The banker's discount on a bill of Rs 1650, due a certain time hence, is Rs 165. Find both the true discount and the banker's gain on this transaction.

Introduction:
Here we are given the banker's discount (BD) on a bill and asked to compute both the true discount (TD) and the banker's gain (BG). This is a standard banker's discount question that tests your understanding of the relationships between BD, TD, BG, and the underlying rate-time product r * t.

Given Data / Assumptions:
Face value of the bill P = Rs 1650. Banker's discount BD = Rs 165. Bill is due after some time t at rate r (simple interest). We must find: True discount (TD) and banker's gain (BG).

Concept / Approach:
Key formulas for banker's discount problems: BD = P * r * t / 100. TD = P * r * t / (100 + r * t). BG = BD − TD. Also, there is a handy relation: BG = TD * r * t / 100. From BD we first find the product r * t, then use that to find TD, and finally compute BG = BD − TD.

Step-by-Step Solution:
Step 1: Use BD to find r * t. BD = P * r * t / 100 ⇒ 165 = 1650 * r * t / 100. 165 = 16.5 * r * t ⇒ r * t = 165 / 16.5 = 10. So r * t = 10. Step 2: Use TD formula. TD = P * r * t / (100 + r * t) = 1650 * 10 / (100 + 10). TD = 16500 / 110 = Rs 150. Step 3: Banker's gain. BG = BD − TD = 165 − 150 = Rs 15.

Verification / Alternative check:
We can also use BG = TD * r * t / 100: BG = 150 * 10 / 100 = 1500 / 100 = Rs 15, which agrees with our earlier result. This confirms that our TD and BG values are consistent with the rate-time product r * t = 10.

Why Other Options Are Wrong:
True discount = Rs 165, banker's gain = Rs 0: This would mean BD = TD, which is never true for positive r and t. True discount = Rs 135, banker's gain = Rs 30: Here BD − TD ≠ 165 − 135 = 30, but this pair does not match the required TD formula for the given P. True discount = Rs 120, banker's gain = Rs 45: The implied BD would be 165, but TD = 120 does not satisfy the correct ratio. True discount = Rs 155, banker's gain = Rs 10: Again, this does not match the exact formula-based computation for the given BD.

Common Pitfalls:
Students often confuse the roles of BD and TD and may try to treat TD as simple interest on the present worth without using the formula TD = P * r * t / (100 + r * t). Another mistake is to assume BG is a fixed percentage of BD, which is not generally true unless derived from the given data.

Final Answer:
The true discount is Rs 150 and the banker's gain is Rs 15.