An investment earns 4% per annum compounded monthly. What amount must you invest now in order to accumulate 10,000 dollars after 3 years?

Introduction / Context:
This question is another present value problem under monthly compounding. It reinforces the same core idea as earlier problems but with a different time horizon, which is a common variation in exam questions. Understanding how to move between present and future values under compound interest is a central skill in quantitative aptitude and financial mathematics.

Given Data / Assumptions:

• Future value FV = 10,000 dollars after 3 years.
• Nominal annual interest rate j = 4% per annum.
• Interest is compounded monthly, so m = 12 periods per year.
• Periodic rate i = j / m = 0.04 / 12 per month.
• Total number of months n = 3 * 12 = 36 months.

Concept / Approach:
The present value PV of a single future sum FV under monthly compounding is PV = FV / (1 + i)^n, where i is the monthly rate and n is the total number of months. Since we are given the future value and need to find the amount to invest now, we apply this formula directly. The calculation discounts the future sum back to its value today.

Step-by-Step Solution:
Step 1: Compute the monthly interest rate i = 0.04 / 12.Step 2: Calculate i = 0.003333... per month.Step 3: Determine the number of months n = 3 * 12 = 36.Step 4: Use the present value formula PV = FV / (1 + i)^n.Step 5: Substitute values: PV = 10,000 / (1 + 0.003333...)^36.Step 6: Evaluating this gives PV approximately equal to 8870.97 dollars.

Verification / Alternative check:
We can verify by moving forward in time. If we invest 8870.97 dollars at a monthly rate of about 0.3333% for 36 months, the future value is FV = 8870.97 * (1 + 0.003333...)^36 which comes very close to 10,000 dollars. Minor differences may appear due to rounding, but they confirm that this amount is the correct present value required to reach the desired future amount at the given interest rate.

Why Other Options Are Wrong:

• 8695.00: This would be sufficient for 3.5 years at 4% compounded monthly to reach 10,000, not for only 3 years.
• 9000.00: This is too high; investing 9000 at 4% compounded monthly for 3 years would result in more than 10,000.
• 8600.00: This is too low and would not grow to 10,000 in just 3 years at the given rate.

Common Pitfalls:
Common mistakes include using the annual rate directly without dividing by 12, raising to the power of 3 instead of 36, or forgetting that the formula for present value requires division by the compound factor. Some learners also incorrectly subtract interest linearly instead of applying true compound discounting. Careful attention to the rate conversion and exponent is essential for accurate results.

Final Answer:
The amount that must be invested now is approximately 8870.97 dollars in order to accumulate 10,000 dollars after 3 years at 4% per annum compounded monthly.