The value of `0.bar(3) + 0.bar(6) + 0. bar( 7 ) + 0. bar( 8 ) ` in fraction will be

Options

A. `2( 3)/( 10)`
B. `2(2)/( 3)`
C. `20. bar( 35)`
D. `5(3)/(10)`

Answer

To solve the problem of finding the value of \(0.\overline{3} + 0.\overline{6} + 0.\overline{7} + 0.\overline{8}\) in fraction form, we will follow these steps:

### Step 1: Convert each repeating decimal to a fraction
1. **Convert \(0.\overline{3}\)**:
- Using the formula \(0.\overline{A} = \frac{A}{9}\), where \(A = 3\):
\[ 0.\overline{3} = \frac{3}{9} \]

2. **Convert \(0.\overline{6}\)**:
- Using the same formula with \(A = 6\):
\[ 0.\overline{6} = \frac{6}{9} \]

3. **Convert \(0.\overline{7}\)**:
- Again using the formula with \(A = 7\):
\[ 0.\overline{7} = \frac{7}{9} \]

4. **Convert \(0.\overline{8}\)**:
- Using the formula with \(A = 8\):
\[ 0.\overline{8} = \frac{8}{9} \]

### Step 2: Add the fractions together
Now we will add all the fractions obtained:
\[ 0.\overline{3} + 0.\overline{6} + 0.\overline{7} + 0.\overline{8} = \frac{3}{9} + \frac{6}{9} + \frac{7}{9} + \frac{8}{9} \]

### Step 3: Combine the fractions
Since all fractions have the same denominator, we can combine them:
\[ = \frac{3 + 6 + 7 + 8}{9} = \frac{24}{9} \]

### Step 4: Simplify the fraction
Now we will simplify \(\frac{24}{9}\):
\[ \frac{24}{9} = \frac{24 \div 3}{9 \div 3} = \frac{8}{3} \]

### Step 5: Convert to mixed fraction (if needed)
To convert \(\frac{8}{3}\) into a mixed fraction:
- Divide 8 by 3, which gives 2 with a remainder of 2.
- Thus, \(\frac{8}{3} = 2 \frac{2}{3}\).

### Final Answer
The value of \(0.\overline{3} + 0.\overline{6} + 0.\overline{7} + 0.\overline{8}\) in fraction form is:
\[ \frac{8}{3} \quad \text{or} \quad 2 \frac{2}{3} \]

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