To find the equal payments that would replace the given payments, we use the concept of the present value (PV) of the given payments and equate it to the PV of the new equal payments. Here's how the solution proceeds step-by-step:

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1. Determine the Present Value of the Given Payments:

Formula for PV:

$PV = \frac{FV}{(1 + i)^n}$

Where:

• $FV$ = Future Value of the payment
• $i$ = Monthly interest rate = $\frac{4.50\%}{12} = 0.00375$
• $n$ = Number of months until the payment is made

For the payment of $50,000 in 6 years ($n = 6 \times 12 = 72$):

$PV_1 = \frac{50,000}{(1 + 0.00375)^{72}}$

For the payment of $92,500 in 8 years ($n = 8 \times 12 = 96$):

$PV_2 = \frac{92,500}{(1 + 0.00375)^{96}}$

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2. Calculate the PV of the Equal Payments:

Let the equal payments be $P$, made at 2 years and 5 years. The PV of these payments is: $PV_{\text{equal}} = P \cdot \frac{1}{(1 + 0.00375)^{24}} + P \cdot \frac{1}{(1 + 0.00375)^{60}}$

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3. Equating Present Values:

The total PV of the given payments equals the PV of the new payments: $PV_1 + PV_2 = P \cdot \left(\frac{1}{(1 + 0.00375)^{24}} + \frac{1}{(1 + 0.00375)^{60}}\right)$

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4. Solve for $P$:

Substitute the known values to find $P$. Let me calculate this. Given data FV1 = 50000 # Future value of first payment FV2 = 92500 # Future value of second payment r_annual = 0.045 # Annual interest rate n_monthly = 12 # Number of compounding periods per year

Calculate monthly interest rate

i = r_annual / n_monthly

Time in months

n1 = 6 * n_monthly # 6 years for first payment n2 = 8 * n_monthly # 8 years for second payment

Calculate PV of given payments

PV1 = FV1 / (1 + i) ** n1 PV2 = FV2 / (1 + i) ** n2 total_PV_given = PV1 + PV2

Time for equal payments in months

n3 = 2 * n_monthly # 2 years for first equal payment n4 = 5 * n_monthly # 5 years for second equal payment

PV factor for equal payments

PV_factor_equal = (1 / (1 + i) ** n3) + (1 / (1 + i) ** n4)

Solve for equal payment P

P = total_PV_given / PV_factor_equal

PV1, PV2, total_PV_given, PV_factor_equal, P### Calculations:

1. Present Value of $50,000 in 6 years: $PV_1 = 38,188.25$

2. Present Value of $92,500 in 8 years: $PV_2 = 64,578.53$

3. Total Present Value of Given Payments: $PV_{\text{total}} = 38,188.25 + 64,578.53 = 102,766.78$

4. Present Value Factor for Equal Payments: $PV_{\text{factor}} = 1.712937$

5. Equal Payment $P$: $P = \frac{102,766.78}{1.712937} = 59,994.48$

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Final Answer:

The equal payments to replace the given payments would be $59,994.48 at both 2 years and 5 years.

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Would you like a detailed explanation or breakdown of any step?
Here are 5 related questions for further exploration:

1. How does changing the interest rate affect the equal payment amount?
2. What if the payments were made annually instead of monthly?
3. How would the solution change if the equal payments were made in 3 and 4 years instead?
4. Can this method be extended to unevenly spaced payments?
5. How does the compounding frequency influence the results?

Tip: For precise calculations, always ensure the interest rate and compounding periods align with the payment intervals.