Suppose you have a student loan of ​$45,000 with an APR of 12​% for 30 years. Complete parts​ (a) through​ (c) below.
a. What are your required monthly​ payments?
The required monthly payment is ​$___. 
​(Do not round until the final answer. Then round to the nearest cent as​ needed.)
b. Suppose you would like to pay the loan off in 15 years instead of 30. What monthly payments will you need to​ make?
The monthly payment required to pay off the loan in 15 years instead of 30 is ​$___.
​(Do not round until the final answer. Then round to the nearest cent as​ needed.)
c. Compare the total amount​ you'll pay over the loan term if you pay the loan off in 15 years  versus 30 years.
Total payments for the 30​-year loanequals​ $___. 
Total payments for the 15​-year loanequals​ $___. 

We can solve this problem using the formula for the monthly payment on a loan, which is derived from the amortization formula:

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

Where:
- $ M $ = monthly payment
- $ P $ = principal loan amount (in this case $45,000)
- $ r $ = monthly interest rate (annual interest rate divided by 12)
- $ n $ = total number of payments (loan term in years multiplied by 12)

Let's calculate the required monthly payments for both 30 years and 15 years.

### Part (a): Required Monthly Payments for a 30-year Loan

For the 30-year loan:
- $ P = 45,000 $
- Annual percentage rate (APR) = 12%, so the monthly interest rate $ r = \frac{12\%}{12} = 0.01 $ (or 1% per month)
- Number of payments $ n = 30 \times 12 = 360 $

Now, let's compute the monthly payment.

### Part (b): Required Monthly Payments for a 15-year Loan

For the 15-year loan:
- $ P = 45,000 $
- Monthly interest rate $ r = 0.01 $
- Number of payments $ n = 15 \times 12 = 180 $

We can now calculate the monthly payment for this loan as well.

### Part (c): Comparison of Total Payments

For this, we'll simply multiply the monthly payment by the total number of payments for both the 30-year and 15-year loans:

- Total payments for the 30-year loan = $ M \times 360 $
- Total payments for the 15-year loan = $ M \times 180 $

Let's now perform the calculations.### Part (a): Required Monthly Payments for a 30-year Loan
The required monthly payment is approximately **$462.88**.

### Part (b): Required Monthly Payments for a 15-year Loan
The monthly payment required to pay off the loan in 15 years is approximately **$540.08**.

### Part (c): Comparison of Total Payments
- Total payments for the 30-year loan: **$166,635.24**
- Total payments for the 15-year loan: **$97,213.61**

### Observations:
By paying off the loan in 15 years, you would save **$69,421.63** in total payments compared to the 30-year option.

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Would you like further details or have any questions?

Here are some related questions:
1. How does the interest rate affect the monthly payment?
2. What happens if you make additional principal payments each month?
3. How would changes in the loan amount impact the monthly payment?
4. What is the difference between a fixed-rate loan and a variable-rate loan?
5. How does the length of the loan term affect the total interest paid?

**Tip:** Making extra payments towards the principal can reduce the overall interest paid and shorten the loan term.

Suppose you have a student loan of ​$45,000 with an APR of 12​% for 30 years. Complete parts​ (a) through​ (c) below.
a. What are your required monthly​ payments?
The required monthly payment is ​$___. 
​(Do not round until the final answer. Then round to the nearest cent as​ needed.)
b. Suppose you would like to pay the loan off in 15 years instead of 30. What monthly payments will you need to​ make?
The monthly payment required to pay off the loan in 15 years instead of 30 is ​$___.
​(Do not round until the final answer. Then round to the nearest cent as​ needed.)
c. Compare the total amount​ you'll pay over the loan term if you pay the loan off in 15 years  versus 30 years.
Total payments for the 30​-year loanequals​ $___.
Total payments for the 15​-year loanequals​ $___.

Calculate the monthly payments and compare the total costs for a $45,000 student loan with a 12% APR over 30 years versus 15 years. Learn how to use the loan amortization formula to determine payments and understand the impact of interest rates and loan terms.

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