The value of `0.1bar23` is:

Options

A. a) 61/495
B. b) 71/485
C. c) 62/719
D. d) 123/999

Answer

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

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

### Step 2: Eliminate the repeating part
To eliminate the repeating part, we can multiply \( x \) by a power of 10. Since the repeating part "23" has 2 digits, we multiply by 100:
\[ 100x = 12.3232323... \]

### Step 3: Multiply by another power of 10
Now, we also need to isolate the non-repeating part. To do this, we multiply \( x \) by 10:
\[ 10x = 1.23232323... \]

### Step 4: Set up the equations
Now we have two equations:
1. \( 100x = 12.3232323...\) (Equation 1)
2. \( 10x = 1.23232323...\) (Equation 2)

### Step 5: Subtract the two equations
Now, we subtract Equation 2 from Equation 1:
\[ 100x - 10x = 12.3232323... - 1.23232323... \]
This simplifies to:
\[ 90x = 11.09 \]

### Step 6: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{11.09}{90} \]

### Step 7: Simplify the fraction
To simplify \( \frac{11.09}{90} \), we can express \( 11.09 \) as \( \frac{1109}{100} \):
\[ x = \frac{1109}{100 \times 90} = \frac{1109}{9000} \]

### Step 8: Final simplification
Now, we can simplify \( \frac{1109}{9000} \) if possible. However, since 1109 is a prime number, we can conclude that:
\[ x = \frac{1109}{9000} \]

### Conclusion
Thus, the value of \( 0.1\overline{23} \) is \( \frac{1109}{9000} \).

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