Mary wants to take out a loan. She can afford to make monthly payments of 200 dollars and wants to pay the loan off after exactly 20 years.
What is the maximum amount that Mary can afford to borrow if the bank charges interest at an annual rate of 10 percent, compounded monthly?

To solve this problem, we need to determine the present value (the amount Mary can borrow) given that she will make monthly payments over 20 years at a 10% annual interest rate, compounded monthly.

### Formula for Present Value of an Annuity:
The formula for the present value $ PV $ of an annuity is:

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

Where:
- $ PV $ is the present value (the loan amount Mary can borrow),
- $ P $ is the monthly payment (200 dollars),
- $ r $ is the monthly interest rate (annual rate divided by 12),
- $ n $ is the total number of payments (years multiplied by 12).

### Given:
- Monthly payment $ P = 200 $,
- Annual interest rate $ = 10\% $, so monthly interest rate $ r = \frac{10\%}{12} = \frac{0.10}{12} = 0.0083333 $,
- Total number of payments $ n = 20 \times 12 = 240 $.

Now we can plug these values into the formula to calculate the maximum loan amount $ PV $.

\[
PV = 200 \times \left( \frac{1 - (1 + 0.0083333)^{-240}}{0.0083333} \right)
\]

Let's calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, you can try calculating the value using the following formula:

\[
PV = 200 \times \left( \frac{1 - (1 + 0.0083333)^{-240}}{0.0083333} \right)
\]

Let me know if you have questions about the process, or if you'd like to try something else!

Would you like details on how to calculate this by hand, or any other questions?

### Here are 5 questions that build on this:
1. How would the maximum loan change if the interest rate increased to 12% annually?
2. What if Mary wanted to pay the loan in 15 years instead of 20?
3. How does compounding monthly affect the total amount paid over the life of the loan compared to yearly compounding?
4. If the interest rate was variable, how could Mary adjust her payments?
5. How much total interest will Mary pay over the life of the loan?

#### Tip:
Always check whether the interest rate is compounded monthly, yearly, or quarterly—this affects the overall cost of the loan significantly!

Mary wants to take out a loan. She can afford to make monthly payments of 200 dollars and wants to pay the loan off after exactly 20 years. What is the maximum amount that Mary can afford to borrow if the bank charges interest at an annual rate of 10 percent, compounded monthly?

Learn how to calculate the maximum loan amount Mary can borrow with monthly payments of $200 over 20 years at a 10% annual interest rate, compounded monthly. This problem involves the present value of an annuity and compound interest, suitable for Grades 11-12.

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