4 years ago, you invested $15,500. In 5 years, you expect to have $38,916. If you expect to earn the same annual return after 5 years from today as the annual return implied from the past and expected values given in the problem, then in how many years from today do you expect to have $49,795? Round final answer to 2 decimal places.

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

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

Where:
- $ A $ is the amount of money after time $ t $
- $ P $ is the initial investment (principal)
- $ r $ is the annual rate of return (as a decimal)
- $ t $ is the number of years

### Step 1: Find the rate of return
First, let's use the information from the past values to find the annual return rate. The initial investment was $15,500, and it grew to $38,916 in 9 years (4 years ago + 5 future years).

We know the formula:

\[
38,916 = 15,500 \left(1 + r\right)^9
\]

Now, solve for $ r $.

\[
\frac{38,916}{15,500} = \left(1 + r\right)^9
\]

\[
2.511 = \left(1 + r\right)^9
\]

Take the 9th root of both sides:

\[
1 + r = \left(2.511\right)^{\frac{1}{9}}
\]

Now, subtract 1 to find $ r $:

\[
r = \left(2.511\right)^{\frac{1}{9}} - 1
\]

### Step 2: Use the rate of return to find the time
Now, we want to know how long it will take for the $38,916 to grow to $49,795 with the same annual return $ r $. We'll use the formula again but with new amounts:

\[
49,795 = 38,916 \left(1 + r\right)^t
\]

Now solve for $ t $:

\[
\frac{49,795}{38,916} = \left(1 + r\right)^t
\]

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

Take the natural logarithm (ln) of both sides to solve for $ t $:

\[
\ln(1.280) = t \ln(1 + r)
\]

Now, solve for $ t $:

\[
t = \frac{\ln(1.280)}{\ln(1 + r)}
\]

Let me calculate both $ r $ and $ t $.The implied annual rate of return is approximately 10.77%. 

To grow from $38,916 to $49,795 with the same return rate, it will take approximately **2.41 years** from today.

Do you want any further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How would the time change if the initial investment were higher?
2. What happens if the annual rate of return is compounded monthly instead of annually?
3. How would a different future target amount affect the time calculation?
4. Can the compound interest formula be used for non-constant rates of return?
5. What impact would fees or taxes have on the expected time to reach $49,795?

**Tip:** Understanding how compound interest works helps you forecast the future value of investments over various time periods.

Find out how long it will take to grow your investment of $38,916 to $49,795 with a constant annual return using compound interest formulas. This problem includes step-by-step calculations for solving with logarithms and compound growth principles. Suitable for Grades 10-12.

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