If Rs. 20 is the true discount on Rs. 260 due after a certain time at a given simple interest rate, what will be the true discount on the same sum due after half of that former time at the same rate of interest?

Introduction / Context:
This question involves the relation between true discount and time when the principal and rate are fixed. Instead of directly giving the rate or time, it gives the true discount for one time period and asks for the true discount when the time is halved. The focus is on correct use of the true discount formula and on understanding that true discount is not directly proportional to time.

Given Data / Assumptions:

• Face value of the bill, S = Rs. 260.
• True discount for some time t, TD1 = Rs. 20.
• We require true discount for time t / 2, called TD2.
• Rate of simple interest remains constant in both cases.
• Standard simple interest and true discount relation applies.

Concept / Approach:
For a sum S due after t years at simple interest rate r percent, true discount is: TD = S * r * t / (100 + r * t) Let k = r * t. Then: TD1 = S * k / (100 + k) For half the time, the effective product is k2 = k / 2 and: TD2 = S * k2 / (100 + k2) We first find k from TD1 and then compute TD2 using k2.

Step-by-Step Solution:
Step 1: Use TD1 = 20, S = 260. 20 = 260 * k / (100 + k). Step 2: Multiply: 20 * (100 + k) = 260 * k. 2000 + 20k = 260k. Step 3: Rearrange: 260k − 20k = 240k = 2000. Step 4: Solve: k = 2000 / 240 = 25 / 3. Step 5: For half the time, k2 = k / 2 = 25 / 6. Step 6: Compute TD2. TD2 = 260 * (25 / 6) / (100 + 25 / 6). Denominator = (600 + 25) / 6 = 625 / 6. TD2 = 260 * (25 / 6) * (6 / 625) = 260 * 25 / 625. Step 7: 260 * 25 = 6500, so TD2 = 6500 / 625 = 10.4 = Rs. 10.40.

Verification / Alternative check:
We can compute the present worth PW1 and PW2 and confirm. For TD1 = 20: PW1 = S − TD1 = 260 − 20 = 240. For TD2: PW2 = S − TD2 = 260 − 10.40 = 249.60. Using PW2 and the true discount formula would recover k2 = 25 / 6, consistent with half the original effective rate time product, confirming correctness.

Why Other Options Are Wrong:
Values like Rs. 9.40, Rs. 9.14, or Rs. 10.14 arise from incorrect algebra or by wrongly assuming that true discount is exactly half when time is halved. That is only true for simple interest and not for true discount, which has a denominator 100 + r * t. Rs. 11.20 would correspond to a different time or rate and does not satisfy the relationship derived from the original discount.

Common Pitfalls:
A frequent mistake is linearly halving the true discount when the time is halved. Because true discount uses the factor 100 + r * t in the denominator, the relation is slightly non linear. Students may also confuse true discount with bank discount or simple interest and omit the denominator adjustment completely. Carefully solving for k and then recomputing for k / 2 avoids these errors.

Final Answer:
The true discount for half the former time is Rs. 10.40.