The present worth of a certain sum due sometime hence is Rs 1600 and the true discount on it is Rs 160. What is the banker's gain (in rupees) on this transaction?

Introduction / Context:
This problem asks us to find the banker's gain when we know the present worth and the true discount on a sum. Banker's gain is the difference between banker's discount and true discount. Given present worth and true discount, we can first determine the face value and the banker's discount and then compute the gain as their difference. The question tests familiarity with the relationships between these three quantities.

Given Data / Assumptions:
- Present worth PW = Rs 1600
- True discount TD = Rs 160
- Face value F of the bill is PW + TD
- Banker's discount BD is calculated on face value at the same rate and time as TD
- Banker's gain BG is defined as BD − TD

Concept / Approach:
First we use F = PW + TD to get the face value. Since true discount is interest on present worth and banker's discount is interest on face value for the same rate and time, the ratio BD / TD equals F / PW. Once BD is known from this ratio, banker's gain BG is simply BD − TD. No explicit knowledge of the rate of interest or the time is required because we always work with proportional relationships.

Step-by-Step Solution:
Step 1: Compute the face value: F = PW + TD = 1600 + 160 = Rs 1760.Step 2: For the same rate and time, BD / TD = F / PW.Step 3: Substitute: BD / 160 = 1760 / 1600.Step 4: Simplify 1760 / 1600 to 11 / 10.Step 5: Therefore BD = 160 × 11 / 10 = 16 × 11 = Rs 176.Step 6: Banker's gain BG = BD − TD = 176 − 160 = Rs 16.

Verification / Alternative check:
We can check that the face value is 10 percent greater than the present worth because TD = 160 and PW = 1600, so TD / PW = 160 / 1600 = 1 / 10. This indicates that the relevant interest fraction is 10 percent. Applying the same percentage to the face value of 1760 gives 1760 × 10 percent = 176 as banker's discount. Subtracting the true discount 160 confirms that the banker's gain is 16 rupees, so our result is consistent.

Why Other Options Are Wrong:
Values like 13, 14 or 15 would correspond to different gaps between BD and TD and would not be compatible with the exact 11 to 10 ratio derived from the present worth and face value. A banker's gain of 12 rupees is also too small. Only 16 rupees keeps the proportional relationships intact and matches the difference between BD and TD for this situation.

Common Pitfalls:
Some learners attempt to guess the rate and time first, which is unnecessary and may lead to mistakes. Others treat true discount as interest on the face value rather than on the present worth, breaking the BD to TD ratio. By directly using F = PW + TD and BD / TD = F / PW, we avoid these complications and reach the answer efficiently.

Final Answer:
The banker's gain on the transaction is 16 rupees.