If the true discount on a certain sum due 6 months hence at 15% simple interest is Rs 120, what is the banker's discount (in rupees) on the same sum for the same time and at the same rate?

Introduction / Context:
Here we are given the true discount on a sum due after a known time at a known rate of simple interest. Our task is to compute the banker's discount for the same sum, time and rate. The question combines knowledge of true discount formulas with the formula for banker's discount and requires careful algebraic manipulation to find the face value first.

Given Data / Assumptions:
- True discount TD = Rs 120
- Time t = 6 months = 0.5 year
- Rate of simple interest r = 15 percent per annum
- Let face value of the bill be F
- Banker's discount BD is to be found for the same rate and time

Concept / Approach:
True discount at simple interest is given by TD = F × r × t / (100 + r × t). Since we know TD, r and t, we can solve this formula for the face value F. Once F is known, banker's discount is calculated as BD = F × r × t / 100, because it is simple interest on the face value for the same period. The difference BD − TD would give banker's gain, but the question only asks for BD.

Step-by-Step Solution:
Step 1: Compute r × t = 15 × 0.5 = 7.5.Step 2: Use the true discount formula TD = F × r × t / (100 + r × t).Step 3: Substitute known values: 120 = F × 7.5 / (100 + 7.5) = F × 7.5 / 107.5.Step 4: Rearrange to find F: F = 120 × 107.5 / 7.5 = 120 × (1075 / 75) = 1720.Step 5: Now compute banker's discount BD = F × r × t / 100 = 1720 × 15 × 0.5 / 100.Step 6: Simplify BD = 1720 × 7.5 / 100 = 12900 / 100 = Rs 129.

Verification / Alternative check:
We can cross check using the ratio BD / TD. Here BD = 129 and TD = 120, so BD / TD = 129 / 120 = 1.075. For r t = 7.5, the ratio BD / TD should be (100 + 7.5) / 100 = 107.5 / 100 = 1.075, which matches exactly. This confirms that the computed face value and banker's discount are consistent with the standard formulas.

Why Other Options Are Wrong:
An answer like 100 or 50 ignores the fact that banker's discount must be slightly larger than the true discount because it is calculated on the face value. A value such as 160 is too large and would violate the relation between TD and BD at the given rate and time. Only Rs 129 satisfies both the algebraic relationship and the inequality BD greater than TD.

Common Pitfalls:
Some learners mistakenly apply the simple interest formula directly to TD instead of to the face value, or they treat 6 months as 6 years when multiplying r by t. Another common error is to forget to add r t in the denominator when using the true discount formula, leading to an incorrect face value. Carefully writing down the formulas and substituting the values step by step helps avoid these mistakes.

Final Answer:
The banker's discount for the given sum is 129 rupees.