Express `0.4bar(23)` in the form of `p/q,` (where `p, q in I, q ne 0)`

Answer

To express \(0.4\overline{23}\) in the form of \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\), we can follow these steps:

### Step 1: Separate the non-repeating and repeating parts
Let \(x = 0.4\overline{23}\). This can be expressed as:
\[ x = 0.4 + 0.0\overline{23} \]

### Step 2: Express the repeating decimal
Let \(y = 0.0\overline{23}\). We can write:
\[ y = 0.0023232323\ldots \]

### Step 3: Multiply to eliminate the repeating part
To eliminate the repeating part, multiply \(y\) by \(1000\) (since the repeating part has two digits):
\[ 1000y = 2.3232323\ldots \]

### Step 4: Set up the equation
Now, we can set up the equation:
\[ 1000y = 2.32 + y \]
Subtract \(y\) from both sides:
\[ 999y = 2.32 \]

### Step 5: Solve for \(y\)
Now, divide both sides by \(999\):
\[ y = \frac{2.32}{999} \]

### Step 6: Convert \(2.32\) to a fraction
Convert \(2.32\) to a fraction:
\[ 2.32 = \frac{232}{100} \]
So,
\[ y = \frac{232}{100 \times 999} = \frac{232}{99900} \]

### Step 7: Substitute back into the expression for \(x\)
Now substitute \(y\) back into the expression for \(x\):
\[ x = 0.4 + y = 0.4 + \frac{232}{99900} \]

### Step 8: Convert \(0.4\) to a fraction
Convert \(0.4\) to a fraction:
\[ 0.4 = \frac{4}{10} = \frac{40}{100} = \frac{3960}{99900} \]

### Step 9: Combine the fractions
Now combine the two fractions:
\[ x = \frac{3960}{99900} + \frac{232}{99900} = \frac{3960 + 232}{99900} = \frac{4192}{99900} \]

### Step 10: Simplify the fraction
Now we need to simplify \(\frac{4192}{99900}\). The GCD of \(4192\) and \(99900\) is \(4\):
\[ \frac{4192 \div 4}{99900 \div 4} = \frac{1048}{24975} \]

Thus, the final answer is:
\[ x = \frac{1048}{24975} \]

### Final Answer
\[ 0.4\overline{23} = \frac{1048}{24975} \]

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