The banker's discount on a bill of Rs 1650 due after some time is Rs 165. What is the bank's gain (in rupees) on this transaction?

Introduction / Context:
This problem combines the ideas of banker's discount, true discount and banker's gain. We are given the banker's discount on a bill and the face value of that bill. Using the relationship between banker's discount and true discount at the same rate and time, we can compute true discount and then obtain the banker's gain as the difference between them. The focus here is on using ratios cleanly without needing the explicit rate or time.

Given Data / Assumptions:
- Face value of the bill F = Rs 1650
- Banker's discount BD = Rs 165
- Rate of interest and time are the same for both BD and true discount TD, but not given explicitly
- Banker's gain BG is defined as BD minus TD

Concept / Approach:
For a bill with face value F, the banker's discount at rate r percent for time t years is BD = F × r × t / 100. The true discount is TD = F × r × t / (100 + r × t). The ratio BD / TD simplifies to (100 + r t) / 100, which also equals F / present worth. However, in this question it is simpler to first compute the product r t using BD and F, then use the TD formula. Once TD is known, banker's gain BG is simply BD minus TD.

Step-by-Step Solution:
Step 1: Use BD = F × r × t / 100 to find r t: 165 = 1650 × r t / 100.Step 2: Simplify: r t = (165 × 100) ÷ 1650 = 1000 ÷ 10 = 10.Step 3: True discount TD = F × r t / (100 + r t) = 1650 × 10 / (100 + 10) = 1650 × 10 / 110.Step 4: Compute TD = 16500 ÷ 110 = Rs 150.Step 5: Banker's gain BG = BD − TD = 165 − 150 = Rs 15.

Verification / Alternative check:
We can check using the relation BD / TD = (100 + r t) / 100. Here BD / TD = 165 / 150 = 11 / 10. Therefore (100 + r t) / 100 = 11 / 10, giving 100 + r t = 110 and r t = 10, which matches the earlier computation. Substituting r t = 10 into the true discount formula again yields TD = 150 and BG = 15, so the value is consistent from both viewpoints.

Why Other Options Are Wrong:
Values like 20 or 18 would correspond to different relationships between BD and TD and would not maintain the correct equality with the given face value and rate time product. Thirteen is too small relative to the difference between a discount of 165 and a true discount close to 150. Only 15 is compatible with the formulas for banker's discount and true discount at some positive rate and time.

Common Pitfalls:
One common error is to treat 165 as the true discount directly and try to find BG without using the proper formulas. Another mistake is mixing up face value and present worth, leading to incorrect calculations of the product r t. Always start by finding r t from BD and F and then compute TD and BG systematically.

Final Answer:
The banker's gain on the bill is 15 rupees.