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To calculate the annual interest rate that turns €1221 into €1701 over 5 years, we can use the formula for compound interest:

\[
A = P \left(1 + \frac{r}{100}\right)^t
\]

Where:
- \(A\) is the final amount (€1701),
- \(P\) is the initial principal (€1221),
- \(r\) is the annual interest rate (the unknown),
- \(t\) is the time in years (5 years).

Now, we plug the known values into the formula:

\[
1701 = 1221 \left(1 + \frac{r}{100}\right)^5
\]

To solve for \(r\), we first divide both sides by 1221:

\[
\left(1 + \frac{r}{100}\right)^5 = \frac{1701}{1221}
\]

Simplifying the right-hand side:

\[
\left(1 + \frac{r}{100}\right)^5 = 1.393
\]

Now, take the 5th root of both sides to isolate \(1 + \frac{r}{100}\):

\[
1 + \frac{r}{100} = \left(1.393\right)^{\frac{1}{5}} = 1.068
\]

Finally, subtract 1 and multiply by 100 to find \(r\):

\[
\frac{r}{100} = 0.068 \quad \Rightarrow \quad r = 6.8\%
\]

So, the annual interest rate is **6.80%**.

Do you want more details or have any questions?

Here are some related questions for further exploration:
1. What happens to the final amount if the interest rate is compounded monthly instead of annually?
2. How much would you need to invest to reach €1701 in 5 years with a 5% annual interest rate?
3. How long would it take to double the initial investment at a 6.8% interest rate?
4. What would be the interest rate if the investment period were 10 years instead of 5?
5. How does simple interest compare with compound interest in this scenario?

**Tip:** When dealing with compound interest, the frequency of compounding (monthly, annually, etc.) can significantly affect the final amount.

What annual interest rate, (to two places of decimal), will turn €1221 into €1701 after 5 years?

Learn how to calculate the annual interest rate required to turn an initial investment of €1221 into €1701 over 5 years using the compound interest formula. Step-by-step solution provided with clear explanations. Suitable for high school students.

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