`0.4bar(23)` is equivalent to the fraction :

Options

A. `(491)/(990)`
B. `(419)/(990)`
C. `(49)/(99)`
D. `(94)/(99)`

Answer

To convert the repeating decimal \(0.4\overline{23}\) into a fraction, we can follow these steps:

### Step 1: Define the repeating decimal
Let \( x = 0.4\overline{23} \).

### Step 2: Eliminate the repeating part
To eliminate the repeating part, we can multiply \( x \) by a power of 10 that moves the decimal point to the right, so the repeating part aligns. Since "23" has two digits, we multiply by 100:
\[ 100x = 42.3\overline{23} \]

### Step 3: Set up another equation
Next, we multiply \( x \) by 10 to shift the decimal point one place to the right:
\[ 10x = 4.23\overline{23} \]

### Step 4: Subtract the two equations
Now we have two equations:
1. \( 100x = 42.3\overline{23} \)
2. \( 10x = 4.23\overline{23} \)

Subtract the second equation from the first:
\[ 100x - 10x = 42.3\overline{23} - 4.23\overline{23} \]
This simplifies to:
\[ 90x = 42.3 - 4.23 \]

### Step 5: Calculate the right-hand side
Now, calculate \( 42.3 - 4.23 \):
\[ 42.3 - 4.23 = 38.07 \]
So we have:
\[ 90x = 38.07 \]

### Step 6: Solve for \( x \)
Now, divide both sides by 90:
\[ x = \frac{38.07}{90} \]

### Step 7: Convert the decimal to a fraction
To convert \( 38.07 \) into a fraction, we can express it as:
\[ 38.07 = \frac{3807}{100} \]
Thus, we have:
\[ x = \frac{3807}{100 \times 90} = \frac{3807}{9000} \]

### Step 8: Simplify the fraction
Now we simplify \( \frac{3807}{9000} \). We can find the greatest common divisor (GCD) of 3807 and 9000, which is 3:
\[ \frac{3807 \div 3}{9000 \div 3} = \frac{1269}{3000} \]

### Step 9: Final simplification
We can check if \( \frac{1269}{3000} \) can be simplified further. The GCD of 1269 and 3000 is 3:
\[ \frac{1269 \div 3}{3000 \div 3} = \frac{423}{1000} \]

Thus, the final answer is:
\[ x = \frac{423}{1000} \]

### Final Answer:
The fraction equivalent to \( 0.4\overline{23} \) is \( \frac{423}{1000} \).

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