`[(0.bar11)+(0.bar22)]xx3` is equal to

Options

A. 3
B. `1.bar9`
C. 1
D. `0.bar3`

Answer

To solve the expression \([(0.\overline{11}) + (0.\overline{22})] \times 3\), we will follow these steps:

### Step 1: Convert \(0.\overline{11}\) to a Fraction
The repeating decimal \(0.\overline{11}\) can be expressed as a fraction. Since "11" has 2 digits repeating, we can use the formula:
\[ 0.\overline{AB} = \frac{AB}{99} \]
where \(AB\) is the repeating part. Here, \(AB = 11\), so:
\[ 0.\overline{11} = \frac{11}{99} \]

### Step 2: Convert \(0.\overline{22}\) to a Fraction
Similarly, for \(0.\overline{22}\), we apply the same formula:
\[ 0.\overline{22} = \frac{22}{99} \]

### Step 3: Add the Two Fractions
Now we will add the two fractions obtained from the previous steps:
\[ 0.\overline{11} + 0.\overline{22} = \frac{11}{99} + \frac{22}{99} = \frac{11 + 22}{99} = \frac{33}{99} \]

### Step 4: Simplify the Sum
The fraction \(\frac{33}{99}\) can be simplified:
\[ \frac{33}{99} = \frac{1}{3} \]

### Step 5: Multiply by 3
Now we multiply the result by 3:
\[ \left(\frac{33}{99}\right) \times 3 = \frac{33 \times 3}{99} = \frac{99}{99} = 1 \]

### Final Answer
Therefore, the value of the expression \([(0.\overline{11}) + (0.\overline{22})] \times 3\) is:
\[ \boxed{1} \]