The true discount on a bill of Rs 540 is Rs 90. What is the banker's discount (in rupees) on this bill at the same rate and for the same time?

Introduction / Context:
This question asks us to find the banker's discount given the true discount and the face value of a bill. The rate of simple interest and the time until maturity are the same for both discounts, but they are not provided explicitly. The key insight is that the ratio of banker's discount to true discount depends only on the relationship between face value and present worth, and not directly on the numerical values of rate and time.

Given Data / Assumptions:
- Face value of the bill F = Rs 540
- True discount TD = Rs 90
- Let present worth be PW
- Banker's discount BD is calculated at the same rate and time as TD
- Banker's gain is BD minus TD but is not directly required here

Concept / Approach:
By definition, true discount TD is the difference between the face value and the present worth, so TD = F − PW. Therefore PW = F − TD. For simple interest, the ratio BD / TD equals F / PW because BD is interest on the face value while TD is interest on the present worth for the same rate and time. Once we compute PW from F and TD, we can obtain this ratio and thus find BD directly from TD without needing rate or time.

Step-by-Step Solution:
Step 1: Compute present worth PW: PW = F − TD = 540 − 90 = Rs 450.Step 2: For the same rate and time, BD and TD are proportional to the principals on which they are calculated.Step 3: Therefore BD / TD = F / PW.Step 4: Substitute values: BD / 90 = 540 / 450 = 6 / 5.Step 5: So BD = 90 × (6 / 5) = 90 × 1.2 = Rs 108.

Verification / Alternative check:
We can interpret the factor 6 / 5 as showing that banker's discount is 20 percent greater than true discount in this case. Since TD is 90, twenty percent of 90 is 18, so BD should be 90 + 18 = 108. This matches the computed result precisely. We could also set up equations using an unknown rate and time, but the proportional method is faster and less error prone.

Why Other Options Are Wrong:
The value 100 is slightly higher than the true discount but does not correspond to the exact ratio F / PW. A value like 120 would make BD one third larger than TD, which contradicts F / PW = 6 / 5. The option 115 is arbitrary and not linked to any simple ratio. The value 90 is simply the true discount itself, not the banker's discount. Only 108 fits the exact relationship between the face value, present worth and the given true discount.

Common Pitfalls:
Some learners mistakenly assume BD equals TD or try to compute both using a guessed rate of interest, which is unnecessary and may lead to inconsistency. Others forget that true discount is based on present worth while banker's discount is based on face value. Remembering the ratio BD / TD = F / PW gives a clean and reliable route to the answer.

Final Answer:
The banker's discount on the bill is 108 rupees.