How much must be deposited today into the following account in order to have $ 40 comma 000 in 7  years for a down payment on a​ house? Assume no additional deposits are made.
An account with monthly compounding and an APR of 4​%
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Part 1
​$
  
enter your response here should be deposited today.

To find out how much must be deposited today, we can use the formula for the present value of a future sum with compound interest. The formula is:

\[
PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}
\]

Where:
- $PV$ is the present value (the amount to deposit today)
- $FV$ is the future value (the amount you want in the future, $40,000)
- $r$ is the annual interest rate (in decimal form, so 4% becomes 0.04)
- $n$ is the number of times interest is compounded per year (monthly compounding means $n = 12$)
- $t$ is the time in years (7 years)

Let's plug in the values and calculate.

\[
PV = \frac{40,000}{(1 + \frac{0.04}{12})^{12 \times 7}}
\]

I'll calculate that now.Approximately **$30,245.43** must be deposited today in order to have $40,000 in 7 years, assuming monthly compounding with an APR of 4%.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How would the required deposit change if the APR increased to 5%?
2. What would the final amount be after 7 years if you deposited $30,245.43 at a 4% APR compounded monthly?
3. How does the frequency of compounding affect the required initial deposit?
4. What if the goal was to reach $50,000 instead of $40,000?
5. How would the required deposit change if the time was reduced to 5 years?

**Tip:** The more frequently interest is compounded, the more interest you'll earn, but you will need a slightly smaller initial deposit to reach your target.

How much must be deposited today into the following account in order to have $40,000 in 7 years for a down payment on a house? Assume no additional deposits are made. An account with monthly compounding and an APR of 4%.

Learn how to calculate how much you need to deposit today to reach $40,000 in 7 years with a 4% APR compounded monthly. This problem covers compound interest concepts and uses the present value formula, ideal for high school students in Grades 10-12.

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