https://file.aiquickdraw.com/mathbot/file/1725809503531ygs2fmp0.jpg

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.

Express 0.34 (repeating) in the form p/q (q ≠ 0).

Learn how to convert the repeating decimal 0.34 into a fraction using algebraic techniques. This step-by-step solution explains key concepts like multiplying by powers of 10 and solving for x. Suitable for students in Grades 7-9.

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