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The problem is asking how many years it will take for a capital to more than double, given a certain interest rate, with three choices: 3%, 5%, and 10%.

We can solve this using the formula for compound interest:

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

Where:
- \( A \) is the final amount (in this case, more than double),
- \( P \) is the initial capital,
- \( r \) is the annual interest rate (3%, 5%, or 10%),
- \( t \) is the time in years.

For the capital to more than double, \( A \geq 2P \), so:

\[
2P = P(1 + \frac{r}{100})^t
\]

Simplifying:

\[
2 = (1 + \frac{r}{100})^t
\]

Now, we can solve for \( t \) by taking the natural logarithm (ln) of both sides:

\[
\ln(2) = t \cdot \ln(1 + \frac{r}{100})
\]

Solving for \( t \):

\[
t = \frac{\ln(2)}{\ln(1 + \frac{r}{100})}
\]

Let me calculate the values of \( t \) for each interest rate (3%, 5%, 10%).Here are the approximate times it takes for the capital to more than double for each interest rate:

- At 3% interest: about 23.45 years
- At 5% interest: about 14.21 years
- At 10% interest: about 7.27 years

So, the minimum time for the capital to more than double is approximately 7.27 years, which corresponds to option **c) 10%**.

Would you like further details on the calculations or any other clarification?

Here are some related questions you might consider:
1. What would happen if the interest was compounded more frequently (e.g., monthly)?
2. How does the time change if the interest rate is increased or decreased by 1%?
3. What if we want the capital to triple instead of double? How long would that take?
4. How does the formula change for continuous compounding?
5. What happens if we withdraw part of the capital each year?

**Tip:** The "Rule of 72" is a simple way to estimate how long it takes for an investment to double, by dividing 72 by the interest rate (e.g., for 10%, it gives roughly 7.2 years).

Ein Kapital wird jährlich mit folgendem Zinssatz verzinst. Wie viele Jahre muss man mindestens warten, damit sich das Kapital mehr als verdoppelt hat? a) 3% b) 5% c) 10%

Learn how to calculate the time it takes for a capital to more than double at different annual interest rates: 3%, 5%, and 10%. This involves exponential growth, compound interest, and logarithmic calculations. Suitable for high school students in Grades 10-12.

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