How much must be deposited today into the following account in order to have $75,000 in 8 years for a down payment on a​ house? Assume no additional deposits are made. An account with annual compounding and an APR of 44​% - ​$________________ should be deposited today.

​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

To calculate the amount that must be deposited today in order to have $75,000 in 8 years with an APR of 44% compounded annually, we will use the formula for the future value of a lump sum compounded periodically:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Since the interest is compounded annually, $ n = 1 $. The formula simplifies to:

\[
A = P \left(1 + r\right)^{t}
\]

To solve for $ P $:

\[
P = \frac{A}{(1 + r)^t}
\]

Substitute the given values:
- $ A = 75,000 $
- $ r = 0.44 $ (44%)
- $ t = 8 $ years

Let's calculate the present value $ P $.You would need to deposit approximately **$4,056.59** today to have $75,000 in 8 years with an APR of 44% compounded annually.

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. How would the deposit amount change if the APR were lower, such as 20%?
2. What would happen to the required deposit if the compounding frequency were semi-annual instead of annual?
3. How much interest will accumulate over the 8 years?
4. How much should be deposited if the goal is to reach $100,000 instead of $75,000?
5. What if you made additional yearly contributions to the account?

**Tip:** High interest rates can significantly decrease the initial deposit required, but also remember that they come with higher risk.

Learn how to calculate the initial deposit needed today to achieve $75,000 in 8 years with an annual APR of 44% compounded annually. Get a step-by-step solution using compound interest formulas.

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