What will come in the place of question mark(?) in each of the following question ?
`(0.bar6+0.bar7+0.bar5+0.bar4)`=?

Options

A. `4 1/4`
B. `3/9`
C. 2
D. `2 4/9`

Answer

To solve the expression \(0.\overline{6} + 0.\overline{7} + 0.\overline{5} + 0.\overline{4}\), we will convert each repeating decimal into a fraction.

### Step 1: Convert \(0.\overline{6}\) to a fraction
Let \(x = 0.\overline{6}\).

Multiplying both sides by 10:
\[ 10x = 6.\overline{6} \]

Now, subtract the original equation from this new equation:
\[ 10x - x = 6.\overline{6} - 0.\overline{6} \]
\[ 9x = 6 \]
\[ x = \frac{6}{9} = \frac{2}{3} \]

### Step 2: Convert \(0.\overline{7}\) to a fraction
Let \(y = 0.\overline{7}\).

Multiplying both sides by 10:
\[ 10y = 7.\overline{7} \]

Subtracting the original equation:
\[ 10y - y = 7.\overline{7} - 0.\overline{7} \]
\[ 9y = 7 \]
\[ y = \frac{7}{9} \]

### Step 3: Convert \(0.\overline{5}\) to a fraction
Let \(z = 0.\overline{5}\).

Multiplying both sides by 10:
\[ 10z = 5.\overline{5} \]

Subtracting the original equation:
\[ 10z - z = 5.\overline{5} - 0.\overline{5} \]
\[ 9z = 5 \]
\[ z = \frac{5}{9} \]

### Step 4: Convert \(0.\overline{4}\) to a fraction
Let \(w = 0.\overline{4}\).

Multiplying both sides by 10:
\[ 10w = 4.\overline{4} \]

Subtracting the original equation:
\[ 10w - w = 4.\overline{4} - 0.\overline{4} \]
\[ 9w = 4 \]
\[ w = \frac{4}{9} \]

### Step 5: Add all the fractions together
Now we can add all the fractions:
\[ 0.\overline{6} + 0.\overline{7} + 0.\overline{5} + 0.\overline{4} = \frac{2}{3} + \frac{7}{9} + \frac{5}{9} + \frac{4}{9} \]

First, convert \(\frac{2}{3}\) to a fraction with a denominator of 9:
\[ \frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9} \]

Now add:
\[ \frac{6}{9} + \frac{7}{9} + \frac{5}{9} + \frac{4}{9} = \frac{6 + 7 + 5 + 4}{9} = \frac{22}{9} \]

### Final Answer
Thus, the value of \(0.\overline{6} + 0.\overline{7} + 0.\overline{5} + 0.\overline{4}\) is:
\[ \frac{22}{9} \]