
Brad decides to purchase a $240,000 house. He wants to finance the entire balance. He has received an APR of 3.4% for a 15-year mortgage. What is Brad’s total cost if he takes all 15 years to pay off the house? Round your answer to the nearest hundredth.
 
Formulas
Regular Payment for Fixed Installment Loans 	PMT=(P⋅rn)[1−(1+rn)−nt]
Number of Fixed Payments Required to Pay Off Credit Card Debt 	R=−log[1−rn(APMT)]log(1+rn)

To solve Brad's mortgage problem, we will use the formula for calculating the regular payment (PMT) for a fixed installment loan. The formula is:

\[
PMT = \frac{P \cdot r_n}{1 - (1 + r_n)^{-n_t}}
\]

Where:
- $ P $ is the principal amount (the loan), which is $240,000.
- $ r $ is the annual interest rate (APR), which is 3.4% or 0.034.
- $ n $ is the number of payments per year (since this is a monthly mortgage, $ n = 12 $).
- $ t $ is the total number of years for the mortgage, which is 15 years.

### Step-by-Step Calculation:

1. **Calculate the monthly interest rate**:
   \[
   r_n = \frac{r}{n} = \frac{0.034}{12} = 0.0028333
   \]

2. **Calculate the total number of payments**:
   \[
   n_t = n \cdot t = 12 \cdot 15 = 180 \text{ payments}
   \]

3. **Substitute the values into the formula**:
   \[
   PMT = \frac{240,000 \cdot 0.0028333}{1 - (1 + 0.0028333)^{-180}}
   \]

Let's now compute the monthly payment (PMT).Brad's monthly mortgage payment (PMT) is approximately \$1,703.96.

### Step 4: Calculate the total cost over 15 years:
The total cost will be the monthly payment multiplied by the total number of payments:

\[
\text{Total Cost} = PMT \times n_t = 1,703.96 \times 180
\]

Let me calculate that for you.Brad's total cost over the 15-year period will be approximately \$306,712.17.

### Summary:
- **Monthly Payment (PMT):** \$1,703.96
- **Total Cost over 15 years:** \$306,712.17

Let me know if you want further details or have any questions. Here are 5 related questions that can expand this information:

1. What if Brad made additional payments? How would that affect the total cost?
2. How would a higher interest rate, say 4%, impact Brad's monthly payments?
3. What would Brad's monthly payment be if he financed the house over 30 years instead of 15?
4. How much would Brad save in total cost if the interest rate were reduced to 3.0%?
5. How would property taxes or insurance affect Brad’s monthly payment?

**Tip:** Making extra payments on your mortgage can reduce the total interest paid over the life of the loan, leading to significant savings.

Brad decides to purchase a $240,000 house. He wants to finance the entire balance. He has received an APR of 3.4% for a 15-year mortgage. What is Brad’s total cost if he takes all 15 years to pay off the house? Round your answer to the nearest hundredth.

Learn how to calculate the total mortgage cost for a $240,000 house financed at 3.4% APR over 15 years. This detailed step-by-step solution covers key financial math concepts, including loan amortization and interest rate calculations, resulting in a total cost of $306,712.17. Suitable for advanced high school math students.

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