Question

What equal payments in 3 years and 5 years would replace payments of $42,500 and $60,000 in 7 years and 9 years, respectively? Assume money can earn 3.36% compounded quarterly. Round to the nearest cent

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Hermann

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To solve this problem, we first need to calculate the present value of the future payments and then determine the equal payments at the 3-year and 5-year marks that have the same total present value when discounted back to the present at the given interest rate of 3.36% compounded quarterly.

### Step 1: Calculate the quarterly interest rate

Given that the annual interest rate is 3.36%, the quarterly interest rate is:

r = \frac{3.36\%}{4} = 0.84\%

Converting to decimal form, we get:

r = 0.0084

### Step 2: Calculate the number of quarters for each payment

- 7 years = 28 quarters

- 9 years = 36 quarters

- 3 years = 12 quarters

- 5 years = 20 quarters

### Step 3: Calculate the present value (PV) of each future payment

The present value formula is given by:

PV = \frac{P}{(1 + r)^n}

whereP is the payment, r is the quarterly interest rate, and n is the number of quarters.

Calculating the present values of the future payments:

- Present value of42,500 due in 7 years is 33,625.56.

- Present value of60,000 due in 9 years is 44,398.58.

### Step 4: Calculate the total present value

Adding the present values of the two future payments gives a total present value of $78,024.14.

### Step 5: Calculate the present value of the payments in 3 years and 5 years

We need to find equal payments that, when discounted back to the present at the interest rate, will match the total present value from step 4.

### Step 6: Solve for the equal payments (P )

Setting up the equation:

P \times \left( \frac{1}{(1 + 0.0084)^{12}} + \frac{1}{(1 + 0.0084)^{20}} \right) = 78024.14

By solving the equation, we find that each equal payment should be approximately $44,573.97.

Therefore, the equal payments required at the 3-year and 5-year marks to match the total present value of the original future payments are approximately\ 44,573.97$ each.

### Step 1: Calculate the quarterly interest rate

Given that the annual interest rate is 3.36%, the quarterly interest rate is:

Converting to decimal form, we get:

### Step 2: Calculate the number of quarters for each payment

- 7 years = 28 quarters

- 9 years = 36 quarters

- 3 years = 12 quarters

- 5 years = 20 quarters

### Step 3: Calculate the present value (PV) of each future payment

The present value formula is given by:

where

Calculating the present values of the future payments:

- Present value of

- Present value of

### Step 4: Calculate the total present value

Adding the present values of the two future payments gives a total present value of $78,024.14.

### Step 5: Calculate the present value of the payments in 3 years and 5 years

We need to find equal payments that, when discounted back to the present at the interest rate, will match the total present value from step 4.

### Step 6: Solve for the equal payments (

Setting up the equation:

By solving the equation, we find that each equal payment should be approximately $44,573.97.

Therefore, the equal payments required at the 3-year and 5-year marks to match the total present value of the original future payments are approximately

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