Question

Abigail wants to take out a loan of $18,000 with interest that compounds monthly. Use the formula A = P(1 + r/n) n*t to find which of these loan terms will have the lowest total cost.
A. 2 years at 7% interest
B. 3 years at 4% interest
C. 4 years at 7% interest
D. 5 years at 5% interest

Answer 1

To determine which loan term has the lowest total cost, we can use the formula for compound interest:

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

Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times interest is compounded per year.
- \( t \) is the time the money is invested for in years.

Given:
- \( P = 18,000 \)

Now let's compute the total amount \( A \) for each option.

### A. 2 years at 7% interest
- \( r = 0.07 \)
- \( n = 12 \) (monthly)
- \( t = 2 \)

\[
A = 18000\left(1 + \frac{0.07}{12}\right)^{12 \times 2}
\]
\[
A = 18000\left(1 + 0.0058333\right)^{24}
\]
\[
A = 18000\left(1.0058333\right)^{24}
\]
\[
A \approx 18000 \times 1.150279
\]
\[
A \approx 20,706.06
\]

### B. 3 years at 4% interest
- \( r = 0.04 \)
- \( n = 12 \) (monthly)
- \( t = 3 \)

\[
A = 18000\left(1 + \frac{0.04}{12}\right)^{12 \times 3}
\]
\[
A = 18000\left(1 + 0.0033333\right)^{36}
\]
\[
A = 18000\left(1.0033333\right)^{36}
\]
\[
A \approx 18000 \times 1.127799
\]
\[
A \approx 20,302.38
\]

### C. 4 years at 7% interest
- \( r = 0.07 \)
- \( n = 12 \) (monthly)
- \( t = 4 \)

\[
A = 18000\left(1 + \frac{0.07}{12}\right)^{12 \times 4}
\]
\[
A = 18000\left(1 + 0.0058333\right)^{48}
\]
\[
A = 18000\left(1.0058333\right)^{48}
\]
\[
A \approx 18000 \times 1.348850
\]
\[
A \approx 24,266.30
\]

### D. 5 years at 5% interest
- \( r = 0.05 \)
- \( n = 12 \) (monthly)
- \( t = 5 \)

\[
A = 18000\left(1 + \frac{0.05}{12}\right)^{12 \times 5}
\]
\[
A = 18000\left(1 + 0.0041667\right)^{60}
\]
\[
A = 18000\left(1.0041667\right)^{60}
\]
\[
A \approx 18000 \times 1.283685
\]
\[
A \approx 23,105.13
\]

### Summary of amounts:
- A. $20,706.06
- B. $20,302.38
- C. $24,266.30
- D. $23,105.13

The loan term with the lowest total cost is **B. 3 years at 4% interest** with a total cost of approximately **$20,302.38**.