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: Define the repeating decimal
Let \( x = 1.\overline{2} \). This means that \( x = 1.22222...\) where the digit '2' repeats indefinitely.

### Step 2: Multiply by 10
To eliminate the repeating part, multiply both sides of the equation by 10:
\[ 10x = 12.22222... \]
This gives us a new equation where the decimal part is still repeating.

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

### Step 4: Subtract the first equation from the second
Now, subtract Equation 1 from Equation 2:
\[ 10x - x = 12.22222... - 1.22222... \]
This simplifies to:
\[ 9x = 11 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 9 to solve for \( x \):
\[ x = \frac{11}{9} \]

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

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