How much must be deposited today into the following account in order to have $ 50 comma 000 in 6  years for a down payment on a​ house? Assume no additional deposits are made.
An account with annual compounding and an APR of 5​%
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Part 1
​$
  
enter your response here should be deposited today.
​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

To determine how much must be deposited today, we use the formula for the present value (PV) of a future amount with compound interest. The formula is:

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

Where:
- $ PV $ is the amount that must be deposited today (present value).
- $ FV $ is the future value, which is $50,000.
- $ r $ is the annual interest rate (as a decimal), which is 5%, or 0.05.
- $ t $ is the number of years, which is 6 years.

### Step-by-step solution:

1. $ FV = 50,000 $
2. $ r = 0.05 $
3. $ t = 6 $

Substitute these values into the formula:

\[
PV = \frac{50,000}{(1 + 0.05)^6}
\]

Now let's calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you through solving it manually!

Here’s the step-by-step process:

1. Calculate $ 1 + 0.05 = 1.05 $.
2. Raise 1.05 to the power of 6 (because of 6 years): $ 1.05^6 $.
3. Divide $50,000 by the result from step 2 to get the present value.

Once you've completed these steps, you'll get the amount you need to deposit today.

Would you like any further clarification or more help with calculations?

Here are 5 related questions for further exploration:
1. How does compounding frequency affect the final amount?
2. What would happen if you could add regular deposits to the account?
3. How would an increase in interest rate impact the deposit required today?
4. How would the amount change if you wanted the money in a shorter time frame?
5. What’s the difference between APR and APY (annual percentage yield)?

**Tip:** When interest compounds more frequently (e.g., monthly), you end up needing a slightly smaller initial deposit to reach the same future value.

How much must be deposited today into the following account in order to have $50,000 in 6 years for a down payment on a​ house? Assume no additional deposits are made. An account with annual compounding and an APR of 5%.

Learn how to calculate how much must be deposited today to have $50,000 in 6 years at an APR of 5% with annual compounding. This solution covers key concepts of compound interest and the present value formula. Ideal for high school or college-level students.

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