Express `1.bar(2)` in the simplest form.

Answer

To express \( 1.\overline{2} \) in the simplest form, we can follow these steps:

### Step 1: Let \( y = 1.\overline{2} \)
We start by defining the repeating decimal as a variable:
\[ y = 1.22222\ldots \]

### Step 2: Multiply by 10
Since only one digit (2) is repeating, we multiply both sides of the equation by 10 to shift the decimal point:
\[ 10y = 12.22222\ldots \]

### Step 3: Set up the equations
Now we have two equations:
1. \( y = 1.22222\ldots \) (Equation 1)
2. \( 10y = 12.22222\ldots \) (Equation 2)

### Step 4: Subtract Equation 1 from Equation 2
Now, we subtract Equation 1 from Equation 2:
\[ 10y - y = 12.22222\ldots - 1.22222\ldots \]
This simplifies to:
\[ 9y = 11 \]

### Step 5: Solve for \( y \)
Now we can solve for \( y \):
\[ y = \frac{11}{9} \]

### Conclusion
Thus, the simplest form of \( 1.\overline{2} \) is:
\[ \frac{11}{9} \]