Questions: One retailer charges 1048 for a certain computer. A firm of tax accountants buys 6 of these computers. It makes a down payment of 1100 and agrees to amortize the balance with monthly payments at 15% interest for 3 years. Prepare an amortization schedule showing the first four monthly payments for this loan. Write the equation that can be used to find the amount of each monthly payment. PMT = (PV * )/(1-(1+)) where PV= (Type integers or decimals. Simplify your answers.)

One retailer charges 1048 for a certain computer. A firm of tax accountants buys 6 of these computers. It makes a down payment of 1100 and agrees to amortize the balance with monthly payments at 15% interest for 3 years. Prepare an amortization schedule showing the first four monthly payments for this loan. Write the equation that can be used to find the amount of each monthly payment. PMT = (PV * )/(1-(1+)) where PV= (Type integers or decimals. Simplify your answers.)
Transcript text: One retailer charges $\$ 1048$ for a certain computer. A firm of tax accountants buys 6 of these computers. It makes a down payment of $\$ 1100$ and agrees to amortize the balance with monthly payments at $15 \%$ interest for 3 years. Prepare an amortization schedule showing the first four monthly payments for this loan. Write the equation that can be used to find the amount of each monthly payment. \[ \text { PMT }=\frac{\mathrm{PV} \cdot \square}{1-(1+\square)} \text { where } \mathrm{PV}= \] (Type integers or decimals. Simplify your answers.)
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Solution

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Solution Steps

To solve this problem, we need to calculate the monthly payment for an amortizing loan. The formula for the monthly payment (PMT) is derived from the present value of an annuity formula. First, calculate the total cost of the computers and subtract the down payment to find the principal amount (PV) to be financed. Then, use the formula for PMT, which involves the principal amount, the monthly interest rate, and the total number of payments. Finally, generate an amortization schedule for the first four payments.

Solution Approach
  1. Calculate the total cost of the computers and subtract the down payment to find the principal amount (PV).
  2. Use the formula for the monthly payment (PMT) of an amortizing loan: \[ \text{PMT} = \frac{\text{PV} \cdot r}{1 - (1 + r)^{-n}} \] where \( r \) is the monthly interest rate and \( n \) is the total number of payments.
  3. Generate the amortization schedule for the first four payments, showing the breakdown of each payment into interest and principal.
Step 1: Calculate Total Cost and Principal

The total cost of the computers is calculated as: \[ \text{Total Cost} = 1048 \times 6 = 6288 \] The principal amount to be financed after the down payment is: \[ \text{Principal} = 6288 - 1100 = 5188 \]

Step 2: Determine Monthly Interest Rate and Total Payments

The annual interest rate is \(15\%\), so the monthly interest rate is: \[ r = \frac{0.15}{12} = 0.0125 \] The total number of payments over 3 years is: \[ n = 3 \times 12 = 36 \]

Step 3: Calculate Monthly Payment

Using the formula for the monthly payment: \[ \text{PMT} = \frac{5188 \times 0.0125}{1 - (1 + 0.0125)^{-36}} \approx 179.8437 \]

Step 4: Generate Amortization Schedule for First Four Payments

The amortization schedule for the first four payments is as follows:

  1. Month 1:

    • Monthly Payment: \(179.8437\)
    • Interest Payment: \(5188 \times 0.0125 \approx 64.85\)
    • Principal Payment: \(179.8437 - 64.85 \approx 114.9937\)
    • Remaining Balance: \(5188 - 114.9937 \approx 5073.0063\)

  2. Month 2:

    • Monthly Payment: \(179.8437\)
    • Interest Payment: \(5073.0063 \times 0.0125 \approx 63.4126\)
    • Principal Payment: \(179.8437 - 63.4126 \approx 116.4311\)
    • Remaining Balance: \(5073.0063 - 116.4311 \approx 4956.5751\)

  3. Month 3:

    • Monthly Payment: \(179.8437\)
    • Interest Payment: \(4956.5751 \times 0.0125 \approx 61.9572\)
    • Principal Payment: \(179.8437 - 61.9572 \approx 117.8865\)
    • Remaining Balance: \(4956.5751 - 117.8865 \approx 4838.6886\)

  4. Month 4:

    • Monthly Payment: \(179.8437\)
    • Interest Payment: \(4838.6886 \times 0.0125 \approx 60.4836\)
    • Principal Payment: \(179.8437 - 60.4836 \approx 119.3601\)
    • Remaining Balance: \(4838.6886 - 119.3601 \approx 4719.3285\)

Final Answer

The monthly payment is approximately \( \boxed{179.8437} \). The amortization schedule for the first four payments is summarized as follows:

  1. Month 1: Remaining Balance \( \approx 5073.0063 \)
  2. Month 2: Remaining Balance \( \approx 4956.5751 \)
  3. Month 3: Remaining Balance \( \approx 4838.6886 \)
  4. Month 4: Remaining Balance \( \approx 4719.3285 \)
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