Express `0.423232323....` in the form of `p/q,` (whre `p, q in I, q ne 0)`

Answer

To express the repeating decimal \(0.423232323...\) in the form of \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\), we can follow these steps:

### Step 1: Define the repeating decimal
Let \(x = 0.423232323...\).

### Step 2: Eliminate the repeating part
To eliminate the repeating part, we can multiply \(x\) by 10 to shift the decimal point:
\[ 10x = 4.23232323... \]
This can also be written as:
\[ 10x = 4.23\overline{23} \]

### Step 3: Multiply by 1000 to shift the decimal point further
Next, we multiply \(x\) by 1000 to shift the decimal point three places to the right:
\[ 1000x = 423.232323... \]
This can be written as:
\[ 1000x = 423.23\overline{23} \]

### Step 4: Set up the equations
Now we have two equations:
1. \(10x = 4.23\overline{23}\) (Equation 1)
2. \(1000x = 423.23\overline{23}\) (Equation 2)

### Step 5: Subtract the first equation from the second
Now, we subtract Equation 1 from Equation 2:
\[ 1000x - 10x = 423.23\overline{23} - 4.23\overline{23} \]
This simplifies to:
\[ 990x = 419 \]

### Step 6: Solve for \(x\)
Now, we can solve for \(x\):
\[ x = \frac{419}{990} \]

### Conclusion
Thus, we have expressed \(0.423232323...\) in the form of \(\frac{p}{q}\) where \(p = 419\) and \(q = 990\).

### Final Answer
\[ 0.423232323... = \frac{419}{990} \]

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