deferment: Find the difference between simple interest and true discount on Rs. 960 due 4 years hence at 5% per annum simple interest.

Introduction / Context:
The difference BD − TD (banker’s discount minus true discount) for a future amount A at rate r and time t is BD − TD = A * (r * t)^2 / (1 + r * t). This follows from BD = A * r * t and TD = A * r * t / (1 + r * t).

Given Data / Assumptions:

• A = 960.
• r = 5% p.a., t = 4 years ⇒ r * t = 0.20.

Concept / Approach:
Either compute BD and TD separately and subtract, or use the identity directly.

Step-by-Step Solution:
BD = A * r * t = 960 * 0.20 = 192.TD = A * (r * t) / (1 + r * t) = 960 * 0.20 / 1.20 = 192 / 1.20 = 160.Difference = BD − TD = 192 − 160 = 32.

Verification / Alternative check:
Identity check: A * (r * t)^2 / (1 + r * t) = 960 * 0.04 / 1.20 = 38.4 / 1.20 = 32, confirming the same result.

Why Other Options Are Wrong:

• 52 and 42 come from arithmetic slips or wrong rate-time usage.
• “None of these” is unnecessary since exact 32 is obtained.

Common Pitfalls:

• Mistaking TD as A − PW without computing PW correctly.
• Applying compound interest accidentally.

Final Answer:
Rs. 32