What is the difference between the banker's discount and the true discount on Rs 8100 for 3 months at 5% per annum simple interest?

Introduction:
This problem asks for the difference between banker's discount (BD) and true discount (TD), which is called the banker's gain (BG). We are given the principal, rate, and time. We can compute BD and TD using their formulas and subtract to obtain the difference.

Given Data / Assumptions:
Face value P = Rs 8100. Rate r = 5% per annum. Time t = 3 months = 3/12 = 1/4 year. We must find BD − TD (the difference, i.e., banker's gain).

Concept / Approach:
For a bill with face value P: BD = P * r * t / 100. TD = P * r * t / (100 + r * t). The difference BD − TD equals the banker's gain BG. We will compute BD and TD numerically and subtract.

Step-by-Step Solution:
Step 1: Compute r * t. t = 3/12 = 1/4 year ⇒ r * t = 5 * (1/4) = 1.25. Step 2: Compute BD. BD = P * r * t / 100 = 8100 * 1.25 / 100. BD = 8100 * 0.0125 = Rs 101.25. Step 3: Compute TD. TD = P * r * t / (100 + r * t) = 8100 * 1.25 / (100 + 1.25). TD = 10125 / 101.25 = Rs 100. Step 4: Difference (banker's gain). BD − TD = 101.25 − 100 = Rs 1.25.

Verification / Alternative check:
We can also use the formula BG = TD * r * t / 100: BG = 100 * 1.25 / 100 = Rs 1.25, which matches our computed difference BD − TD, confirming the result.

Why Other Options Are Wrong:
Rs 2, Rs 2.25, Rs 0.5, Rs 3: None of these values are equal to TD * r * t / 100 with TD = 100 and r * t = 1.25, and therefore they are inconsistent with the exact formulas for BD and TD.

Common Pitfalls:
Some learners mistakenly apply simple interest on the present worth rather than the face value, or they forget to convert 3 months into 1/4 year. Another common mistake is rounding too early instead of working with exact fractions until the final step.

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