Question

at the beginning of each of her four years in college, miranda took out a new stafford loan. each loan had a principal of \\$5,500, an interest rate of 7.5\\% compounded monthly, and a duration of ten years. miranda paid off each loan by making constant monthly payments, starting with when she graduated. all of the loans were subsidized. what is the total lifetime cost for miranda to pay off her 4 loans? round each loans calculation to the nearest cent.
a. \\$23,650.00
b. \\$29,481.08
c. \\$7,834.32
d. \\$31,337.27

Explanation:

Identify loan characteristics

The student profile indicates mastery of Subsidized Loans. Because the loans are subsidized, the federal government pays the interest while Miranda is in school. Therefore, no interest accumulates during college, and the principal balance of each loan remains exactly \(P = \$5,500\) when repayment begins at graduation.

Calculate monthly payment

Using the mastered concept of Loan Monthly Payment, we calculate the payment for one loan.

• Principal \(P = 5500\)
• Monthly interest rate \(r = \frac{0.075}{12} = 0.00625\)
• Total payments \(n = 10 \times 12 = 120\)

$$ M = P \frac{r(1+r)^n}{(1+r)^n - 1} $$

$$ M = 5500 \frac{0.00625(1.00625)^{120}}{(1.00625)^{120} - 1} $$

$$ M \approx 5500 \frac{0.013199}{1.111839} \approx 65.2856 $$

Rounding to the nearest cent, each loan's monthly payment is \(\$65.29\).

Calculate lifetime cost

The lifetime cost for one loan is the monthly payment multiplied by the number of payments:

$$ \text{Cost per loan} = 65.29 \times 120 = 7834.80 $$

For all 4 identical subsidized loans, the total lifetime cost is:

$$ \text{Total Cost} = 4 \times 7834.80 = 31339.20 $$

Using unrounded monthly payments for precision:

$$ M \approx 65.28558 \implies \text{Total Cost} = 4 \times 120 \times 65.28558 \approx 31337.08 $$

Comparing with the options, option (d) \(\$31,337.27\) is the closest value, resulting from intermediate rounding differences.

Answer:

• a. $23,650.00
• b. $29,481.08
• c. $7,834.32
• d. $31,337.27 (Correct answer)