The given repeating decimal is $0.\overline{36}$. To convert this into a rational number, follow these steps:

Step 1: Assign the repeating decimal to a variable

Let: $x = 0.\overline{36}$

Step 2: Multiply by a power of 10

Since the repeating part has two digits (36), multiply by $100$ to shift the decimal point: $100x = 36.\overline{36}$

Step 3: Subtract the original equation

$100x - x = (36.\overline{36}) - (0.\overline{36})$

$99x = 36$

Step 4: Solve for $x$

$x = \frac{36}{99}$

Now, simplify the fraction by dividing both numerator and denominator by their greatest common divisor (GCD), which is 9:

$x = \frac{36 \div 9}{99 \div 9} = \frac{4}{11}$

Final Answer:

$0.\overline{36} = \frac{4}{11}$

Would you like a further explanation on any step?

Here are five related questions to deepen understanding:

1. How do you convert $0.\overline{7}$ into a fraction?
2. What is the difference between terminating and repeating decimals?
3. How can you convert a repeating decimal with three repeating digits into a fraction?
4. Why do all repeating decimals represent rational numbers?
5. What is the decimal representation of $\frac{5}{9}$?

Tip: The method of multiplying by a power of 10 is key to converting repeating decimals to fractions.