The value of `0.1bar(23)` is :

Options

A. `(61)/(495)`
B. `(71)/(485)`
C. `(62)/(719)`
D. `(123)/(999)`

Answer

To find the value of \(0.1\overline{23}\), we can follow these steps:

### Step 1: Define the variable
Let \( x = 0.1\overline{23} \). This means that \( x = 0.123232323...\)

### Step 2: Multiply by 10
To eliminate the decimal, multiply both sides by 10:
\[ 10x = 1.23232323... \]

### Step 3: Multiply by 1000
Next, multiply both sides by 1000 to shift the decimal point further:
\[ 1000x = 123.23232323... \]

### Step 4: Set up the equation
Now, we have two equations:
1. \( 10x = 1.23232323...\)
2. \( 1000x = 123.23232323...\)

### Step 5: Subtract the first equation from the second
Subtract the first equation from the second:
\[ 1000x - 10x = 123.23232323... - 1.23232323... \]
This simplifies to:
\[ 990x = 122 \]

### Step 6: Solve for \( x \)
Now, divide both sides by 990:
\[ x = \frac{122}{990} \]

### Step 7: Simplify the fraction
To simplify \( \frac{122}{990} \), we find the greatest common divisor (GCD) of 122 and 990, which is 2:
\[ x = \frac{122 \div 2}{990 \div 2} = \frac{61}{495} \]

### Conclusion
Thus, the value of \( 0.1\overline{23} \) is:
\[ \boxed{\frac{61}{495}} \]