One retailer charges \( \$ 1028 \) for a certain computer. A firm of tax accountants buys 7 of these computers. It makes a down payment of \( \$ 1100 \) and agrees to amortize the balance with monthly payments at \( 15 \% \) interest for 5 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 \( =\frac{\mathrm{PV} \cdot \square}{1-(1+\square)} \) where PV \( =\square \) (Type integers or decimals. Simplify your answers.)

To solve this problem, we need to follow these steps: 1. **Calculate the total cost of the computers.** 2. **Determine the amount financed after the down payment.** 3. **Identify the monthly interest rate and the number of payments.** 4. **Use the amortization formula to find the monthly payment.** 5. **Prepare the amortization schedule for the first four payments.** ### Step 1: Calculate the total cost of the computers The cost of one computer is \( \$ 1028 \). The firm buys 7 computers, so the total cost (PV) is: \[ \text{Total Cost} = 1028 \times 7 \] ### Step 2: Determine the amount financed after the down payment The firm makes a down payment of \( \$ 1100 \). The amount financed (loan amount) is: \[ \text{Loan Amount} = \text{Total Cost} - \text{Down Payment} \] ### Step 3: Identify the monthly interest rate and the number of payments The annual interest rate is \( 15\% \), so the monthly interest rate \( r \) is: \[ r = \frac{15\%}{12} = \frac{0.15}{12} \] The total number of payments \( n \) over 5 years is: \[ n = 5 \times 12 \] ### Step 4: Use the amortization formula to find the monthly payment The formula for the monthly payment \( PMT \) is given by: \[ PMT = \frac{PV \cdot r}{1 - (1 + r)^{-n}} \] Where: - \( PV \) is the loan amount, - \( r \) is the monthly interest rate, - \( n \) is the total number of payments. ### Step 5: Prepare the amortization schedule for the first four payments Now, let's perform the calculations step by step. 1. Calculate the total cost of the computers. 2. Calculate the loan amount. 3. Calculate the monthly interest rate and number of payments. 4. Calculate the monthly payment using the formula. Let's perform these calculations. Calculate the value by following steps: - step0: Calculate: \(5\times 12\) - step1: Multiply the numbers: \(60\) Calculate or simplify the expression \( 1028 * 7 \). Calculate the value by following steps: - step0: Calculate: \(1028\times 7\) - step1: Multiply the numbers: \(7196\) Calculate or simplify the expression \( (0.15 / 12) \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.15}{12}\) - step1: Convert the expressions: \(\frac{\frac{3}{20}}{12}\) - step2: Multiply by the reciprocal: \(\frac{3}{20}\times \frac{1}{12}\) - step3: Reduce the numbers: \(\frac{1}{20}\times \frac{1}{4}\) - step4: Multiply the fractions: \(\frac{1}{20\times 4}\) - step5: Multiply: \(\frac{1}{80}\) Calculate or simplify the expression \( 7196 - 1100 \). Calculate the value by following steps: - step0: Calculate: \(7196-1100\) - step1: Subtract the numbers: \(6096\) Calculate or simplify the expression \( \frac{6096 * 0.0125}{1 - (1 + 0.0125)^{-60}} \). Calculate the value by following steps: - step0: Calculate: \(\frac{6096\times 0.0125}{1-\left(1+0.0125\right)^{-60}}\) - step1: Add the numbers: \(\frac{6096\times 0.0125}{1-1.0125^{-60}}\) - step2: Convert the expressions: \(\frac{6096\times 0.0125}{1-\left(\frac{81}{80}\right)^{-60}}\) - step3: Multiply the numbers: \(\frac{76.2}{1-\left(\frac{81}{80}\right)^{-60}}\) - step4: Subtract the numbers: \(\frac{76.2}{\frac{81^{60}-80^{60}}{81^{60}}}\) - step5: Convert the expressions: \(\frac{\frac{381}{5}}{\frac{81^{60}-80^{60}}{81^{60}}}\) - step6: Multiply by the reciprocal: \(\frac{381}{5}\times \frac{81^{60}}{81^{60}-80^{60}}\) - step7: Multiply the fractions: \(\frac{381\times 81^{60}}{5\left(81^{60}-80^{60}\right)}\) - step8: Multiply: \(\frac{381\times 81^{60}}{5\times 81^{60}-5\times 80^{60}}\) ### Summary of Calculations 1. **Total Cost of Computers:** \[ \text{Total Cost} = 1028 \times 7 = 7196 \] 2. **Loan Amount:** \[ \text{Loan Amount} = 7196 - 1100 = 6096 \] 3. **Monthly Interest Rate:** \[ r = \frac{15\%}{12} = 0.0125 \] 4. **Number of Payments:** \[ n = 5 \times 12 = 60 \] 5. **Monthly Payment Calculation:** Using the formula: \[ PMT = \frac{PV \cdot r}{1 - (1 + r)^{-n}} = \frac{6096 \cdot 0.0125}{1 - (1 + 0.0125)^{-60}} \approx 145.02 \] ### Amortization Schedule for the First Four Payments To create the amortization schedule, we will calculate the interest and principal portions of each payment for the first four months. 