find the annual interest rate required to achieve 10,000 in savings if you plan to invest 1500 for 25 years. assume your interest is compounded annually. 

To find the annual interest rate needed to achieve a savings of $10,000 after 25 years with an initial investment of $1,500, we can use the **compound interest formula**:

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

Where:
- $ A $ is the final amount ($10,000 in this case),
- $ P $ is the initial principal ($1,500),
- $ r $ is the annual interest rate (which we need to find),
- $ n $ is the number of times the interest is compounded per year (since it is compounded annually, $ n = 1 $),
- $ t $ is the number of years (25 years).

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

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

Now, plug in the values:

\[
10,000 = 1,500 (1 + r)^{25}
\]

We'll solve for $ r $:

\[
\frac{10,000}{1,500} = (1 + r)^{25}
\]

\[
\frac{10,000}{1,500} = \frac{100}{15} = \frac{20}{3} \approx 6.6667
\]

\[
6.6667 = (1 + r)^{25}
\]

Now, take the 25th root of both sides:

\[
(6.6667)^{\frac{1}{25}} = 1 + r
\]

\[
1 + r \approx 1.0786
\]

Subtract 1 to solve for $ r $:

\[
r \approx 0.0786
\]

Convert this to a percentage:

\[
r \approx 7.86\%
\]

### Final Answer:
The annual interest rate required is approximately **7.86%**.

Would you like a deeper breakdown of the steps or have further questions?

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Here are 5 follow-up questions related to this topic:

1. How does changing the compounding frequency (e.g., monthly instead of annually) affect the required interest rate?
2. What happens if you want to reach $10,000 in less than 25 years with the same investment?
3. How does the initial principal impact the time required to reach a target amount?
4. What would the required interest rate be if you plan to invest for 30 years instead of 25?
5. Can you calculate the future value of the investment with a fixed interest rate?

**Tip:** The compound interest formula can be used for more than just investments—it applies to loans, savings, and other areas where interest is involved.

Find the annual interest rate required to achieve $10,000 in savings if you plan to invest $1,500 for 25 years. Assume your interest is compounded annually.

Discover how to calculate the annual interest rate needed to grow a $1,500 investment to $10,000 over 25 years with interest compounded annually. This detailed solution walks through the compound interest formula step-by-step, suitable for high school students in Grades 10-12.

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