On a bill due 6 months hence, the banker's discount at 6% per annum is Rs. 18.54. For the same bill, time period and rate, what is the corresponding true discount?

Introduction / Context:
This question in commercial arithmetic asks you to find the true discount when the banker's discount is known. Both discounts are calculated for the same bill, time and rate. The key challenge is to work backwards from banker's discount to face value and then apply the true discount formula. Understanding this process is useful for handling bank-related questions about bills of exchange and discounting.

Given Data / Assumptions:

• Banker's discount BD = Rs. 18.54.
• Time = 6 months = 0.5 years.
• Rate of interest R = 6% per annum.
• We assume simple interest and a single payment bill.

Concept / Approach:
We use two important formulas:

• Banker's discount: BD = S * R * T / 100.
• True discount: TD = S * (R * T) / (100 + R * T).

First we use BD to find the face value S of the bill. Then we use S in the true discount formula, with the same rate and time, to find TD.

Step-by-Step Solution:
Step 1: Let S be the face value. With R = 6% and T = 0.5 years, R * T = 3.Step 2: Banker's discount BD = S * R * T / 100 = S * 6 * 0.5 / 100 = S * 3 / 100.Step 3: Given BD = 18.54, so 18.54 = S * 3 / 100.Step 4: Solving for S: S = 18.54 * 100 / 3 = 618.Step 5: Use true discount formula: TD = S * (R * T) / (100 + R * T) = 618 * 3 / (100 + 3).Step 6: TD = 618 * 3 / 103 = 1854 / 103 = 18.Step 7: Therefore, the true discount is Rs. 18.

Verification / Alternative check:
We can also compute present worth P = S − TD = 618 − 18 = 600. Then simple interest on 618 for half a year at 6% is 618 * 3 / 100 = 18.54, which matches the banker's discount given in the question. The difference BD − TD = 18.54 − 18 = 0.54 is exactly the banker's gain, which should equal TD^2 / P when expressed appropriately, providing an additional theoretical check on consistency.

Why Other Options Are Wrong:
Rs. 24 and Rs. 36 are larger than the banker's discount, which cannot happen for positive time and rate since true discount is always less than banker's discount. Rs. 12 and Rs. 20 do not fit the formulas when substituted back; they produce incorrect present worth or fail to give BD = 18.54 at 6% for 6 months. Only Rs. 18 satisfies both the BD and TD relationships.

Common Pitfalls:

• Using the true discount formula directly with BD in place of S.
• Forgetting to convert 6 months to 0.5 years when computing R * T.
• Confusing present worth with sum due and mixing the roles of S and P.
• Assuming TD is always roughly equal to BD without doing the actual calculation.

Final Answer:
The true discount on the bill is Rs. 18.