To express \( 8.2\overline{2} \) (which means 8.2222... repeating) in the form of \( \frac{p}{q} \), let's follow the steps:

1. Let \( x = 8.2\overline{2} \), so \( x = 8.2222\ldots \).

2. Multiply both sides by 10 to move the decimal point one position to the right:
   \[
   10x = 82.2222\ldots
   \]

3. Subtract the original \( x = 8.2222\ldots \) from the equation \( 10x = 82.2222\ldots \):
   \[
   10x - x = 82.2222\ldots - 8.2222\ldots
   \]
   \[
   9x = 74
   \]

4. Solve for \( x \):
   \[
   x = \frac{74}{9}
   \]

Thus, \( 8.2\overline{2} = \frac{74}{9} \).

Would you like a detailed explanation of any step? 

Here are 5 related questions:
1. How do you convert other repeating decimals to fractions?
2. What is the difference between terminating and repeating decimals?
3. How can you simplify fractions like \( \frac{74}{9} \)?
4. How can repeating decimals be represented as geometric series?
5. What are continued fractions and how do they relate to decimals?

**Tip:** When converting repeating decimals to fractions, use algebraic manipulation by multiplying and subtracting to eliminate the repeating part.

Learn how to convert the repeating decimal 8.2 (8.222...) into a fraction in the form of p/q. This step-by-step guide covers key algebraic concepts, including manipulating repeating decimals using subtraction and algebra to find the exact fraction. Suitable for students in Grades 8-10.

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