Question
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.
Answers
GPT-4o mini
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 \( \frac{r}{n} = \frac{0.055}{12} \):
\[
\frac{0.055}{12} \approx 0.00458333
\]
2. Next, calculate \( (1 + \frac{r}{n}) \):
\[
1 + \frac{0.055}{12} \approx 1.00458333
\]
3. Raise it to the power of \( -n \cdot l \):
\( n \cdot l = 12 \cdot 30 = 360 \)
\[
\left(1.00458333\right)^{-360} \approx 0.174731
\]
4. Calculate \( 1 - \left(1.00458333\right)^{-360} \):
\[
1 - 0.174731 \approx 0.825269
\]
5. 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**:
\[
\text{Total Payments} = \text{PMT} \times n \cdot l = 1563 \times 360 \approx 563,280
\]
2. **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**.
\[
\text{PMT} = \frac{P \left( \frac{r}{n} \right)}{1 - \left(1 + \frac{r}{n}\right)^{-n \cdot l}}
\]
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 \( \frac{r}{n} = \frac{0.055}{12} \):
\[
\frac{0.055}{12} \approx 0.00458333
\]
2. Next, calculate \( (1 + \frac{r}{n}) \):
\[
1 + \frac{0.055}{12} \approx 1.00458333
\]
3. Raise it to the power of \( -n \cdot l \):
\( n \cdot l = 12 \cdot 30 = 360 \)
\[
\left(1.00458333\right)^{-360} \approx 0.174731
\]
4. Calculate \( 1 - \left(1.00458333\right)^{-360} \):
\[
1 - 0.174731 \approx 0.825269
\]
5. 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**:
\[
\text{Total Payments} = \text{PMT} \times n \cdot l = 1563 \times 360 \approx 563,280
\]
2. **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**.