The present value of a bill due at the end of 2 years is Rs 1250. If the same bill were due at the end of 2 years and 11 months, its present worth would be Rs 1200. Assuming simple interest in both cases, what is the sum (amount of the bill)?

Introduction / Context:
This true discount problem gives the present value of the same bill for two different maturity times. Based on these two present values and the assumption of simple interest, we must determine both the rate and the final amount of the bill. The options, however, list only amounts, so our key output is the sum of the bill. This type of question tests the ability to set up and solve equations involving present value under simple interest.

Given Data / Assumptions:
- Present value when the bill is due in 2 years is Rs 1250.
- Present value when the bill is due in 2 years and 11 months is Rs 1200.
- Interest is at the same simple rate in both cases.
- Let the amount of the bill be A and the rate of interest be r percent per annum.

Concept / Approach:
Under simple interest, the amount A due after t years from present value P is A = P * (1 + r * t / 100). Equivalently, P = A / (1 + r * t / 100). In this question we have two present values 1250 and 1200 for the same amount A but at two different times, namely t1 = 2 years and t2 = 2 years and 11 months (which is 35/12 years). We can write two equations in terms of A and r, then equate the two expressions for A and solve for r, and finally compute A.

Step-by-Step Solution:
Step 1: Let A be the amount of the bill and r be the annual simple interest rate. Step 2: For the bill due in 2 years, present value P1 = 1250, so A = 1250 * (1 + 2r/100). Step 3: For the bill due in 2 years and 11 months, which is 35/12 years, present value P2 = 1200, so A = 1200 * (1 + (35/12) * r / 100). Step 4: Since both expressions represent the same amount A, set them equal: 1250 * (1 + 2r/100) = 1200 * (1 + (35/12) * r / 100). Step 5: Solving this equation gives r = 5 percent per annum. Step 6: Substitute r = 5 in A = 1250 * (1 + 2r/100) = 1250 * (1 + 10/100) = 1250 * 1.10 = Rs 1375.

Verification / Alternative check:
Check with the second expression. Using r = 5 percent and t = 35/12 years, we get A = 1200 * (1 + (35/12) * 5 / 100). The factor (35/12) * 5 / 100 = 35 / 240 ≈ 0.14583, so the total multiplier is approximately 1.14583. Then A = 1200 * 1.14583 ≈ 1375. This matches the amount obtained from the first case, confirming that A = Rs 1375 is correct.

Why Other Options Are Wrong:
Options Rs 1175, Rs 1475 and Rs 1575 do not simultaneously satisfy both present value conditions when checked using the same rate. Only Rs 1375 yields present values of 1250 and 1200 for the two respective periods at a consistent simple interest rate of 5 percent per annum.

Common Pitfalls:
Some learners incorrectly treat the difference in present values as interest for the extra 11 months, ignoring the implied change in time for both cases. Another mistake is to forget to convert 2 years and 11 months correctly into a fraction of a year, which should be 35/12 years. Always write clear equations for present value in each scenario and solve them systematically.

Final Answer:
The sum (amount of the bill) is Rs 1375.