Someone needs to borrow 

​$10 comma 00010,000 

to buy a car and the person has determined that monthly payments of 

​$200200 

are affordable. The bank offers a 

33​-year 

loan at 

66​% 

​APR, a 

44​-year 

loan at 

6.56.5​%, 

or a 

55​-year 

loan at 

77​% 

APR. Which loan best meets the​ person's needs? Explain.

Question content area bottom

Part 1

Which loan best meets the​ person's needs?

​(Round to the nearest cent as​ needed.)

A.

The third loan best meets the​ person's needs because the monthly payment of 

​$enter your response here 

is less than the maximum budgeted amount of 

​$200200 

per month.

B.

The second loan best meets the​ person's needs because the monthly payment of 

​$enter your response here 

is less than the maximum budgeted amount of 

​$200200 

per month.

C.

The first loan best meets the​ person's needs because the monthly payment of 

​$enter your response here 

is less than the maximum budgeted amount of 

​$200 

per month.

D.

None of the loans meet the​ person's needs.

To determine which loan best meets the person's needs, we need to calculate the monthly payments for each of the three loan options using the formula for the monthly payment on an amortizing loan:

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

Where:
- $ M $ is the monthly payment.
- $ P $ is the loan principal (amount borrowed).
- $ r $ is the monthly interest rate (APR divided by 12).
- $ n $ is the total number of payments (loan term in years multiplied by 12).

Given:
- Principal $ P = 10,000 $
- Maximum affordable payment $ M = 200 $
  
### Loan 1: 3-year loan at 6% APR
- Annual interest rate $ r = 0.06 $
- Monthly interest rate $ r = \frac{0.06}{12} = 0.005 $
- Total number of payments $ n = 3 \times 12 = 36 $

### Loan 2: 4-year loan at 6.5% APR
- Annual interest rate $ r = 0.065 $
- Monthly interest rate $ r = \frac{0.065}{12} $
- Total number of payments $ n = 4 \times 12 = 48 $

### Loan 3: 5-year loan at 7% APR
- Annual interest rate $ r = 0.07 $
- Monthly interest rate $ r = \frac{0.07}{12} $
- Total number of payments $ n = 5 \times 12 = 60 $

Let's calculate the monthly payments for each option.The calculated monthly payments for each loan are as follows:

1. **3-year loan at 6% APR:** $304.22
2. **4-year loan at 6.5% APR:** $237.15
3. **5-year loan at 7% APR:** $198.01

The only loan option where the monthly payment is less than the maximum affordable payment of $200 is the **5-year loan at 7% APR** with a payment of $198.01.

### Conclusion:
**Option A** is correct: The third loan best meets the person's needs because the monthly payment of **$198.01** is less than the maximum budgeted amount of $200 per month.

Would you like any further details or have any questions?

Here are 5 follow-up questions to consider:
1. What happens to the total interest paid over the life of each loan?
2. How does extending the loan term affect the monthly payment?
3. What are the pros and cons of choosing a longer loan term?
4. How would a higher principal amount affect the loan options?
5. What is the impact of a different interest rate on the affordability of each loan?

**Tip:** A lower monthly payment often comes with higher total interest paid over the life of the loan, so it's important to consider both factors when choosing a loan.

Discover which loan option is best for purchasing a car based on affordability. Calculate monthly payments for loans with different APRs and terms. Example includes a 3-year loan at 6% APR, 4-year loan at 6.5% APR, and 5-year loan at 7% APR. Find out how to determine the most suitable loan based on monthly budget constraints.

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