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?

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.