1. **First Payment:** - Interest for the first month: \[ \text{Interest} = \text{Loan Amount} \times r = 6096 \times 0.0125 \] - Principal repayment: \[ \text{Principal} = PMT - \text{Interest} \] - Remaining balance after the first payment: \[ \text{Remaining Balance} = \text{Loan Amount} - \text{Principal} \] 2. **Repeat for the next three payments.** Let's calculate the first payment details and then proceed to the next three payments. Calculate the value by following steps: - step0: Calculate: \(6096\times 0.0125\) - step1: Multiply the numbers: \(76.2\) Calculate or simplify the expression \( 145.023414 - (6096 * 0.0125) \). Calculate the value by following steps: - step0: Calculate: \(145.023414-\left(6096\times 0.0125\right)\) - step1: Multiply the numbers: \(145.023414-76.2\) - step2: Subtract the numbers: \(68.823414\) Calculate or simplify the expression \( 145.023414 - ((6096 - (145.023414 - 76.2)) * 0.0125) \). Calculate the value by following steps: - step0: Calculate: \(145.023414-\left(\left(6096-\left(145.023414-76.2\right)\right)\times 0.0125\right)\) - step1: Subtract the numbers: \(145.023414-\left(\left(6096-68.823414\right)\times 0.0125\right)\) - step2: Subtract the numbers: \(145.023414-\left(6027.176586\times 0.0125\right)\) - step3: Multiply the numbers: \(145.023414-75.339707325\) - step4: Subtract the numbers: \(69.683706675\) Calculate or simplify the expression \( (6096 - (145.023414 - 76.2) - ((6096 - (145.023414 - 76.2)) * 0.0125)) \). Calculate the value by following steps: - step0: Calculate: \(6096-\left(145.023414-76.2\right)-\left(\left(6096-\left(145.023414-76.2\right)\right)\times 0.0125\right)\) - step1: Subtract the numbers: \(6096-68.823414-\left(\left(6096-\left(145.023414-76.2\right)\right)\times 0.0125\right)\) - step2: Subtract the numbers: \(6096-68.823414-\left(\left(6096-68.823414\right)\times 0.0125\right)\) - step3: Subtract the numbers: \(6096-68.823414-\left(6027.176586\times 0.0125\right)\) - step4: Multiply the numbers: \(6096-68.823414-75.339707325\) - step5: Subtract the numbers: \(5951.836878675\) Calculate or simplify the expression \( (6096 - (145.023414 - 76.2)) * 0.0125 \). Calculate the value by following steps: - step0: Calculate: \(\left(6096-\left(145.023414-76.2\right)\right)\times 0.0125\) - step1: Subtract the numbers: \(\left(6096-68.823414\right)\times 0.0125\) - step2: Subtract the numbers: \(6027.176586\times 0.0125\) - step3: Multiply the numbers: \(75.339707325\) Calculate or simplify the expression \( 6096 - (145.023414 - 76.2) \). Calculate the value by following steps: - step0: Calculate: \(6096-\left(145.023414-76.2\right)\) - step1: Subtract the numbers: \(6096-68.823414\) - step2: Subtract the numbers: \(6027.176586\) ### Amortization Schedule for the First Four Payments 1. **First Payment:** - **Interest:** \[ \text{Interest} = 6096 \times 0.0125 = 76.2 \] - **Principal Repayment:** \[ \text{Principal} = 145.02 - 76.2 = 68.82 \] - **Remaining Balance:** \[ \text{Remaining Balance} = 6096 - 68.82 = 6027.18 \] 2. **Second Payment:** - **Interest:** \[ \text{Interest} = 6027.18 \times 0.0125 \approx 75.34 \] - **Principal Repayment:** \[ \text{Principal} = 145.02 - 75.34 \approx 69.68 \] - **Remaining Balance:** \[ \text{Remaining Balance} = 6027.18 - 69.68 \approx 5957.50 \] 3. **Third Payment:** - **Interest:** \[ \text{Interest} = 5957.50 \times 0.0125 \approx 74.46 \] - **Principal Repayment:** \[ \text{Principal} = 145.02 - 74.46 \approx 70.56 \] - **Remaining Balance:** \[ \text{Remaining Balance} = 5957.50 - 70.56 \approx 5886.94 \] 4. **Fourth Payment:** - **Interest:** \[ \text{Interest} = 5886.94 \times 0.0125 \approx 73.59 \] - **Principal Repayment:** \[ \text{Principal} = 145.02 - 73.59 \approx 71.43 \] - **Remaining Balance:** \[ \text{Remaining Balance} = 5886.94 - 71.43 \approx 5815.51 \] ### Amortization Schedule Summary | Payment Number | Payment Amount | Interest | Principal | Remaining Balance | |----------------|----------------|----------|-----------|-------------------| | 1 | 145.02 | 76.20 | 68.82 | 6027.18 | | 2 | 145.02 | 75.34 | 69.68 | 5957.50 | | 3 | 145.02 | 74.46 | 70.56 | 5886.94 | | 4 | 145.02 | 73.59 | 71.43 | 5815.51 | ### Final Equation for Monthly Payment The equation to find the amount of each monthly payment is: \[ PMT = \frac{6096 \cdot 0.0125}{1 - (1 + 0.0125)^{-60}} \] This completes the amortization schedule and the calculations for the loan.