Hal has just graduated from four years of college. For the last two years, he took out a Stafford loan to pay for his tuition. Each loan had a duration of ten years and interest compounded monthly, and Hal will pay each of them back by making monthly payments, starting as he graduates. Hal's loans are detailed in the table below.

| Year | Loan Amount ([tex]$) | Interest Rate (%) | Subsidized? |
|-------|------------------|-------------------|-------------|
| Junior| 4,048 | 5.9 | N |
| Senior| 5,295 | 7.6 | Y |

Once all of his loans are paid off, what will Hal's total lifetime cost be? Round all dollar values to the nearest cent.

A. $[/tex]9,023.28
B. [tex]$8,467.20
C. $[/tex]11,498.40
D. $13,615.20



Answer :

To determine Hal's total lifetime cost of his college loans, we need to follow a structured approach to calculate the monthly payments for each loan and then the total cost over the lifetime of the loan. Here are the steps involved:

### Step-by-Step Solution:

1. Identify the loan details:
- Junior year loan amount: [tex]\(\$4048\)[/tex]
- Senior year loan amount: [tex]\(\$5295\)[/tex]
- Interest rate for the junior year loan: [tex]\(5.9\%\)[/tex]
- Interest rate for the senior year loan: [tex]\(7.6\%\)[/tex]
- Loan duration: 10 years
- Compounds monthly, meaning there are [tex]\(12\)[/tex] payments per year.

2. Convert annual interest rates to monthly interest rates:
- Monthly interest rate for junior year loan: [tex]\(\frac{5.9}{100 \times 12}\)[/tex]
- Monthly interest rate for senior year loan: [tex]\(\frac{7.6}{100 \times 12}\)[/tex]

3. Determine the number of payments:
- Total number of monthly payments over 10 years: [tex]\(10 \times 12 = 120\)[/tex]

4. Calculate the monthly payment for each loan using the installment loan formula:
For a loan with principal [tex]\(P\)[/tex], monthly interest rate [tex]\(r\)[/tex], and total payments [tex]\(n\)[/tex], the monthly payment [tex]\(M\)[/tex] is given by:
[tex]\[ M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \][/tex]

For the Junior year loan (Principal [tex]\(P = \$4048\)[/tex], Monthly rate [tex]\(r\)[/tex], Payments [tex]\(n = 120\)[/tex]):
[tex]\[ M_{\text{junior}} = \frac{4048 \cdot 0.0049167 \cdot (1 + 0.0049167)^{120}}{(1 + 0.0049167)^{120} - 1} \approx \$44.74 \][/tex]
- Monthly payment: [tex]\(\$44.74\)[/tex]

For the Senior year loan (Principal [tex]\(P = \$5295\)[/tex], Monthly rate [tex]\(r\)[/tex], Payments [tex]\(n = 120\)[/tex]):
[tex]\[ M_{\text{senior}} = \frac{5295 \cdot 0.0063333 \cdot (1 + 0.0063333)^{120}}{(1 + 0.0063333)^{120} - 1} \approx \$63.13 \][/tex]
- Monthly payment: [tex]\(\$63.13\)[/tex]

5. Calculate the total cost over the loan duration for each loan:
- Total cost for junior year loan: [tex]\(M_{\text{junior}} \times 120 = 44.74 \times 120 = 5368.57\)[/tex]
- Total cost for senior year loan: [tex]\(M_{\text{senior}} \times 120 = 63.13 \times 120 = 7575.51\)[/tex]

6. Sum the total costs for both loans to get the total lifetime cost:
[tex]\[ \text{Total lifetime cost} = 5368.57 + 7575.51 = 12944.09 \][/tex]

So, Hal's total lifetime cost for his loans is [tex]\(\$12,944.09\)[/tex].

Therefore, the correct answer is:

d. [tex]\(\$13,615.20\)[/tex]

(Note: There seems to be a discrepancy between the expected answer options and the calculated result. The given correct calculated value is \[tex]$12,944.09, yet the closest option seems to be \(\$[/tex]11,498.40\) instead of [tex]\(\$13,615.20\)[/tex]. Please double-check the provided options.)