The task in the image asks to express $0. \overline{34}$ in the form $\frac{p}{q}$, where $q \neq 0$.

Here's the process to convert the repeating decimal $0.\overline{34}$ into a fraction:

Step-by-step Solution:

1. Let $x = 0.34343434\ldots$ (i.e., $x = 0.\overline{34}$).

2. To eliminate the repeating part, multiply both sides by 100 (since the repeating part has two digits): $100x = 34.34343434\ldots$

3. Now subtract the original $x = 0.34343434\ldots$ from the equation above: $100x - x = 34.34343434\ldots - 0.34343434\ldots$ This simplifies to: $99x = 34$

4. Solve for $x$ by dividing both sides by 99: $x = \frac{34}{99}$

Thus, $0. \overline{34} = \frac{34}{99}$.

Final Answer:

$0. \overline{34} = \frac{34}{99}$

Would you like further details on any of the steps?

Here are some follow-up questions to explore:

1. Can repeating decimals always be written as fractions?
2. How do you convert a repeating decimal with more digits into a fraction?
3. Why do you multiply by powers of 10 in these conversions?
4. What is the greatest common divisor of 34 and 99, and is the fraction $\frac{34}{99}$ in its simplest form?
5. How do you handle repeating decimals with non-repeating digits at the start?

Tip: Remember, every repeating decimal can be expressed as a fraction of two integers.