How much money should be deposited today in an account that earns 4.5 % compounded monthly so that it will accumulate to $ 11,000 in 4 ​years?

To determine how much money should be deposited today to accumulate $11,000 in 4 years with an interest rate of 4.5% compounded monthly, we can use the formula for compound interest:

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

Where:
- $A$ is the future value (what we want to accumulate, $11,000).
- $P$ is the present value (the initial deposit, which we're solving for).
- $r$ is the annual interest rate (4.5%, or 0.045).
- $n$ is the number of times the interest is compounded per year (monthly compounding means $n = 12$).
- $t$ is the time in years (4 years).

We need to solve for $P$. Rearranging the formula to solve for $P$:

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

Now, plugging in the values:

\[
P = \frac{11,000}{\left(1 + \frac{0.045}{12}\right)^{12 \times 4}}
\]

Let's calculate this:

\[
P = \frac{11,000}{\left(1 + 0.00375\right)^{48}}
\]

\[
P = \frac{11,000}{\left(1.00375\right)^{48}}
\]

\[
P = \frac{11,000}{1.197992}
\]

\[
P \approx 9,181.49
\]

So, you would need to deposit approximately **$9,181.49** today to accumulate $11,000 in 4 years at an interest rate of 4.5%, compounded monthly.

Do you want more details or have any questions about this?

Here are 5 related questions:
1. How would the answer change if the interest were compounded annually instead of monthly?
2. What happens to the required initial deposit if the interest rate increases to 5%?
3. How would the formula adjust for a quarterly compounding period?
4. How much interest would be earned after 4 years on the initial deposit?
5. What if the investment goal was $15,000 instead of $11,000?

**Tip:** When dealing with compound interest, the more frequently the interest is compounded, the greater the accumulated amount will be.

How much money should be deposited today in an account that earns 4.5% compounded monthly so that it will accumulate to $11,000 in 4 years?

Learn how to calculate the initial deposit required to accumulate $11,000 in 4 years with a 4.5% interest rate compounded monthly. Using the compound interest formula, this problem walks through the steps to solve for the deposit amount, suitable for high school students studying finance or algebra.

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