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

Introduction / Context:
This question involves the concepts of present worth, true discount and banker's gain in commercial arithmetic. When a bill is discounted at a bank, the bank uses banker's discount, while true discount is the mathematically correct deduction based on present worth. The difference between these two is called banker's gain.

Given Data / Assumptions:

Present worth (P) of the sum = Rs. 1600.
True discount (TD) = Rs. 160.
Amount due at maturity (A) = P + TD = 1600 + 160 = Rs. 1760.
We are required to find the banker's gain (BG).
Rate of interest and time are the same for both types of discount, but their exact values are not needed.

Concept / Approach:
True discount is the difference between the future amount and its present worth. Banker's discount is simple interest on the amount due (A). Banker's gain is defined as:
BG = Banker's discount - True discount. When the present worth P and true discount TD are known, there is a direct formula:
BG = TD^2 / P This comes from the relationship between simple interest on present worth and on the amount due.

Step-by-Step Solution:
Step 1: Note the given values. P = Rs. 1600, TD = Rs. 160. Step 2: Apply the banker's gain formula. BG = TD^2 / P. Step 3: Substitute the numbers. BG = 160 * 160 / 1600. BG = 25600 / 1600. BG = 16.

Verification / Alternative check:
Let the simple interest rate * time factor be k. Then TD = P * k = 1600k = 160, so k = 0.1. Amount A = 1760, so banker's discount BD = A * k = 1760 * 0.1 = 176. Banker's gain BG = BD - TD = 176 - 160 = 16, confirming the result from the shortcut formula.

Why Other Options Are Wrong:
13, 14 and 15 are all less than the computed banker's gain. None of them satisfies BD - TD when calculated consistently using the same rate and time derived from the given true discount and present worth.

Common Pitfalls:
A common mistake is to treat the given true discount as if it were banker's discount or to attempt guesswork with the rate of interest. Another frequent error is forgetting that BG depends on TD and present worth, not directly on the amount alone. Using BG = TD^2 / P when P and TD are known ensures a quick and reliable solution.

Final Answer:
The banker's gain is Rs. 16.