Evaluate `1.bar2xx0.bar3`=

Options

A. `0.bar407`
B. `0.0bar36`
C. `1.bar13`
D. `0.bar037`

Answer

To evaluate \( 1.\overline{2} \times 0.\overline{3} \), we will first convert the repeating decimals into fractions.

### Step 1: Convert \( 1.\overline{2} \) into a fraction
Let \( x = 1.\overline{2} \).

This can be expressed as:
\[ x = 1.2222\ldots \]

To eliminate the repeating part, we multiply both sides by 10:
\[ 10x = 12.2222\ldots \]

Now, we subtract the first equation from the second:
\[ 10x - x = 12.2222\ldots - 1.2222\ldots \]
\[ 9x = 11 \]

Dividing both sides by 9 gives:
\[ x = \frac{11}{9} \]

### Step 2: Convert \( 0.\overline{3} \) into a fraction
Let \( y = 0.\overline{3} \).

This can be expressed as:
\[ y = 0.3333\ldots \]

Again, we multiply both sides by 10:
\[ 10y = 3.3333\ldots \]

Now, we subtract the first equation from the second:
\[ 10y - y = 3.3333\ldots - 0.3333\ldots \]
\[ 9y = 3 \]

Dividing both sides by 9 gives:
\[ y = \frac{1}{3} \]

### Step 3: Multiply the fractions
Now we have:
\[ 1.\overline{2} = \frac{11}{9} \quad \text{and} \quad 0.\overline{3} = \frac{1}{3} \]

We can now multiply these two fractions:
\[ 1.\overline{2} \times 0.\overline{3} = \frac{11}{9} \times \frac{1}{3} = \frac{11 \times 1}{9 \times 3} = \frac{11}{27} \]

### Step 4: Convert the result back to decimal
To express \( \frac{11}{27} \) as a decimal, we perform the division:
\[ 11 \div 27 = 0.407407\ldots \]

This decimal is repeating, so we can write it as:
\[ 0.\overline{407} \]

### Final Answer
Thus, the final answer is:
\[ 1.\overline{2} \times 0.\overline{3} = 0.\overline{407} \]

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