What will come in place of question mark (?) .
`0.bar5 + 0.bar3 ` of `0.bar(11) `= ?

Options

A. 528
B. 891
C. 530
D. None of these

Answer

To solve the equation \(0.\overline{5} + 0.\overline{3} = 0.\overline{11} = ?\), we will convert the repeating decimals into fractions and then perform the addition.

### Step 1: Convert \(0.\overline{5}\) into a fraction
Let \(x = 0.\overline{5}\).

Multiply both sides by 10:
\[ 10x = 5.\overline{5} \]

Now, subtract the original equation from this new equation:
\[ 10x - x = 5.\overline{5} - 0.\overline{5} \]
\[ 9x = 5 \]
\[ x = \frac{5}{9} \]

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

Multiply both sides by 10:
\[ 10y = 3.\overline{3} \]

Now, subtract the original equation from this new equation:
\[ 10y - y = 3.\overline{3} - 0.\overline{3} \]
\[ 9y = 3 \]
\[ y = \frac{3}{9} = \frac{1}{3} \]

### Step 3: Convert \(0.\overline{11}\) into a fraction
Let \(z = 0.\overline{11}\).

Multiply both sides by 100:
\[ 100z = 11.\overline{11} \]

Now, subtract the original equation from this new equation:
\[ 100z - z = 11.\overline{11} - 0.\overline{11} \]
\[ 99z = 11 \]
\[ z = \frac{11}{99} = \frac{1}{9} \]

### Step 4: Add the fractions
Now we need to add \(0.\overline{5}\) and \(0.\overline{3}\):
\[ 0.\overline{5} + 0.\overline{3} = \frac{5}{9} + \frac{1}{3} \]

To add these fractions, we need a common denominator. The common denominator for \(9\) and \(3\) is \(9\):
\[ \frac{1}{3} = \frac{3}{9} \]

Now we can add:
\[ \frac{5}{9} + \frac{3}{9} = \frac{5 + 3}{9} = \frac{8}{9} \]

### Step 5: Conclusion
Thus, we have:
\[ 0.\overline{5} + 0.\overline{3} = \frac{8}{9} \]

Now, we can check if this equals \(0.\overline{11}\):
\[ 0.\overline{11} = \frac{1}{9} \]

### Final Answer
The value that comes in place of the question mark is:
\[ \frac{8}{9} \text{ or } 0.\overline{8} \]