The value of `0.bar(2)+0.bar(3) +0.bar(4)+0.bar(9) +0.bar(39)` is

Options

A. `0.bar(57)`
B. `1 20/33`
C. `2 1/3`
D. `2 13/33 `

Answer

To solve the problem \(0.\overline{2} + 0.\overline{3} + 0.\overline{4} + 0.\overline{9} + 0.\overline{39}\), we need to convert each repeating decimal into a fraction.

### Step 1: Convert \(0.\overline{2}\) to a fraction
The repeating decimal \(0.\overline{2}\) can be expressed as:
\[ 0.\overline{2} = \frac{2}{9} \]

### Step 2: Convert \(0.\overline{3}\) to a fraction
The repeating decimal \(0.\overline{3}\) can be expressed as:
\[ 0.\overline{3} = \frac{3}{9} = \frac{1}{3} \]

### Step 3: Convert \(0.\overline{4}\) to a fraction
The repeating decimal \(0.\overline{4}\) can be expressed as:
\[ 0.\overline{4} = \frac{4}{9} \]

### Step 4: Convert \(0.\overline{9}\) to a fraction
The repeating decimal \(0.\overline{9}\) can be expressed as:
\[ 0.\overline{9} = \frac{9}{9} = 1 \]

### Step 5: Convert \(0.\overline{39}\) to a fraction
The repeating decimal \(0.\overline{39}\) can be expressed as:
\[ 0.\overline{39} = \frac{39}{99} = \frac{13}{33} \quad \text{(after simplifying)} \]

### Step 6: Sum all the fractions
Now we can sum all the fractions we have found:
\[ \frac{2}{9} + \frac{1}{3} + \frac{4}{9} + 1 + \frac{13}{33} \]

To add these fractions, we need a common denominator. The least common multiple of \(9\), \(3\), and \(33\) is \(99\).

### Step 7: Convert each fraction to have a common denominator of \(99\)
- Convert \(\frac{2}{9}\):
\[ \frac{2}{9} = \frac{2 \times 11}{9 \times 11} = \frac{22}{99} \]
- Convert \(\frac{1}{3}\):
\[ \frac{1}{3} = \frac{1 \times 33}{3 \times 33} = \frac{33}{99} \]
- Convert \(\frac{4}{9}\):
\[ \frac{4}{9} = \frac{4 \times 11}{9 \times 11} = \frac{44}{99} \]
- Convert \(1\):
\[ 1 = \frac{99}{99} \]
- Convert \(\frac{13}{33}\):
\[ \frac{13}{33} = \frac{13 \times 3}{33 \times 3} = \frac{39}{99} \]

### Step 8: Add the fractions
Now we can add all the fractions:
\[ \frac{22}{99} + \frac{33}{99} + \frac{44}{99} + \frac{99}{99} + \frac{39}{99} = \frac{22 + 33 + 44 + 99 + 39}{99} = \frac{237}{99} \]

### Step 9: Simplify the fraction
Now we simplify \(\frac{237}{99}\):
\[ \frac{237 \div 9}{99 \div 9} = \frac{26.3333}{11} \quad \text{(not a whole number)} \]
Since \(237\) and \(99\) share a common factor of \(9\):
\[ \frac{237}{99} = \frac{79}{33} \]

### Final Answer
Thus, the value of \(0.\overline{2} + 0.\overline{3} + 0.\overline{4} + 0.\overline{9} + 0.\overline{39}\) is:
\[ \frac{79}{33} \]