On a sum of Rs. 8100 for 3 months at 5% per annum, what is the difference between the banker's discount and the true discount?

Introduction / Context:
This question focuses on the difference between banker's discount and true discount, a quantity commonly referred to as the banker's gain. You are given the face value, time and rate of interest and must compute both discounts and then find their difference. It checks your ability to handle short-duration interest calculations and correctly apply the formulas for commercial discounting.

Given Data / Assumptions:

• Face value S = Rs. 8100.
• Time = 3 months = 0.25 years.
• Rate R = 5% per annum.
• Simple interest is assumed.
• We need BD − TD (the banker's gain).

Concept / Approach:
We will compute both discounts explicitly:

• Banker's discount BD = S * R * T / 100.
• True discount TD = S * (R * T) / (100 + R * T).

Then the difference between them, BD − TD, gives the required answer. Since the time is short (3 months), the difference will be relatively small.

Step-by-Step Solution:
Step 1: Convert time: T = 3 months = 3 / 12 = 0.25 years.Step 2: Compute R * T = 5 * 0.25 = 1.25.Step 3: Banker's discount: BD = S * R * T / 100 = 8100 * 5 * 0.25 / 100.Step 4: BD = 8100 * 1.25 / 100 = 101.25.Step 5: True discount: TD = S * (R * T) / (100 + R * T) = 8100 * 1.25 / (100 + 1.25).Step 6: TD = 10125 / 101.25 = 100.Step 7: Difference (banker's gain) = BD − TD = 101.25 − 100 = 1.25.

Verification / Alternative check:
We can also use the formula banker's gain = TD^2 / P, where P is present worth. Here P = S − TD = 8100 − 100 = 8000. Then TD^2 / P = 100^2 / 8000 = 10000 / 8000 = 1.25. This matches the directly computed difference BD − TD, confirming the correctness of the value Rs. 1.25.

Why Other Options Are Wrong:
Rs. 2 and Rs. 2.25 are larger than the theoretical banker's gain based on the short time and low rate. Rs. 0.50 and Rs. 3 do not satisfy the exact formulas when substituted back. Only Rs. 1.25 is consistent with both direct computation and the banker's gain relation involving true discount and present worth.

Common Pitfalls:

• Forgetting to convert 3 months into years when calculating R * T.
• Using 5 as R * T instead of 1.25, which leads to large errors.
• Mixing up true discount and simple interest on present worth.
• Assuming the difference will be negligible without actually computing it.

Final Answer:
The difference between the banker's discount and the true discount is Rs. 1.25.