The banker's gain on a bill due 1 year hence at 12% per annum is Rs 6. What is the true discount on the bill (in rupees)?

Introduction / Context:
This question focuses on banker's gain and true discount for a bill under simple interest conditions. Banker's gain is the difference between banker's discount and true discount. Knowing the rate and time, we can express both BD and TD in terms of the face value of the bill and then relate banker's gain to that face value. With banker's gain given, we can work backwards to find the true discount.

Given Data / Assumptions:
- Time until the bill is due t = 1 year
- Rate of simple interest r = 12 percent per annum
- Banker's gain BG = Rs 6
- Let face value of the bill be F
- We need to find true discount TD on this bill

Concept / Approach:
For a bill of face value F at rate r percent for time t years, banker's discount is BD = F × r × t / 100. True discount is TD = F × r × t / (100 + r × t). Banker's gain is BG = BD − TD. For known r t, BG becomes a constant fraction of F. We can find F from BG, then compute TD from its formula. Finally, we report TD as the answer.

Step-by-Step Solution:
Step 1: Compute r × t = 12 × 1 = 12.Step 2: Banker's discount BD = F × 12 / 100 = 3F / 25.Step 3: True discount TD = F × 12 / (100 + 12) = F × 12 / 112 = 3F / 28.Step 4: Banker's gain BG = BD − TD = 3F / 25 − 3F / 28.Step 5: Simplify BG = 3F × (1 / 25 − 1 / 28) = 3F × (28 − 25) / (25 × 28) = 3F × 3 / 700 = 9F / 700.Step 6: Given BG = 6, so 9F / 700 = 6 which implies F = 6 × 700 / 9 = 1400 / 3.Step 7: Now compute TD = 3F / 28 = 3 × (1400 / 3) / 28 = 1400 / 28 = Rs 50.

Verification / Alternative check:
We can check the result by computing BD as well. With F = 1400 / 3, BD = 3F / 25 = 3 × (1400 / 3) / 25 = 1400 / 25 = Rs 56. True discount TD is 50, so banker's gain BG = BD − TD = 56 − 50 = Rs 6, which matches the given banker's gain. This confirms that the computed true discount is correct even though the face value is a fraction.

Why Other Options Are Wrong:
Values such as 47, 58 or 62 would lead to a banker's gain different from Rs 6 when combined with the required BD at 12 percent for 1 year. For example, if TD were 62, the relationship between BD, TD and BG would not satisfy the equations derived from the rate and time. Only 50 fits consistently with the given banker's gain and interest conditions.

Common Pitfalls:
Students often assume that the face value must be a round integer and then force BD and TD into those values, which can cause errors. Others confuse banker's gain with true discount or neglect to compute both BD and TD before subtracting. It is essential to set up the algebra carefully, solve for the face value first and then compute the true discount from its standard formula.

Final Answer:
The true discount on the bill is 50 rupees.