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. What is the time period (in months)?

Introduction / Context:
This question links the banker's discount on one sum to the true discount on another sum when both are calculated at the same rate of simple interest and for the same time period. By equating these two quantities and using the standard formulas, we can solve for the time. It is a good test of understanding how BD and TD depend differently on the face value, rate and time.

Given Data / Assumptions:
- Face value for banker's discount F1 = Rs 1800
- Face value for true discount F2 = Rs 1872
- Rate of simple interest r = 12 percent per annum
- Time period t is the same for both and is to be found
- Banker's discount on Rs 1800 equals the true discount on Rs 1872

Concept / Approach:
Banker's discount is calculated on the face value as BD = F1 × r × t / 100. True discount on F2 is given by TD = F2 × r × t / (100 + r × t). The condition BD = TD gives an equation in terms of r, t and the two face values. We can cancel the common factor r × t and solve for r t, then convert the resulting time in years to months.

Step-by-Step Solution:
Step 1: Write BD on Rs 1800: BD = 1800 × r × t / 100.Step 2: Write TD on Rs 1872: TD = 1872 × r × t / (100 + r × t).Step 3: Since BD = TD, we have 1800 × r × t / 100 = 1872 × r × t / (100 + r × t).Step 4: Cancel r × t from both sides (it is non zero), giving 1800 / 100 = 1872 / (100 + r × t).Step 5: Simplify 1800 / 100 to 18, so 18 = 1872 / (100 + r × t).Step 6: Cross multiply: 18 × (100 + r × t) = 1872 which gives 1800 + 18 r t = 1872.Step 7: Therefore 18 r t = 72 and r t = 4.Step 8: With r = 12 percent, 12 × t = 4, so t = 4 ÷ 12 = 1 / 3 year.Step 9: Convert to months: time = (1 / 3) × 12 = 4 months.

Verification / Alternative check:
To verify, substitute t = 4 months = 1 / 3 year back into the formulas. For Rs 1800, BD = 1800 × 12 × (1 / 3) / 100 = 1800 × 4 / 100 = Rs 72. For Rs 1872, r t = 4, so TD = 1872 × 4 / (100 + 4) = 7488 / 104 = Rs 72. Both discounts are equal at Rs 72, which confirms that a time of 4 months is correct.

Why Other Options Are Wrong:
If t were 3 months, time in years would be 1 / 4 and r t would be 3, not 4, which would break the equality between BD and TD. Values such as 5, 6 or 7 months would also produce different r t products and no longer satisfy the derived equation. Only 4 months leads to BD and TD being numerically equal for the given sums.

Common Pitfalls:
Students sometimes forget to cancel r × t from both sides and instead try to solve for r and t separately, which is unnecessary. Another mistake is to misconvert years to months or to treat 4 as the time directly in years instead of as the product r t. Carefully distinguishing between the product r t and the actual time in years is crucial.

Final Answer:
The required time period is 4 months.