Pat receives a series of four annual federally subsidized student loans, each for $5500 at 6.9%. To defray rising costs for her senior year, 3 years after acquiring the first loan she takes out a private student loan for $4200 at 7.4% interest with a term of 10 years and capitalizes the interest for her last year of college. She graduates 9 months after getting the private loan. Payments on all loans are deferred until 6 months after graduation. Find her monthly payment.
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Part 1 of 3
(a) Find the monthly payment on the federally subsidized loans. Round your answer to two decimal places, if necessary.
The monthly payment on the federally subsidized loans is

Let's calculate the monthly payment on the federally subsidized loans. 

We are given:
- Loan amount per year = $5500
- Number of loans = 4
- Interest rate = 6.9% (annual rate)
- Payment is deferred until 6 months after graduation.

Since all loans have a deferral period and we're calculating after graduation, we treat the loans as starting repayment at the same time, so they can be summed together. To calculate the monthly payment, we'll use the amortization formula for each loan.

The amortization formula for calculating the monthly payment $ M $ is:

\[
M = \frac{P \cdot r(1 + r)^n}{(1 + r)^n - 1}
\]

Where:
- $ P $ is the total loan amount (the sum of all loans),
- $ r $ is the monthly interest rate (annual interest rate divided by 12),
- $ n $ is the total number of monthly payments (loan term in years multiplied by 12).

### Step 1: Calculate the total loan amount
The total amount Pat borrowed through federally subsidized loans is:

\[
P = 5500 \times 4 = 22000 \, \text{USD}
\]

### Step 2: Calculate the monthly interest rate
The annual interest rate is 6.9%, so the monthly interest rate $ r $ is:

\[
r = \frac{6.9\%}{12} = \frac{0.069}{12} \approx 0.00575
\]

### Step 3: Define the loan term in months
The problem doesn't specify the loan term, so we'll assume a standard repayment term for student loans, which is 10 years (120 months).

\[
n = 10 \times 12 = 120 \, \text{months}
\]

### Step 4: Calculate the monthly payment

Now, let's plug these values into the amortization formula to find the monthly payment $ M $.

\[
M = \frac{22000 \cdot 0.00575(1 + 0.00575)^{120}}{(1 + 0.00575)^{120} - 1}
\]

Let's compute this.The monthly payment on the federally subsidized loans is approximately **$254.31**.

Would you like any additional details or clarification?

Here are some related questions to further your understanding:
1. How does the interest rate affect monthly payments over the loan term?
2. What happens if Pat wants to pay off her loans faster by increasing monthly payments?
3. How would changes in the loan term (e.g., from 10 years to 15 years) impact her monthly payment?
4. If interest rates were lower, how much would the monthly payment decrease?
5. How does loan deferment affect the total amount repaid over time?

**Tip**: Making extra payments, even small ones, can significantly reduce the total interest paid over the life of a loan.

Pat receives a series of four annual federally subsidized student loans, each for $5500 at 6.9%. To defray rising costs for her senior year, 3 years after acquiring the first loan she takes out a private student loan for $4200 at 7.4% interest with a term of 10 years and capitalizes the interest for her last year of college. She graduates 9 months after getting the private loan. Payments on all loans are deferred until 6 months after graduation. Find her monthly payment.

Learn how to calculate the monthly payment for Pat's federally subsidized student loans at 6.9% interest using amortization and compound interest formulas. This detailed solution includes loan terms, interest rate conversion, and the amortization formula. Ideal for students studying finance or economics.

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