The banker's discount on Rs. 1800 at 12% per annum is equal to the true discount on Rs. 1872 for the same time and at the same rate. For what length of time is this equality true?

Introduction / Context:
This problem connects banker's discount on one sum with true discount on another sum, both calculated for the same time and interest rate. You must determine the time period that makes these two quantities equal. It is a good test of your ability to form equations using the standard formulas of discount and then solve for time.

Given Data / Assumptions:

• Sum 1: S1 = Rs. 1800 with banker's discount BD1.
• Sum 2: S2 = Rs. 1872 with true discount TD2.
• Rate of interest R = 12% per annum.
• Time = T years (same for both sums, unknown).
• Condition: BD1 = TD2.

Concept / Approach:
We use separate formulas:

• Banker's discount on S1: BD1 = S1 * R * T / 100.
• True discount on S2: TD2 = S2 * (R * T) / (100 + R * T).

We equate BD1 and TD2, then solve for T. Once T is found in years, we convert it to months for the final answer.

Step-by-Step Solution:
Step 1: Compute BD1 in terms of T: BD1 = 1800 * 12 * T / 100 = 216T.Step 2: Let x = R * T = 12T for convenience.Step 3: True discount on S2: TD2 = 1872 * x / (100 + x).Step 4: Since BD1 = TD2, we have 216T = 1872 * x / (100 + x).Step 5: Substitute x = 12T into the equation: 216T = 1872 * 12T / (100 + 12T).Step 6: For T ≠ 0, cancel T from both sides: 216 = 1872 * 12 / (100 + 12T).Step 7: Multiply both sides by (100 + 12T): 216(100 + 12T) = 1872 * 12.Step 8: Compute right side: 1872 * 12 = 22464.Step 9: Expand left side: 21600 + 2592T = 22464.Step 10: Rearranging: 2592T = 22464 − 21600 = 864.Step 11: T = 864 / 2592 = 1/3 year.Step 12: Convert to months: (1/3) year = 12 / 3 = 4 months.

Verification / Alternative check:
Take T = 1/3 year. Then BD1 = 1800 * 12 * (1/3) / 100 = 1800 * 4 / 100 = 72. For the second bill, R * T = 12 * (1/3) = 4. True discount TD2 = 1872 * 4 / (100 + 4) = 7488 / 104 = 72. Since both BD1 and TD2 equal Rs. 72, the time T = 1/3 year or 4 months is confirmed as correct.

Why Other Options Are Wrong:
Any time other than 4 months will break the equality BD1 = TD2. Substituting 3 months, 5 months or 6 months into the formulas leads to different numerical values for BD1 and TD2. Thus, those options do not satisfy the basic condition given in the question.

Common Pitfalls:

• Not cancelling T correctly and ending up with a more complicated equation than necessary.
• Confusing the formulas for true discount and simple interest.
• Making arithmetic mistakes when multiplying or subtracting large numbers.
• Forgetting to convert the final answer from years to months.

Final Answer:
The required time is 4 months.