To express $0.\overline{45}$ (where 45 repeats indefinitely) as a fraction, let's follow these steps:

Step 1: Let $x = 0.\overline{45}$

This means that: $x = 0.45454545\ldots$

Step 2: Multiply both sides of the equation by 100 to move the decimal point two places to the right

$100x = 45.454545\ldots$

Step 3: Subtract the original equation from this new equation

$100x - x = 45.454545\ldots - 0.454545\ldots$

This simplifies to: $99x = 45$

Step 4: Solve for $x$

$x = \frac{45}{99}$

Step 5: Simplify the fraction

Both 45 and 99 can be divided by their greatest common divisor (GCD), which is 9: $\frac{45}{99} = \frac{45 \div 9}{99 \div 9} = \frac{5}{11}$

Final Answer:

$0.\overline{45} = \frac{5}{11}$

Would you like further clarification or examples?

Follow-up questions:

1. How do you convert other repeating decimals like $0.\overline{3}$ to a fraction?
2. Can any repeating decimal be expressed as a fraction?
3. What is the decimal equivalent of the fraction $\frac{5}{11}$?
4. How do you handle fractions with recurring decimals involving more than one digit?
5. How do you simplify fractions in general?

Tip:

When working with repeating decimals, multiplying by powers of 10 (like 100, 1000) is key to eliminating the repeating part.