If Rs. 10 is allowed as true discount on a bill of Rs. 110 due after a certain time at simple interest, what will be the true discount on the same sum when it is due after double that time, assuming the same rate of interest?

Introduction / Context:
This problem tests understanding of true discount under simple interest and how it changes when the time is doubled while the principal and rate of interest remain constant. Instead of giving the rate directly, the question provides one true discount and asks for another, so we have to work backwards to find the effective rate time product and then recompute the new true discount.

Given Data / Assumptions:

• Face value (amount due) of the bill, S = Rs. 110.
• True discount on this bill for some time t is TD1 = Rs. 10.
• The same sum is now considered for double the original time, that is 2t.
• The rate of simple interest remains the same in both cases.
• We have to find the new true discount TD2 for time 2t.

Concept / Approach:
For simple interest, if a sum S is due after time t at rate r percent per annum, and true discount TD is allowed, then: TD = S * r * t / (100 + r * t) Let k = r * t, which represents the effective rate time product. We can use the first situation to find k and then use k2 = 2k for the doubled time. The new true discount is then: TD2 = S * k2 / (100 + k2)

Step-by-Step Solution:
Step 1: Use TD1 = 10 for S = 110 and unknown k. 10 = 110 * k / (100 + k) Step 2: Multiply both sides: 10 * (100 + k) = 110 * k. 1000 + 10k = 110k. Step 3: Rearrange to get 100k = 1000 so k = 10. Step 4: For double the time, k2 = 2k = 20. Step 5: New true discount TD2 = 110 * 20 / (100 + 20) = 2200 / 120. Step 6: Simplify 2200 / 120 = 220 / 12 = 55 / 3 ≈ 18.33.

Verification / Alternative check:
We can also compute the present worth PW1 for the first case and PW2 for the second case, and then calculate TD2 = S − PW2. For k = 10, we have: PW1 = S * 100 / (100 + 10) = 11000 / 110 = Rs. 100. So the effective rate time product of 10 percent is confirmed. For k2 = 20: PW2 = 110 * 100 / (100 + 20) = 11000 / 120 ≈ 91.67. TD2 = 110 − 91.67 ≈ 18.33. This matches the earlier result.

Why Other Options Are Wrong:
Values like Rs. 18.20, Rs. 18.00, Rs. 18.30, and Rs. 19.00 do not match the exact fraction 55 divided by 3. They arise from rounding or incorrect substitution of time into the true discount formula, or from using simple interest instead of true discount directly. The correct calculation gives 18.33 recurring, which corresponds to option stating Rs. 18.33.

Common Pitfalls:
Learners sometimes double the discount from Rs. 10 to Rs. 20 when the time doubles, forgetting that true discount is not directly proportional to time because of the denominator term 100 + r * t. Others confuse true discount with simple interest and use SI = P * r * t / 100 without adjusting for due amount. Always use the correct true discount formula and remember that r * t is treated as a combined factor first, then applied to the fraction S * r * t / (100 + r * t).

Final Answer:
The true discount for double the time is Rs. 18.33.