The value of ` 0.bar2 + 0.bar3 + 0.bar(32)` is :

Options

A. `0. bar(87)`
B. `0.bar(77)`
C. `0.bar(82)`
D. `0.bar(86)`

Answer

To solve the problem of finding the value of \( 0.\overline{2} + 0.\overline{3} + 0.\overline{32} \), we will follow these steps:

### Step 1: Convert the repeating decimals to fractions
1. **Convert \( 0.\overline{2} \)**:
- Let \( x = 0.\overline{2} \).
- Then, \( 10x = 2.\overline{2} \).
- Subtracting the first equation from the second gives:
\[ 10x - x = 2.\overline{2} - 0.\overline{2} \implies 9x = 2 \implies x = \frac{2}{9} \]
- Therefore, \( 0.\overline{2} = \frac{2}{9} \).

2. **Convert \( 0.\overline{3} \)**:
- Let \( y = 0.\overline{3} \).
- Then, \( 10y = 3.\overline{3} \).
- Subtracting gives:
\[ 10y - y = 3.\overline{3} - 0.\overline{3} \implies 9y = 3 \implies y = \frac{3}{9} = \frac{1}{3} \]
- Therefore, \( 0.\overline{3} = \frac{1}{3} \).

3. **Convert \( 0.\overline{32} \)**:
- Let \( z = 0.\overline{32} \).
- Then, \( 100z = 32.\overline{32} \).
- Subtracting gives:
\[ 100z - z = 32.\overline{32} - 0.\overline{32} \implies 99z = 32 \implies z = \frac{32}{99} \]
- Therefore, \( 0.\overline{32} = \frac{32}{99} \).

### Step 2: Add the fractions
Now we need to add \( \frac{2}{9} + \frac{1}{3} + \frac{32}{99} \).

1. **Find a common denominator**:
- The least common multiple (LCM) of \( 9, 3, \) and \( 99 \) is \( 99 \).

2. **Convert each fraction to have a denominator of 99**:
- Convert \( \frac{2}{9} \):
\[ \frac{2}{9} = \frac{2 \times 11}{9 \times 11} = \frac{22}{99} \]
- Convert \( \frac{1}{3} \):
\[ \frac{1}{3} = \frac{1 \times 33}{3 \times 33} = \frac{33}{99} \]
- \( \frac{32}{99} \) remains the same.

3. **Add the fractions**:
\[ \frac{22}{99} + \frac{33}{99} + \frac{32}{99} = \frac{22 + 33 + 32}{99} = \frac{87}{99} \]

### Step 3: Simplify the result
1. **Simplify \( \frac{87}{99} \)**:
- The greatest common divisor (GCD) of \( 87 \) and \( 99 \) is \( 9 \).
- Therefore:
\[ \frac{87 \div 9}{99 \div 9} = \frac{9}{11} \]

### Final Result
The value of \( 0.\overline{2} + 0.\overline{3} + 0.\overline{32} \) is \( \frac{9}{11} \).