7700 dollars is placed in an account with an annual interest rate of 5.5%. To the nearest year, how long will it take for the account value to reach 26500 dollars?

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

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

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

However, in this case, we assume the interest is compounded annually ($n = 1$), so the formula simplifies to:

\[
A = P \left(1 + r\right)^t
\]

We need to find $t$ (time in years) when the account grows from $7,700 to $26,500 with an annual interest rate of 5.5% ($r = 0.055$).

\[
26500 = 7700 \left(1 + 0.055\right)^t
\]

Dividing both sides by 7700:

\[
\frac{26500}{7700} = \left(1.055\right)^t
\]

\[
3.44156 = \left(1.055\right)^t
\]

To solve for $t$, take the natural logarithm (ln) of both sides:

\[
\ln(3.44156) = t \cdot \ln(1.055)
\]

\[
t = \frac{\ln(3.44156)}{\ln(1.055)}
\]

Now, calculate $t$:

\[
t \approx \frac{1.236}{0.05357} \approx 23.08
\]

So, it will take approximately **23 years** for the account value to reach $26,500.

Do you want more details or have any questions?

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Here are 8 related questions:

1. How would the time change if the interest rate increased to 6%?
2. What would be the final amount after 23 years if the principal was $10,000 instead?
3. How long would it take to double the initial investment at 5.5% interest?
4. What is the formula for continuously compounded interest, and how does it differ from regular compounding?
5. If the interest were compounded quarterly instead of annually, how would this affect the time to reach $26,500?
6. What would the time be if the target amount were $50,000 instead of $26,500?
7. How would the time change if the initial investment were increased to $10,000?
8. How does inflation impact the real value of money in long-term investments?

**Tip:** Remember that logarithms are handy tools for solving equations where the variable is an exponent, like in compound interest problems.

Learn how to calculate the time required for $7,700 to grow to $26,500 at an annual interest rate of 5.5%. Detailed step-by-step solution using compound interest formula and logarithms.

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