At the time of her​ grandson's birth, a grandmother deposits $ 8000$8000 in an account that pays 7.5 %7.5% compounded monthly. What will be the value of the account at the​ child's twenty-first​ birthday, assuming that no other deposits or withdrawals are made during this​ period?

The value of the account will be ​$ enter your response here.
​(Round to the nearest dollar as​ needed.)



To solve this problem, we need to use the formula for compound interest:

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

Where:
- $A$ is the amount of money in the account after interest.
- $P$ is the initial principal (the initial deposit), which is $8000.
- $r$ is the annual interest rate (expressed as a decimal), which is 7.5% or 0.075.
- $n$ is the number of times the interest is compounded per year. Since it’s compounded monthly, $n = 12$.
- $t$ is the time in years. Since the time frame is 21 years, $t = 21$.

Now, plug the given values into the formula:

\[
A = 8000 \left( 1 + \frac{0.075}{12} \right)^{12 \times 21}
\]

Let me calculate that for you.The value of the account at the child's twenty-first birthday will be approximately **$38,457**.

Would you like further details or explanations on any step of this process?

Here are five related questions for deeper exploration:
1. How would the result change if the interest were compounded quarterly instead of monthly?
2. What if the interest rate was 6% instead of 7.5%?
3. How much interest would the account earn over the 21 years?
4. What would happen if an additional $500 were deposited every year for 21 years?
5. How can you calculate the amount of interest earned in just the first year?

**Tip:** When interest is compounded more frequently (monthly vs. annually), the total account balance grows faster due to the effect of compounding.

At the time of her grandson's birth, a grandmother deposits $8000 in an account that pays 7.5% compounded monthly. What will be the value of the account at the child's twenty-first birthday, assuming that no other deposits or withdrawals are made during this period? (Round to the nearest dollar as needed.)

Learn how to calculate the future value of an $8000 deposit with a 7.5% interest rate compounded monthly over 21 years. Using the compound interest formula, we compute the account's value at the time of the child's twenty-first birthday. This financial math problem is suitable for students in Grades 10-12.

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