6500 dollars is placed in an account with an annual interest rate of 8%. To the nearest tenth of a year, how long will it take for the account value to reach 20900 dollars?

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

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

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

In this case, it is not explicitly mentioned whether the interest is compounded or not, so we'll assume the interest is compounded annually (which means $n = 1$).

Thus, the formula simplifies to:

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

Substituting the given values:
- $A = 20900$ (the future value).
- $P = 6500$ (the initial deposit).
- $r = 0.08$ (8% interest rate).
- $t$ is what we need to find.

Now, plug these into the formula:

\[
20900 = 6500 \left( 1 + 0.08 \right)^t
\]

This simplifies to:

\[
20900 = 6500 \times (1.08)^t
\]

Now divide both sides by 6500:

\[
\frac{20900}{6500} = (1.08)^t
\]

\[
3.2154 \approx (1.08)^t
\]

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

\[
\ln(3.2154) = \ln((1.08)^t)
\]

\[
\ln(3.2154) = t \cdot \ln(1.08)
\]

Now solve for $t$:

\[
t = \frac{\ln(3.2154)}{\ln(1.08)}
\]

Let me calculate that.The time it will take for the account value to reach $20,900 is approximately 15.2 years.

Would you like further details or have any questions about the process?

Here are 5 related questions to explore:
1. How would the result change if the interest was compounded monthly?
2. What would the final amount be if the interest rate was reduced to 6%?
3. How long would it take if the initial deposit was $10,000 instead of $6,500?
4. What if the goal amount was $50,000 with the same 8% interest rate?
5. How can inflation affect the real value of your investment over time?

**Tip:** Remember, the more frequently interest is compounded, the faster the money grows!

Learn how to calculate the time it takes for $6,500 to grow to $20,900 at an annual interest rate of 8%. This compound interest problem uses exponential growth and logarithms. Step-by-step explanation provided, suitable for Grades 10-12.

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