The present worth of a certain sum due sometime hence is Rs. 3400 and the true discount on it is Rs. 340. What is the banker's gain on this transaction?

Introduction / Context:
This problem again brings together present worth, true discount and banker's gain. You are given the present worth and true discount of a certain sum and asked to determine the banker's gain, which is the difference between banker's discount and true discount. Such questions reinforce the useful algebraic connections between these quantities in commercial arithmetic.

Given Data / Assumptions:

• Present worth P = Rs. 3400.
• True discount TD = Rs. 340.
• Let BG be the banker's gain.
• Simple interest assumptions apply and time and rate are the same for BD and TD.

Concept / Approach:
We use the well-known relation between banker's gain, true discount and present worth:

• Banker's gain: BG = TD^2 / P.

This formula lets us compute banker's gain directly from TD and P, without explicitly dealing with time and rate.

Step-by-Step Solution:
Step 1: Use BG = TD^2 / P.Step 2: Substitute TD = 340 and P = 3400: BG = 340^2 / 3400.Step 3: Compute 340^2 = 115600.Step 4: Now BG = 115600 / 3400.Step 5: Divide: 115600 ÷ 3400 = 34.Step 6: Therefore, the banker's gain is Rs. 34.

Verification / Alternative check:
We can reconstruct the face value S and banker's discount BD to check consistency. Face value S = P + TD = 3400 + 340 = 3740. Using BD = TD + BG, we get BD = 340 + 34 = 374. Another relation is BD = S * TD / P; substituting S = 3740, TD = 340 and P = 3400 gives BD = 3740 * 340 / 3400 = 374, which matches. The difference BD − TD = 374 − 340 = 34 equals BG, confirming the answer.

Why Other Options Are Wrong:
Values like Rs. 21, Rs. 17, Rs. 18 or Rs. 27 do not satisfy the equation BG = TD^2 / P. Plugging these values into BG * P and comparing with TD^2 reveals inconsistencies. Only Rs. 34 makes TD^2 = BG * P hold exactly, so it is the correct banker's gain.

Common Pitfalls:

• Confusing banker's gain with true discount and simply answering Rs. 340.
• Attempting to guess BG instead of using the TD^2 / P relation.
• Arithmetic errors when squaring 340 or dividing by 3400.
• Mixing up present worth and sum due, leading to wrong formulas.

Final Answer:
The banker's gain on this transaction is Rs. 34.