To find the equivalent single payment today (present value) that would replace the two future payments of $6,500 in 4 months and $6,500 in 8 months, we need to discount these future payments to their present values using the 3% annual interest rate.

Step 1: Convert the annual interest rate to a monthly rate.

• The interest rate is 3% per year.
• The monthly interest rate is:
  $\text{Monthly Rate} = \frac{3\%}{12} = 0.25\% \text{ per month}$

In decimal form, the monthly rate is $0.0025$.

Step 2: Calculate the present value of each payment.

• The first payment of $6,500 is due in 4 months. Its present value is: $PV_1 = \frac{6500}{(1 + 0.0025)^4}$

• The second payment of $6,500 is due in 8 months. Its present value is: $PV_2 = \frac{6500}{(1 + 0.0025)^8}$

Step 3: Sum the present values to find the total equivalent payment today.

Let me compute this.The equivalent single payment you would need to make today, instead of paying $6,500 in 4 months and $6,500 in 8 months, is approximately $12,806.85.

Would you like more details or explanations on any part of this calculation?

Here are five related questions to expand your understanding:

1. How does the present value change if the interest rate increases?
2. What if the payments were made at different intervals, say 6 and 12 months?
3. How would compound interest affect the equivalent payment today?
4. What is the difference between present value and future value in financial terms?
5. How does inflation impact the real value of future payments?

Tip: Present value allows you to compare the value of money at different times by accounting for interest, helping you make informed financial decisions.