​ (Solving for n​)  How many years will it take for ​$500 to grow to ​$1 comma 039.50 if​ it's invested at 5 percent compounded​ annually?
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Part 1
The number of years it will take for ​$500 to grow to ​$1 comma 039.50 at 5 percent compounded annually is 
  
enter your response here years. ​ (Round to one decimal​ place.)

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

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

Where:
- $A$ is the future value (the amount after compounding, in this case $1,039.50),
- $P$ is the initial principal (in this case $500),
- $r$ is the annual interest rate (in decimal form, so 5% = 0.05),
- $n$ is the number of times the interest is compounded per year (for annual compounding, $n = 1$),
- $t$ is the time in years (what we are solving for).

Since $n = 1$, the formula simplifies to:

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

We are given:
- $A = 1039.50$,
- $P = 500$,
- $r = 0.05$.

Now we substitute the known values into the equation:

\[
1039.50 = 500(1 + 0.05)^t
\]

Let's solve for $t$. 

\[
1039.50 = 500(1.05)^t
\]

Dividing both sides by 500:

\[
\frac{1039.50}{500} = (1.05)^t
\]

\[
2.079 = (1.05)^t
\]

Now we can take the natural logarithm (ln) of both sides to solve for $t$:

\[
\ln(2.079) = t \cdot \ln(1.05)
\]

Finally, we solve for $t$:

\[
t = \frac{\ln(2.079)}{\ln(1.05)}
\]

Let me calculate that.It will take approximately **15.0 years** for $500 to grow to $1,039.50 at a 5% interest rate compounded annually.

Would you like more details or have any questions?

Here are some related questions:
1. How does compounding frequency affect the growth of investments?
2. What is the formula for continuous compounding, and how would the result differ?
3. If the interest rate were 6%, how would that change the time required?
4. How can you calculate the future value for monthly compounding?
5. What if the initial amount were $1,000 instead of $500?

**Tip:** Always round intermediate calculations only at the final step to maintain accuracy.

How many years will it take for $500 to grow to $1,039.50 if it's invested at 5 percent compounded annually?

Find out how long it takes for $500 to grow to $1,039.50 at a 5% annual interest rate, compounded annually. Learn how to use the compound interest formula and logarithms to solve this problem. This solution is ideal for high school students in Grades 10-12.

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