To address your questions step by step, we will calculate the interest rate as a decimal, find the monthly payments and total paid for both the 30-year and 15-year loans, and finally determine the savings from choosing the 15-year loan over the 30-year loan.
Step 1: Interest Rate as Decimal for the 30-Year Loan
The interest rate for the 30-year loan is 7.5%. As a decimal, this is calculated as:
\[ \text{Interest Rate} = \frac{7.5}{100} = 0.075 \]
Step 2: Monthly Payment for the 30-Year Loan
To calculate the monthly payment for a fixed-rate mortgage, we can use the formula:
\[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \]
Where:
- \( M \) = monthly payment
- \( P \) = loan principal ($300,000)
- \( r \) = monthly interest rate (annual rate / 12)
- \( n \) = total number of payments (loan term in months)
For the 30-year loan:
- \( P = 300,000 \)
- \( r = \frac{0.075}{12} = 0.00625 \)
- \( n = 30 \times 12 = 360 \)
Plugging in these values:
\[ M = 300,000 \frac{0.00625(1 + 0.00625)^{360}}{(1 + 0.00625)^{360} - 1} \]
Calculating \( (1 + 0.00625)^{360} \):
\[ (1 + 0.00625)^{360} \approx 7.6136 \quad \text{(using a calculator)} \]
Now substitute back into the formula:
\[ M = 300,000 \frac{0.00625 \times 7.6136}{7.6136 - 1} \]
\[ M = 300,000 \frac{0.047604}{6.6136} \approx 300,000 \times 0.007196 \approx 2158.72 \]
Step 3: Total Paid for the 30-Year Loan
The total amount paid over the term of the loan is given by:
\[ \text{Total Paid} = M \times n \]
So for the 30-year loan:
\[ \text{Total Paid} = 2158.72 \times 360 \approx 776,155.20 \]
Step 4: Monthly Payment for the 15-Year Loan
Using the same formula for the 15-year loan with \( r = \frac{0.0725}{12} = \frac{0.0725}{12} = 0.00604167 \) and \( n = 15 \times 12 = 180 \):
\[ M = 300,000 \frac{0.00604167(1 + 0.00604167)^{180}}{(1 + 0.00604167)^{180} - 1} \]
Calculating \( (1 + 0.00604167)^{180} \):
\[ (1 + 0.00604167)^{180} \approx 2.8678 \quad \text{(using a calculator)} \]
Substituting:
\[ M = 300,000 \frac{0.00604167 \times 2.8678}{2.8678 - 1} \]
\[ M = 300,000 \frac{0.01729}{1.8678} \approx 300,000 \times 0.009252 \approx 2775.61 \]
Step 5: Total Paid for the 15-Year Loan
\[ \text{Total Paid} = M \times n = 2775.61 \times 180 \approx 500,193.80 \]
Step 6: Savings by Choosing 15-Year Loan Instead of 30-Year Loan
\[ \text{Savings} = \text{Total Paid (30 years)} - \text{Total Paid (15 years)} \]
Calculating: \[ \text{Savings} = 776,155.20 - 500,193.80 \approx 275,961.40 \]
Summary of Results
- Interest Rate (30 years): 0.075
- Monthly Payment (30 years): Approx. $2158.72
- Total Paid (30 years): Approx. $776,155.20
- Monthly Payment (15 years): Approx. $2775.61
- Total Paid (15 years): Approx. $500,193.80
- Savings by choosing 15-year loan: Approx. $275,961.40
These results help compare the two loan options effectively.
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