Question

Kayla wants to take out a mortgage of $90,000 with interest that compounds monthly. Use the formula A= P(1 + r/n) n*t to find which of these loans will have the lowest total cost.
A) 20 years at 6% interest
B) 25 years at 5.5% interest
C) 30 years at 4.5% interest
D) 35 years at 4% interest

Answer 1

To calculate the total cost of each mortgage option, we'll use the formula:

\[
A = P \left( 1 + \frac{r}{n} \right)^{n \cdot t}
\]

Where:
- \(A\) = the total amount (principal + interest) after time t,
- \(P\) = principal amount (initial loan amount),
- \(r\) = annual interest rate (as a decimal),
- \(n\) = number of times that interest is compounded per year,
- \(t\) = number of years.

Given that Kayla wants to take out a mortgage of \(P = 90,000\), we will calculate \(A\) for each loan option.

### Option A: 20 years at 6% interest
- \(r = 0.06\)
- \(n = 12\) (monthly compounding)
- \(t = 20\)

\[
A = 90000 \left( 1 + \frac{0.06}{12} \right)^{12 \cdot 20}
\]

Calculating:

\[
A = 90000 \left( 1 + 0.005 \right)^{240} \approx 90000 \times (3.207135) \approx 288641.99
\]

### Option B: 25 years at 5.5% interest
- \(r = 0.055\)
- \(n = 12\)
- \(t = 25\)

\[
A = 90000 \left( 1 + \frac{0.055}{12} \right)^{12 \cdot 25}
\]

Calculating:

\[
A = 90000 \left( 1 + 0.0045833 \right)^{300} \approx 90000 \times (3.775116) \approx 338763.82
\]

### Option C: 30 years at 4.5% interest
- \(r = 0.045\)
- \(n = 12\)
- \(t = 30\)

\[
A = 90000 \left( 1 + \frac{0.045}{12} \right)^{12 \cdot 30}
\]

Calculating:

\[
A = 90000 \left( 1 + 0.00375 \right)^{360} \approx 90000 \times (4.467744) \approx 402696.96
\]

### Option D: 35 years at 4% interest
- \(r = 0.04\)
- \(n = 12\)
- \(t = 35\)

\[
A = 90000 \left( 1 + \frac{0.04}{12} \right)^{12 \cdot 35}
\]

Calculating:

\[
A = 90000 \left( 1 + 0.0033333 \right)^{420} \approx 90000 \times (5.426471) \approx 488732.81
\]

### Summary of Total Costs:
- A) 20 years at 6% interest: **$288,641.99**
- B) 25 years at 5.5% interest: **$338,763.82**
- C) 30 years at 4.5% interest: **$402,696.96**
- D) 35 years at 4% interest: **$488,732.81**

### Conclusion
The loan option with the lowest total cost is **Option A: 20 years at 6% interest** with a total payment of **$288,641.99**.