`[(5.bar(88)-4.bar(58))-(0.bar(64)+0.bar(36))]` is equal to

Options

A. `1.bar(01)`
B. `1.bar(30)`
C. `1.bar(19)`
D. `0.bar(29)`

Answer

To solve the expression \([(5.\overline{88}-4.\overline{58})-(0.\overline{64}+0.\overline{36})]\), we will follow these steps:

### Step 1: Convert repeating decimals to fractions
1. **Convert \(5.\overline{88}\)**:
- Let \(x = 5.\overline{88}\).
- Then, \(100x = 588.\overline{88}\).
- Subtracting these gives:
\[ 100x - x = 588.\overline{88} - 5.\overline{88} \implies 99x = 583 \implies x = \frac{583}{99}. \]
- Therefore, \(5.\overline{88} = \frac{583}{99}\).

2. **Convert \(4.\overline{58}\)**:
- Let \(y = 4.\overline{58}\).
- Then, \(100y = 458.\overline{58}\).
- Subtracting gives:
\[ 100y - y = 458.\overline{58} - 4.\overline{58} \implies 99y = 454 \implies y = \frac{454}{99}. \]
- Therefore, \(4.\overline{58} = \frac{454}{99}\).

3. **Convert \(0.\overline{64}\)**:
- Let \(z = 0.\overline{64}\).
- Then, \(100z = 64.\overline{64}\).
- Subtracting gives:
\[ 100z - z = 64.\overline{64} - 0.\overline{64} \implies 99z = 64 \implies z = \frac{64}{99}. \]
- Therefore, \(0.\overline{64} = \frac{64}{99}\).

4. **Convert \(0.\overline{36}\)**:
- Let \(w = 0.\overline{36}\).
- Then, \(100w = 36.\overline{36}\).
- Subtracting gives:
\[ 100w - w = 36.\overline{36} - 0.\overline{36} \implies 99w = 36 \implies w = \frac{36}{99}. \]
- Therefore, \(0.\overline{36} = \frac{36}{99}\).

### Step 2: Substitute back into the expression
Now substituting these values back into the original expression:
\[ \left(\frac{583}{99} - \frac{454}{99}\right) - \left(\frac{64}{99} + \frac{36}{99}\right) \]

### Step 3: Simplify the expression
1. **Calculate the first part**:
\[ \frac{583 - 454}{99} = \frac{129}{99}. \]

2. **Calculate the second part**:
\[ \frac{64 + 36}{99} = \frac{100}{99}. \]

3. **Combine the results**:
\[ \frac{129}{99} - \frac{100}{99} = \frac{29}{99}. \]

### Step 4: Final result
The final result is:
\[ \frac{29}{99} = 0.\overline{29}. \]

### Conclusion
Thus, the value of the expression \([(5.\overline{88}-4.\overline{58})-(0.\overline{64}+0.\overline{36})]\) is \(0.\overline{29}\).

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