The true discount on a bill of face value Rs 540 is Rs 90. What is the banker's discount on this bill for the same rate and time?

Introduction:
We are told the true discount (TD) on a bill and asked to compute the banker's discount (BD). This is a typical banker's discount question that uses the relationship between the present worth, the face value, and the true discount, and then uses the ratio between BD and TD.

Given Data / Assumptions:
Face value P = Rs 540. True discount TD = Rs 90. Rate and time are the same for both BD and TD (exact values need not be found explicitly).

Concept / Approach:
Let r be the rate percent and t be the time in years. Then: TD = P * r * t / (100 + r * t). BD = P * r * t / 100. Let x = r * t. Then: TD = P * x / (100 + x). We can find x from the TD equation and then compute BD = P * x / 100.

Step-by-Step Solution:
Step 1: Set TD equation. 90 = 540 * x / (100 + x). 90(100 + x) = 540x. 9000 + 90x = 540x. 9000 = 450x ⇒ x = 9000 / 450 = 20. So r * t = x = 20. Step 2: Compute BD. BD = P * x / 100 = 540 * 20 / 100. BD = 540 * 0.2 = Rs 108.

Verification / Alternative check:
We can also check the ratio BD / TD: BD / TD = 108 / 90 = 1.2. From theory: BD / TD = (100 + x) / 100 = (100 + 20) / 100 = 120 / 100 = 1.2. Both values match, confirming the correctness of BD = Rs 108.

Why Other Options Are Wrong:
Rs 115, Rs 100, Rs 120, Rs 96: Each of these amounts would change the ratio BD / TD away from 1.2 and thus violate the standard formula relationship between BD, TD, and r * t.

Common Pitfalls:
Learners sometimes confuse face value with present worth, or they try to guess the rate and time separately instead of working with x = r * t. It is more efficient to solve for x and use it directly to compute BD.

Final Answer:
The banker's discount on the bill is Rs 108.