The true discount on a certain sum due 6 months (half a year) hence at 15% per annum is Rs 120. What is the banker's discount on the same sum for the same time and at the same rate?

Introduction:
This is a classic relationship question between true discount (TD) and banker's discount (BD). We are given the true discount and asked to compute the banker's discount for the same sum, the same time, and the same simple interest rate.

Given Data / Assumptions:
True discount TD = Rs 120. Time t = 6 months = 0.5 year. Rate r = 15% per annum. We must find banker's discount BD.

Concept / Approach:
For a bill with face value P, at rate r and time t: TD = P * r * t / (100 + r * t). BD = P * r * t / 100. The ratio BD / TD is: BD / TD = (100 + r * t) / 100. Therefore: BD = TD * (100 + r * t) / 100. We will compute r * t, plug it into this formula, and obtain BD directly without needing the actual face value P.

Step-by-Step Solution:
Step 1: Compute r * t. r * t = 15 * 0.5 = 7.5. Step 2: Use BD = TD * (100 + r * t) / 100. BD = 120 * (100 + 7.5) / 100. BD = 120 * 107.5 / 100. BD = 120 * 1.075 = Rs 129.

Verification / Alternative check:
We can check the ratio: BD / TD = 129 / 120 = 1.075. From theory, BD / TD = (100 + r * t) / 100 = (100 + 7.5) / 100 = 107.5 / 100 = 1.075. The two values match, confirming our calculation.

Why Other Options Are Wrong:
Rs 50 and Rs 100: These are too small and do not satisfy the theoretical ratio between BD and TD. Rs 160: Too large; it would imply an incorrect effective (100 + r * t) value. Rs 120: This would make BD equal to TD, which is impossible for positive rate and time.

Common Pitfalls:
A frequent error is to treat BD and TD as equal or to attempt to compute BD using only simple interest on present worth. The safe method is always to use the ratio BD / TD = (100 + r * t) / 100 once TD is given.

Final Answer:
The banker's discount on the bill is Rs 129.