You want to buy a new car. You can afford payments of $250 per month and can borrow the money at an interest rate of 3.1% compounded monthly for 5 years.

How much are you able to borrow?

To determine how much you can borrow, we can use the formula for the present value of an annuity, which is often used for calculating loans:

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

Where:
- $PV$ = present value (amount you can borrow)
- $P$ = monthly payment (\$250)
- $r$ = monthly interest rate (annual rate divided by 12)
- $n$ = total number of payments (months)

### Step-by-step breakdown:

1. **Monthly interest rate $r$:**
   \[
   r = \frac{3.1\%}{12} = \frac{0.031}{12} = 0.0025833
   \]

2. **Total number of payments $n$:**
   Since the loan is for 5 years, and there are 12 months in a year:
   \[
   n = 5 \times 12 = 60
   \]

3. **Substitute into the formula:**
   \[
   PV = 250 \times \left[ \frac{1 - (1 + 0.0025833)^{-60}}{0.0025833} \right]
   \]

Now I'll calculate the value.You are able to borrow approximately \$13,878.74.

Would you like further details or have any questions about the calculations?

Here are some related questions to explore:
1. How would a higher interest rate affect the loan amount you can borrow?
2. What if you could afford payments of \$300 per month instead?
3. How does increasing the loan term to 7 years change the amount you can borrow?
4. What if the interest was compounded quarterly instead of monthly?
5. How much interest will you pay over the life of the loan?

**Tip:** The longer the loan term, the lower your monthly payments, but the more interest you pay overall.

You want to buy a new car. You can afford payments of $250 per month and can borrow the money at an interest rate of 3.1% compounded monthly for 5 years. How much are you able to borrow?

Calculate how much you can borrow for a car loan with $250 monthly payments, 3.1% interest compounded monthly, and a 5-year term. Learn how to use the present value of annuities formula to solve loan-related problems. Ideal for high school students studying financial mathematics.

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