TD is less than SI by a known amount: The true discount on a certain sum due after 2 1/2 years at 6% per annum is less than the simple interest on the same sum and period by Rs. 81. Find the sum due (face value).

Introduction / Context:
For a future sum S at rate r and time t (simple interest), the difference SI − TD equals S * (r t)^2 / (1 + r t). When this difference is given, we can solve for S directly. Here, “TD is less than SI by Rs. 81” means SI − TD = 81.

Given Data / Assumptions:

• r = 6% per annum = 0.06.
• t = 2.5 years ⇒ r t = 0.06 * 2.5 = 0.15.
• SI − TD = Rs. 81.

Concept / Approach:
Use SI − TD = S * (r t)^2 / (1 + r t). Rearrange to S = (SI − TD) * (1 + r t) / (r t)^2 and substitute the numbers.

Step-by-Step Solution:

Verification / Alternative check:
Back-check: 4,140 * 0.0195652 ≈ 81, so the relation holds.

Why Other Options Are Wrong:
4,240, 4,150, and 4,250 are off the exact computation; 4,000 is also incorrect.

Common Pitfalls:
Misinterpreting “less than” and using TD − SI, or forgetting the denominator (1 + r t) in the difference formula.

Final Answer:
Rs. 4140