We can solve this problem using the formula for compound interest:

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

Where:

• $A$ is the final amount in the account, which is $7,626.
• $P$ is the principal (initial deposit) that we need to find.
• $r$ is the annual interest rate (in decimal form), which is 4.2% or 0.042.
• $n$ is the number of times the interest is compounded per year, which is 12 (since it's compounded monthly).
• $t$ is the time in years.

First, we need to convert 6 years and 9 months into a decimal form for $t$.

$t = 6 + \frac{9}{12} = 6.75 \text{ years}$

Now, substitute the values into the compound interest formula:

$7,626 = P \left(1 + \frac{0.042}{12}\right)^{12 \times 6.75}$

Simplify the expression and solve for $P$:

$7,626 = P \left(1 + \frac{0.042}{12}\right)^{81}$

Let's compute this step by step.The initial amount deposited in Emily's account was approximately $5,746.31.

Would you like a breakdown of how this was calculated or further explanation on any part of the process?

Here are 5 related questions to deepen your understanding:

1. What would happen if the interest rate was compounded quarterly instead of monthly?
2. How would the final amount change if the interest rate increased to 5%?
3. What would the initial deposit be if the account grew to $8,000 under the same conditions?
4. How much interest was earned on the deposit over the 6 years and 9 months?
5. How does the time period affect the final amount when compounding interest?

Tip: Remember that the more frequently interest is compounded, the faster the account balance grows, even if the interest rate remains the same.