The value of `0.5bar6 -0.7bar(23)+0.3bar9 xx 0.bar7` is :

Options

A. A) `0.1bar(58)`
B. B) `0.bar(158)`
C. C) `0.bar(154)`
D. D) `0.1bar(54)`

Answer

To solve the expression \(0.5\overline{6} - 0.7\overline{23} + 0.3\overline{9} \times 0.\overline{7}\), we will convert the repeating decimals into fractions and then perform the arithmetic operations step by step.

### Step 1: Convert the repeating decimals into fractions

1. **Convert \(0.5\overline{6}\)**:
Let \(x = 0.5\overline{6}\).
\[ x = 0.56666\ldots \]
Multiply by 10:
\[ 10x = 5.6666\ldots \]
Subtract the first equation from the second:
\[ 10x - x = 5.6666\ldots - 0.56666\ldots \]
\[ 9x = 5.1 \implies x = \frac{5.1}{9} = \frac{51}{90} \]

2. **Convert \(0.7\overline{23}\)**:
Let \(y = 0.7\overline{23}\).
\[ y = 0.7232323\ldots \]
Multiply by 1000:
\[ 1000y = 723.2323\ldots \]
Subtract:
\[ 1000y - 10y = 723.2323\ldots - 7.2323\ldots \]
\[ 990y = 716 \implies y = \frac{716}{990} = \frac{358}{495} \]

3. **Convert \(0.3\overline{9}\)**:
Let \(z = 0.3\overline{9}\).
\[ z = 0.39999\ldots \]
Multiply by 10:
\[ 10z = 3.9999\ldots \]
Subtract:
\[ 10z - z = 3.9999\ldots - 0.39999\ldots \]
\[ 9z = 3.6 \implies z = \frac{3.6}{9} = \frac{4}{10} = \frac{2}{5} \]

4. **Convert \(0.\overline{7}\)**:
Let \(w = 0.\overline{7}\).
\[ w = 0.77777\ldots \]
Multiply by 10:
\[ 10w = 7.7777\ldots \]
Subtract:
\[ 10w - w = 7.7777\ldots - 0.7777\ldots \]
\[ 9w = 7 \implies w = \frac{7}{9} \]

### Step 2: Substitute the fractions into the expression

Now we substitute the fractions back into the expression:
\[ \frac{51}{90} - \frac{358}{495} + \left(\frac{2}{5} \times \frac{7}{9}\right) \]

### Step 3: Calculate \(0.3\overline{9} \times 0.\overline{7}\)

Calculate \( \frac{2}{5} \times \frac{7}{9} \):
\[ \frac{2 \times 7}{5 \times 9} = \frac{14}{45} \]

### Step 4: Find a common denominator for the entire expression

The common denominator for \(90\), \(495\), and \(45\) is \(990\).

1. Convert \( \frac{51}{90} \):
\[ \frac{51 \times 11}{90 \times 11} = \frac{561}{990} \]

2. Convert \( \frac{358}{495} \):
\[ \frac{358 \times 2}{495 \times 2} = \frac{716}{990} \]

3. Convert \( \frac{14}{45} \):
\[ \frac{14 \times 22}{45 \times 22} = \frac{308}{990} \]

### Step 5: Combine the fractions

Now we can combine:
\[ \frac{561}{990} - \frac{716}{990} + \frac{308}{990} = \frac{561 - 716 + 308}{990} = \frac{153}{990} \]

### Step 6: Simplify the result

The fraction \( \frac{153}{990} \) can be simplified by finding the GCD of \(153\) and \(990\), which is \(9\):
\[ \frac{153 \div 9}{990 \div 9} = \frac{17}{110} \]

### Final Answer
Thus, the final value of the expression is:
\[ \frac{17}{110} \]