True discount on Rs 440 becomes Rs 40 for a certain time. If the same sum becomes due at double that time, what is the new true discount (approx., to 2 decimals)?

Introduction / Context:
In commercial arithmetic, true discount (TD) is the rebate for paying before the due date based on the present worth concept. For a fixed rate and sum, TD depends nonlinearly on time through the factor 1 + r t / 100.

Given Data / Assumptions:

• Face value (amount due at maturity) S = Rs 440.
• At time t, TD₁ = Rs 40.
• We seek TD₂ at time 2t.
• Simple interest framework; rate r is constant (unknown but cancels).

Concept / Approach:
For simple interest, present worth PW = S / (1 + k), where k = r t / 100. Thus TD = S − PW = S · k/(1 + k). For doubled time we use 2k in place of k.

Step-by-Step Solution:

Verification / Alternative check:
Using percentage arithmetic: at t the effective fraction = k/(1 + k) = 0.1/1.1 ≈ 0.0909; at 2t it is 0.2/1.2 ≈ 0.1667. Scaling Rs 440 gives ≈ Rs 40 and ≈ Rs 73.33, respectively.

Why Other Options Are Wrong:
72.00 and 72.80 understate the computed Rs 73.33; 73.20 is a rounding undershoot.

Common Pitfalls:
Doubling time does not double true discount because the denominator 1 + k also changes.

Final Answer:
Rs 73.32