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Use the formula PMT = (P(r/n))/[1 - (1 + r/n) ^ (- n * l)] to determine the regular payment amount, rounded to the nearest dollar. The price of the home is $350,000. The bank requires a 20% down payment. The cost of the home is financed with a 30-year fixed rate mortage at an interest rate of 5.5%. Find the total cost of interest over 30 years. Showing your work is required.

1 answer

To calculate the regular payment amount (PMT) using the formula:

\[ \text{PMT} = \frac{P \left( \frac{r}{n} \right)}{1 - \left(1 + \frac{r}{n}\right)^{-n \cdot l}} \]

We need to determine several variables:

  1. Home Price (H) = $350,000
  2. Down Payment Percentage = 20%
  3. Loan Amount (P) = Home Price - Down Payment
  4. Interest Rate (r) = 5.5% = 0.055
  5. Number of Payments per Year (n) = 12 (monthly payments)
  6. Loan Term (l) = 30 years

Step 1: Calculate Down Payment and Loan Amount

First, we need to calculate the down payment and the amount of the loan.

\[ \text{Down Payment} = H \times \text{Down Payment Percentage} = 350,000 \times 0.20 = 70,000 \]

\[ P = H - \text{Down Payment} = 350,000 - 70,000 = 280,000 \]

Step 2: Substitute Variables into the PMT Formula

Now we can substitute the values into the PMT formula:

  • \( P = 280,000 \)
  • \( r = 0.055 \)
  • \( n = 12 \)
  • \( l = 30 \)

The formula becomes:

\[ \text{PMT} = \frac{280,000 \left( \frac{0.055}{12} \right)}{1 - \left(1 + \frac{0.055}{12}\right)^{-12 \times 30}} \]

Step 3: Calculate Each Component

  1. Calculate \( \frac{r}{n} = \frac{0.055}{12} \):

\[ \frac{0.055}{12} \approx 0.00458333 \]

  1. Next, calculate \( (1 + \frac{r}{n}) \):

\[ 1 + \frac{0.055}{12} \approx 1.00458333 \]

  1. Raise it to the power of \( -n \cdot l \):

\( n \cdot l = 12 \cdot 30 = 360 \)

\[ \left(1.00458333\right)^{-360} \approx 0.174731 \]

  1. Calculate \( 1 - \left(1.00458333\right)^{-360} \):

\[ 1 - 0.174731 \approx 0.825269 \]

  1. Substitute everything into the PMT formula:

\[ \text{PMT} = \frac{280,000 \cdot 0.00458333}{0.825269} \approx \frac{1288.33333}{0.825269} \approx 1562.63 \]

Step 4: Round to the Nearest Dollar

Thus, the monthly payment (PMT) is:

\[ \text{PMT} \approx 1563 \]

Step 5: Calculate Total Payments and Total Interest

  1. Total Payments Over 30 years:

\[ \text{Total Payments} = \text{PMT} \times n \cdot l = 1563 \times 360 \approx 563,280 \]

  1. Total Interest Paid:

\[ \text{Total Interest} = \text{Total Payments} - \text{Loan Amount} = 563,280 - 280,000 = 283,280 \]

Summary

  • The regular monthly payment amount is $1,563.
  • The total cost of interest over 30 years is $283,280.