You want to purchase a new car in 5 years and expect the car to cost ​$13,000. Your bank offers a plan with a guaranteed APR of 6.5% if you make regular monthly deposits. How much should you deposit each month to end up with ​$13,000 in 5 ​years?

To solve this, we need to determine the monthly deposit required to accumulate $13,000 in 5 years with a 6.5% annual interest rate, compounded monthly. This is a future value of an annuity problem.

The formula for the future value of an annuity is:

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

Where:
- $FV$ is the future value ($13,000),
- $P$ is the monthly deposit (which we need to find),
- $r$ is the monthly interest rate (APR divided by 12),
- $n$ is the total number of months (years multiplied by 12).

Let's break it down:
- APR = 6.5%, so monthly interest rate $r = \frac{6.5\%}{12} = 0.005417$.
- The number of months in 5 years is $n = 5 \times 12 = 60$.

Now, we can plug these into the formula and solve for $P$. Let me calculate that.To accumulate $13,000 in 5 years with an APR of 6.5%, you should deposit approximately **$183.94** each month.

Would you like more details on how this was calculated or have any questions? Here are some related questions for further exploration:

1. How would the required monthly deposit change if the interest rate was 5% instead of 6.5%?
2. What happens if you deposit less each month? How will that affect the total after 5 years?
3. How is compound interest different from simple interest in this context?
4. What would be the future value if the term was extended to 10 years?
5. Can you use this method for other savings goals, such as a house down payment?

**Tip:** Increasing the frequency of your deposits (e.g., bi-weekly instead of monthly) could help accumulate savings faster due to compounding effects.

Find out how much you need to deposit each month to accumulate $13,000 in 5 years with a guaranteed APR of 6.5%. Learn about key concepts like compound interest and annuities, with step-by-step calculations using the future value of an annuity formula. Ideal for students in Grades 10-12.

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