The banker's discount on a bill due 6 months hence at 6% per annum is Rs 18.54. What is the true discount on this bill?

Introduction:
This question gives the banker's discount (BD) and asks us to find the true discount (TD) for the same time and rate. We use the known ratio BD / TD in terms of the rate-time product r * t to compute TD directly.

Given Data / Assumptions:
Banker's discount BD = Rs 18.54. Rate r = 6% per annum. Time t = 6 months = 0.5 year. We must find true discount TD.

Concept / Approach:
For a bill with face value P at rate r and time t: BD = P * r * t / 100. TD = P * r * t / (100 + r * t). Hence: BD / TD = (100 + r * t) / 100. So: TD = BD * 100 / (100 + r * t). We will compute r * t and then use this relationship to find TD.

Step-by-Step Solution:
Step 1: Compute r * t. r * t = 6 * 0.5 = 3. Step 2: Use TD = BD * 100 / (100 + r * t). TD = 18.54 * 100 / (100 + 3). TD = 18.54 * 100 / 103. TD = 1854 / 103 = Rs 18.

Verification / Alternative check:
We can check the ratio BD / TD: BD / TD = 18.54 / 18 = 1.03. From theory: BD / TD = (100 + r * t) / 100 = (100 + 3) / 100 = 103 / 100 = 1.03, which matches perfectly.

Why Other Options Are Wrong:
Rs 24, Rs 12, Rs 36, Rs 15: None of these values satisfy the exact ratio BD / TD = 103 / 100 with BD = Rs 18.54. Using any of them would either make BD too large or too small relative to TD under the given rate and time.

Common Pitfalls:
A common error is to subtract some arbitrary amount from BD to guess TD, or to use simple interest on the present worth without knowing it. The right method is to remember and apply the ratio BD / TD = (100 + r * t) / 100.

Final Answer:
The true discount on the bill is Rs 18.