If John had inherited $100000 and wanted it to grow into $500000 in 25 years, what rate of annual interest compounded monthly would he need to earn to reach his goal? 

To find the rate of interest John would need for his $100,000 to grow to $500,000 in 25 years with monthly compounding, we can use the compound interest formula:

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

Where:
- $ A $ is the future value ($500,000),
- $ P $ is the initial principal ($100,000),
- $ r $ is the annual interest rate (which we are solving for),
- $ n $ is the number of times interest is compounded per year (12 for monthly compounding),
- $ t $ is the time in years (25 years).

We need to solve for $ r $. Plugging in the known values:

\[
500,000 = 100,000 \left(1 + \frac{r}{12}\right)^{12 \times 25}
\]

This simplifies to:

\[
5 = \left(1 + \frac{r}{12}\right)^{300}
\]

Now, we'll solve for $ r $ step by step.

### Step 1: Take the 300th root of both sides to isolate $ \frac{r}{12} $:

\[
5^{\frac{1}{300}} = 1 + \frac{r}{12}
\]

### Step 2: Subtract 1 from both sides:

\[
5^{\frac{1}{300}} - 1 = \frac{r}{12}
\]

### Step 3: Multiply by 12 to isolate $ r $:

\[
r = 12 \left(5^{\frac{1}{300}} - 1\right)
\]

Now, I'll calculate the value of $ r $.The annual interest rate that John would need, compounded monthly, is approximately **6.46%** to grow his $100,000 to $500,000 in 25 years.

Do you want further details or have any questions?

Here are 5 related questions to explore:
1. How would the rate change if the interest was compounded annually instead of monthly?
2. What if John wanted to reach $1,000,000 in 25 years—what would the required interest rate be?
3. How does the compounding frequency affect the total amount earned over time?
4. How long would it take for John's investment to grow to $500,000 at 5% annual interest, compounded monthly?
5. What is the difference between simple interest and compound interest growth for long-term investments?

**Tip:** The more frequent the compounding, the higher the effective interest rate, leading to greater returns over time.

If John had inherited $100000 and wanted it to grow into $500000 in 25 years, what rate of annual interest compounded monthly would he need to earn to reach his goal?

Discover how to calculate the annual interest rate required for $100,000 to grow into $500,000 over 25 years, compounded monthly. This solution uses the compound interest formula and includes step-by-step calculations for an interest rate of approximately 6.46%. Suitable for high school students in Grades 10-12.

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