The true discount on a bill due 9 months hence at 12% simple interest per annum is Rs. 540. What is the present worth of the bill?

Introduction / Context:
In this question the true discount on a bill is known, along with the time to maturity and the rate of simple interest. We are asked for the present worth of the bill. This involves first finding the face value of the bill using the true discount formula and then subtracting the true discount to get the present worth. Such problems are common in banking related aptitude topics.

Given Data / Assumptions:

• True discount TD = Rs. 540.
• Time to maturity t = 9 months = 9 / 12 year = 3 / 4 year.
• Rate of simple interest r = 12 percent per annum.
• Face value (amount of the bill at maturity), S, is unknown.
• Present worth PW = S − TD must be found.

Concept / Approach:
True discount TD for sum S due after t years at rate r percent is: TD = S * r * t / (100 + r * t) We first find r * t, then solve for S using the given true discount. Finally, we calculate the present worth PW = S − TD. The option list clearly suggests that 6000, 6500, 7000, or 1600 are likely candidates, so accurate calculation is important.

Step-by-Step Solution:
Step 1: Convert time to years: t = 9 months = 9 / 12 = 3 / 4 year. Step 2: Compute r * t = 12 * 3 / 4 = 36 / 4 = 9. Step 3: Use TD formula: 540 = S * 9 / (100 + 9) = S * 9 / 109. Step 4: Rearrange: S = 540 * 109 / 9. Step 5: Simplify 540 / 9 = 60, so S = 60 * 109 = 6540. Step 6: Present worth PW is S − TD = 6540 − 540 = 6000.

Verification / Alternative check:
We can recompute TD from S = 6540 to verify. With r * t = 9: TD = 6540 * 9 / (100 + 9) = 6540 * 9 / 109. Compute: 6540 * 9 = 58860, and 58860 / 109 = 540. This matches the given true discount. Then the present worth PW = 6540 − 540 = 6000, confirming the correctness of the answer.

Why Other Options Are Wrong:
Values such as Rs. 6500 or Rs. 7000 would be closer to the face value S and not to the present worth. Rs. 1600 and Rs. 5400 are much too small to be the present worth when the discount is only Rs. 540 for less than a year at 12 percent. Only Rs. 6000 is consistent with the derived face value and discount.

Common Pitfalls:
Students often forget to convert months to years correctly and may use t = 9 instead of 3 / 4, causing very large or very small answers. Another common mistake is to misinterpret Rs. 540 as the present worth rather than the true discount. Always distinguish between S (amount due), TD (true discount), and PW (present worth), and apply the formula TD = S * r * t / (100 + r * t) systematically.

Final Answer:
The present worth of the bill is Rs. 6000.