At the time of her​ grandson's birth, a grandmother deposits $6000 in an account that pays 8% compounded monthly. What will be the value of the account at the​ child's twenty-first​ birthday, assuming that no other deposits or withdrawals are made during this​ period?The value of the account will be ​$

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

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

Where:
- $A$ = the future value of the investment/loan, including interest.
- $P$ = the principal investment amount (the initial deposit or loan amount).
- $r$ = the annual interest rate (decimal).
- $n$ = the number of times that interest is compounded per year.
- $t$ = the time the money is invested or borrowed for, in years.

### Given:
- $P = 6000$
- $r = 8\% = 0.08$
- $n = 12$ (compounded monthly)
- $t = 21$ years

### Plugging in the values:

\[
A = 6000 \left(1 + \frac{0.08}{12}\right)^{12 \times 21}
\]

Simplifying:

\[
A = 6000 \left(1 + 0.00667\right)^{252}
\]

\[
A = 6000 \left(1.00667\right)^{252}
\]

Now, let's calculate $A$:

\[
A \approx 6000 \times 4.9189 = 29,513.40
\]

### The value of the account at the child’s twenty-first birthday will be approximately **$29,513.40**.

Would you like more details on any step of this process or have any questions?

### Related questions:
1. What if the interest rate was compounded quarterly instead of monthly?
2. How would the final value change if the interest rate were 6% instead of 8%?
3. What is the formula for simple interest, and how does it compare to compound interest?
4. How much would the final value be if the deposit were $10,000 instead of $6,000?
5. How much total interest would be earned by the time the child turns 21?

### Tip:
When interest is compounded more frequently, the final amount grows larger because interest is calculated on previously earned interest more often.

At the time of her grandson's birth, a grandmother deposits $6000 in an account that pays 8% 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 $

Learn how to calculate compound interest for a $6000 deposit with an 8% interest rate compounded monthly over 21 years. This detailed solution covers the compound interest formula and shows the future value of the account at the child's 21st birthday, approximately $29,513.40. Ideal for high school students in Grades 10-12.

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