`2.bar3+3.bar4-4.bar8=`________

Options

A. `0.bar7`
B. `1.bar2`
C. `0.bar8`
D. `1.bar9`

Answer

To solve the expression \(2.\overline{3} + 3.\overline{4} - 4.\overline{8}\), we will first convert the repeating decimals into fractions, perform the arithmetic operations, and then convert the result back into a decimal with a repeating part if necessary.

### Step 1: Convert \(2.\overline{3}\) into a fraction
Let \(x = 2.\overline{3}\).
Then, \(10x = 23.\overline{3}\).

Subtracting these two equations:
\[ 10x - x = 23.\overline{3} - 2.\overline{3} \]
\[ 9x = 21 \implies x = \frac{21}{9} = \frac{7}{3} \]

### Step 2: Convert \(3.\overline{4}\) into a fraction
Let \(y = 3.\overline{4}\).
Then, \(10y = 34.\overline{4}\).

Subtracting these two equations:
\[ 10y - y = 34.\overline{4} - 3.\overline{4} \]
\[ 9y = 31 \implies y = \frac{31}{9} \]

### Step 3: Convert \(4.\overline{8}\) into a fraction
Let \(z = 4.\overline{8}\).
Then, \(10z = 48.\overline{8}\).

Subtracting these two equations:
\[ 10z - z = 48.\overline{8} - 4.\overline{8} \]
\[ 9z = 44 \implies z = \frac{44}{9} \]

### Step 4: Substitute the fractions into the expression
Now we substitute the fractions back into the original expression:
\[ 2.\overline{3} + 3.\overline{4} - 4.\overline{8} = \frac{7}{3} + \frac{31}{9} - \frac{44}{9} \]

### Step 5: Find a common denominator
The common denominator for \(3\) and \(9\) is \(9\). We convert \(\frac{7}{3}\) to have a denominator of \(9\):
\[ \frac{7}{3} = \frac{7 \times 3}{3 \times 3} = \frac{21}{9} \]

### Step 6: Combine the fractions
Now we can combine the fractions:
\[ \frac{21}{9} + \frac{31}{9} - \frac{44}{9} = \frac{21 + 31 - 44}{9} = \frac{8}{9} \]

### Step 7: Convert the result back to decimal
The fraction \(\frac{8}{9}\) can be expressed as a decimal:
\[ \frac{8}{9} = 0.\overline{8} \]

### Final Answer
Thus, the final answer is:
\[ \boxed{0.\overline{8}} \]