2. How much was deposited in Emily’s account if it grew to $7,626 with
a 4.2% interest rate, compounded monthly, after 6 years and 9
months? (Hint: For t, don’t just put in 6. Put in a value for t that
represents 6 years, 9 months as a decimal. In other words, what is “9
months” as a decimal?) 

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.

How much was deposited in Emily’s account if it grew to $7,626 with a 4.2% interest rate, compounded monthly, after 6 years and 9 months? (Hint: For t, don’t just put in 6. Put in a value for t that represents 6 years, 9 months as a decimal. In other words, what is “9 months” as a decimal?)

Discover how to calculate the initial deposit in Emily’s account that grew to $7,626 at a 4.2% interest rate compounded monthly over 6 years and 9 months. This example covers essential financial concepts like compound interest, converting months to decimal form, and solving for the principal. Suitable for high school students.

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