The property taxes of Mr. and Mrs. Espedido amount to 2450 dollars and are due on July 1. If they pay eight months earlier and the city can earn 6% compounded monthly on surplus funds, what amount should the city accept as an early payment?

Introduction / Context:
This question is about present value, which is the reverse of the usual compound interest calculation. The city is entitled to receive 2450 dollars on July 1, but if the taxes are paid eight months early, the city can invest the money and earn interest. Therefore, the city should accept a smaller amount now that will grow to 2450 dollars in eight months at the given rate. This is a standard time value of money problem involving monthly compounding.

Given Data / Assumptions:

• Future value (tax due on July 1) F = 2450 dollars.
• Time difference for early payment = 8 months.
• Nominal annual interest rate r = 6% compounded monthly.
• We want the present value P that will grow to 2450 in 8 months at this rate.
• Number of compounding periods n = 8 months.

Concept / Approach:
With monthly compounding, the monthly rate i is r / 12 in decimal form. The relationship between present value P and future value F after n months is F = P * (1 + i) ^ n. To find P when F is known, we rearrange this equation to P = F / ((1 + i) ^ n). This gives the amount that, if received now and invested at the monthly rate, would accumulate to the required future amount in n months.

Step-by-Step Solution:
Step 1: Convert the nominal annual rate to monthly rate: r = 6% per year, so i = 0.06 / 12 = 0.005 per month. Step 2: The number of months until the due date is n = 8. Step 3: Use the present value formula P = F / ((1 + i) ^ n). Step 4: Compute the monthly growth factor: 1 + i = 1 + 0.005 = 1.005. Step 5: Compute (1.005) ^ 8, which is approximately 1.04068. Step 6: Now compute P = 2450 / 1.04068, which is about 2354.17. Step 7: Therefore, the city should accept approximately 2354.17 dollars if the taxes are paid eight months early.

Verification / Alternative check:
To verify, we can check that investing 2354.17 at 0.5% per month for eight months indeed reaches about 2450. The future value is 2354.17 * (1.005) ^ 8, which returns very close to 2450 when calculated. Also, note that the present value must be slightly less than 2450 because receiving money earlier is more valuable than receiving it later at a positive interest rate. The discount of roughly 95.83 dollars over eight months at a moderate rate of 6% per year is reasonable and consistent with the logic of present value.

Why Other Options Are Wrong:
The value 2354 is close but is a rounded down approximation that does not match the more precise calculation. The option 2376 is too high and would grow to more than 2450 in eight months, putting the city ahead of what is appropriate for an equivalent payment. The option 2389 is also slightly high and does not correspond to the correct growth factor. The option 2400 is very close to the full tax amount but does not adjust sufficiently for the eight months of potential earnings that the city has by receiving the funds early.

Common Pitfalls:
Students may mistakenly treat this as a future value problem and multiply instead of dividing by the compounding factor, which yields an amount larger than 2450, contradicting the idea of early payment. Others may use simple interest rather than monthly compounding, which leads to a slightly different present value. Forgetting to convert the annual rate to a monthly rate is another frequent problem. The key is to recognise that present value calculations require dividing the future amount by the appropriate growth factor based on both rate and time.

Final Answer:
The amount the city should accept for the property taxes if paid eight months early, with an earning rate of 6% compounded monthly, is approximately 2354.17 dollars, which is option A.