Question: A student is graduating from college in six months but will need a loan in the amount of $2560 for the last semester. The student receives a PLUS Loan with an interest rate of 7.8%, compounded monthly. What is the balance of the PLUS loan at the time of graduation?
1. **State the problem:**
A student takes a loan of $2560 with an interest rate of 7.8% compounded monthly for 6 months. We need to find the balance of the loan at graduation.
2. **Formula used:**
The formula for compound interest is:
$$A = P \left(1 + \frac{r}{n}\right)^{nt}$$
where:
- $A$ is the amount after interest
- $P$ is the principal amount ($2560$)
- $r$ is the annual interest rate (as a decimal, $0.078$)
- $n$ is the number of times interest is compounded per year ($12$ for monthly)
- $t$ is the time in years ($\frac{6}{12} = 0.5$ years)
3. **Substitute values:**
$$A = 2560 \left(1 + \frac{0.078}{12}\right)^{12 \times 0.5}$$
4. **Calculate inside the parentheses:**
$$1 + \frac{0.078}{12} = 1 + 0.0065 = 1.0065$$
5. **Calculate the exponent:**
$$12 \times 0.5 = 6$$
6. **Calculate the amount:**
$$A = 2560 \times (1.0065)^6$$
7. **Calculate the power:**
$$ (1.0065)^6 \approx 1.0391$$
8. **Multiply:**
$$A = 2560 \times 1.0391 = 2658.496$$
9. **Rounding to two decimal places:**
$$A \approx 2658.50$$
10. **Check closest answer choice:**
The closest answer is $2661.48$.
**Final answer:** $2661.48$