https://file.aiquickdraw.com/mathbot/file/1730046830950qkiu8edj.jpg

To solve this problem, we need to determine the maximum loan amount you can afford with a monthly payment of $350 over a 4-year period at an interest rate of 6%.

We can use the formula for the present value of an annuity, which is appropriate for fixed monthly payments on a loan:
\[
PV = \frac{P \cdot \left(1 - (1 + r)^{-n}\right)}{r}
\]
where:
- $ PV $ is the loan amount (present value),
- $ P $ is the monthly payment,
- $ r $ is the monthly interest rate,
- $ n $ is the total number of payments.

### Step-by-Step Solution:

1. **Identify the known values:**
   - Monthly payment, $ P = 350 $.
   - Annual interest rate = 6%, so the monthly interest rate $ r = \frac{6\%}{12} = 0.5\% = 0.005 $.
   - Loan term = 4 years, so $ n = 4 \times 12 = 48 $ months.

2. **Plug the values into the formula:**
   \[
   PV = \frac{350 \cdot \left(1 - (1 + 0.005)^{-48}\right)}{0.005}
   \]

3. **Calculate the loan amount:**
   Let's break this calculation down:
   - $ (1 + 0.005)^{-48} \approx 0.778394 $.
   - $ 1 - 0.778394 = 0.221606 $.
   - $ 350 \times 0.221606 = 77.5621 $.
   - Finally, $ \frac{77.5621}{0.005} \approx 15,512.42 $.

So, the largest loan amount you can afford is approximately **$15,512.42**.

Would you like a detailed breakdown of each calculation step, or do you have any further questions?

---

### Related Questions:
1. How would the loan amount change if the interest rate was higher, say 8%?
2. What would the monthly payment be for a $20,000 loan at 6% interest over 4 years?
3. How does extending the loan term affect the amount you can borrow?
4. What is the effect of making a higher monthly payment on the loan amount you can afford?
5. How does an interest rate compounded quarterly affect the loan amount compared to monthly compounding?

### Tip:
When calculating loans, always convert annual interest rates to monthly rates if payments are made monthly. This ensures accuracy in your results.

You can afford a $350 per month car payment. You've found a 4-year loan at 6% interest. How big of a loan can you afford?

Determine the maximum loan amount you can afford with a $350 monthly payment over 4 years at 6% interest. This problem involves calculating the present value of an annuity and understanding the impact of monthly payments and interest rates on loan affordability. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation