To express the repeating decimal \( 1.\overline{2} \) as a fraction, we can follow these steps:
1. Let \( x \) represent the repeating decimal:
[tex]\[
x = 1.2222\ldots
\][/tex]
2. Multiply both sides of the equation by 10 to move the decimal point one place to the right:
[tex]\[
10x = 12.2222\ldots
\][/tex]
3. Subtract the original equation \( x = 1.2222\ldots \) from this new equation to eliminate the repeating part:
[tex]\[
10x - x = 12.2222\ldots - 1.2222\ldots
\][/tex]
4. Perform the subtraction:
[tex]\[
9x = 11
\][/tex]
5. Solve for \( x \) by dividing both sides by 9:
[tex]\[
x = \frac{11}{9}
\][/tex]
Therefore, \( 1.\overline{2} \) as a fraction is \(\frac{11}{9}\).
To verify, let's convert \(\frac{11}{9}\) back to a decimal:
[tex]\[
\frac{11}{9} = 1.2222\ldots
\][/tex]
This confirms that our fraction [tex]\(\frac{11}{9}\)[/tex] is correct.
