Effect of halving the time on true discount: True discount on Rs. 260 due after a certain time is Rs. 20. What will be the true discount on the same sum due after half that time, at the same simple-interest rate?

Introduction / Context:
True discount (TD) relates a future amount A, the rate r, and time t by TD = A * (r * t) / (1 + r * t). Problems often ask how TD changes if time is altered while A and r remain fixed. This one halves the time and seeks the new TD.

Given Data / Assumptions:

• A = 260.
• At time t, TD = 20.
• Rate r unchanged; we need TD at time t/2.

Concept / Approach:
Let x = r * t (with r in per annum as a fraction). Then 20 = 260 * x / (1 + x). Solve x first; then compute TD for x/2: TD’ = 260 * (x/2) / (1 + x/2).

Step-by-Step Solution:
20 (1 + x) = 260 x ⇒ 20 + 20x = 260x ⇒ 20 = 240x ⇒ x = 1/12.For half time: x/2 = 1/24.TD’ = 260 * (1/24) / (1 + 1/24) = 260 * (1/24) / (25/24) = 260 * (1/25) = 10.

Verification / Alternative check:
Compute present worths: PW = 260 − 20 = 240 for time t; for t/2, PW’ = 260 − 10 = 250. Ratio PW changes align with halving the discount factor x → x/2, confirming the result.

Why Other Options Are Wrong:

• Rs. 10.40 and Rs. 15.20 arise from misusing proportionality (TD is not directly proportional to t due to the denominator 1 + r * t).
• Rs. 13 is arbitrary without basis in the formula.

Common Pitfalls:

• Treating TD as simply A * r * t (that is banker’s discount) instead of true discount with the (1 + r * t) denominator.

Final Answer:
Rs. 10