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The problem asks to write the fraction that represents the repeating decimals for:

**a) \( 0.\overline{36} \)**  
**b) \( 0.\overline{142} \)**

### Solution for a) \( 0.\overline{36} \):

Let's set \( x = 0.\overline{36} \).

1. Multiply by 100 (since the repeating part has 2 digits):
   \[
   100x = 36.\overline{36}
   \]
2. Subtract the original equation \( x = 0.\overline{36} \) from this new equation:
   \[
   100x - x = 36.\overline{36} - 0.\overline{36}
   \]
   \[
   99x = 36
   \]
3. Solve for \( x \):
   \[
   x = \frac{36}{99}
   \]
4. Simplify the fraction:
   \[
   x = \frac{4}{11}
   \]

So, \( 0.\overline{36} = \frac{4}{11} \).

### Solution for b) \( 0.\overline{142} \):

Let \( y = 0.\overline{142} \).

1. Multiply by 1000 (since the repeating part has 3 digits):
   \[
   1000y = 142.\overline{142}
   \]
2. Subtract the original equation \( y = 0.\overline{142} \) from this new equation:
   \[
   1000y - y = 142.\overline{142} - 0.\overline{142}
   \]
   \[
   999y = 142
   \]
3. Solve for \( y \):
   \[
   y = \frac{142}{999}
   \]

So, \( 0.\overline{142} = \frac{142}{999} \).

### Final Answers:
- \( 0.\overline{36} = \frac{4}{11} \)
- \( 0.\overline{142} = \frac{142}{999} \)

Would you like more details or have any other questions?

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Here are 5 related questions to consider:
1. How do you convert a non-repeating decimal into a fraction?
2. What is the general method to convert any repeating decimal into a fraction?
3. Can repeating decimals always be expressed as rational numbers?
4. How would the method change for a repeating decimal like \( 0.0\overline{5} \)?
5. What is the difference between terminating and repeating decimals in terms of fraction representation?

**Tip:** Always check if the resulting fraction can be simplified after converting from a repeating decimal.

Write the fraction that represents the repeating decimal for: a) 0.3636... b) 0.142142...

Learn how to convert the repeating decimals 0.3636... and 0.142142... into fractions. This problem explains key concepts such as converting repeating decimals to fractions and simplifying them for Grades 7-10.

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