What equal payments in 3 years and 5 years would replace payments of $32,500 and $100,000 in 7 years and 8 years, respectively? Assume money can earn 4.68% compounded quarterly. Use 8 years as the focal date.

Round to the nearest cent

To solve this, we will use the concept of time value of money and the equivalence of payments. The main idea is that the present values (PV) or future values (FV) of the original payments must equal the future values of the new payments, all calculated at a common focal date, which in this case is 8 years from now.

### Step 1: Calculate the future values of the original payments

The payments are:

- $32,500 in 7 years
- $100,000 in 8 years

We will bring the $32,500 forward to the 8-year mark using the future value formula for compound interest:

\[
FV = PV \times (1 + i)^n
\]

Where:
- $ PV = 32,500 $ (the payment at year 7)
- $ i = \frac{4.68\%}{4} = 0.0117 $ (quarterly interest rate, since it's compounded quarterly)
- $ n = 4 \times (8 - 7) = 4 $ (the number of quarters between years 7 and 8)

Now, calculate the future value of the $32,500 payment at the 8-year mark:

\[
FV = 32,500 \times (1 + 0.0117)^4
\]

Next, the $100,000 payment is already at the 8-year mark, so its future value remains $100,000.

### Step 2: Calculate the total future value of the payments at 8 years

Once we find the future value of the $32,500 payment, we will sum it with the $100,000.

### Step 3: Find the present values of the new equal payments

Now, let's find the present values of the equal payments that are to be made in years 3 and 5. We'll discount these payments back to the 8-year mark using the present value formula:

\[
PV = FV \times (1 + i)^{-n}
\]

For the payment in year 3:
- $ FV = X $ (the unknown payment)
- $ n = 4 \times (8 - 3) = 20 $ quarters

For the payment in year 5:
- $ FV = X $
- $ n = 4 \times (8 - 5) = 12 $ quarters

### Step 4: Set up the equation and solve for $ X $

The total future value of the original payments at 8 years must equal the total present value of the new equal payments:

\[
32,500 \times (1 + 0.0117)^4 + 100,000 = X \times (1 + 0.0117)^{-20} + X \times (1 + 0.0117)^{-12}
\]

Now we solve for $ X $.

---

Let me calculate these values for you.The future value of the $32,500 payment at the 8-year mark is approximately **$34,047.90**. 

The equal payments that would replace the payments of $32,500 in 7 years and $100,000 in 8 years, if made at years 3 and 5, would each be approximately **$80,646.98**.

Would you like further details on how these calculations were made or have any other questions?

Here are 5 related questions to expand on this:
1. How would the equal payments change if the interest rate were different?
2. What is the impact of using monthly compounding instead of quarterly compounding?
3. How do you calculate present and future values with varying interest rates?
4. Can you explain how the time value of money principle applies to other financial scenarios?
5. What happens if the payment dates are changed to non-integer years, like 3.5 years?

**Tip:** When using compound interest, the frequency of compounding periods (monthly, quarterly, etc.) can significantly affect the total amount paid or received.

What equal payments in 3 years and 5 years would replace payments of $32,500 and $100,000 in 7 years and 8 years, respectively? Assume money can earn 4.68% compounded quarterly. Use 8 years as the focal date.

Find out how to calculate equal payments in 3 and 5 years to replace payments of $32,500 and $100,000 in 7 and 8 years, using compound interest at 4.68% quarterly. The problem covers key concepts of time value of money, compound interest, and payment equivalence, suitable for advanced high school and undergraduate students.

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