Convert 0.137`bar(3206)` in the `p/(q)` form of rational number.

Answer

To convert the repeating decimal \(0.137\overline{3206}\) into the form \(\frac{p}{q}\), we can follow these steps:

### Step 1: Define the repeating decimal
Let \(x = 0.137320632063206...\)

### Step 2: Eliminate the repeating part
Since the repeating part "3206" has 4 digits, we will multiply \(x\) by \(10^4 = 10000\):
\[ 10000x = 13732.0632063206... \]

### Step 3: Shift the decimal point
Next, we will multiply \(x\) by \(10^3 = 1000\) to shift the decimal point three places to the right:
\[ 1000x = 137.320632063206... \]

### Step 4: Set up the equation
Now we have two equations:
1. \(10000x = 13732.0632063206...\) (Equation 1)
2. \(1000x = 137.320632063206...\) (Equation 2)

### Step 5: Subtract the two equations
Subtract Equation 2 from Equation 1:
\[ 10000x - 1000x = 13732.0632063206... - 137.320632063206... \]
This simplifies to:
\[ 9000x = 13594.7425742574... \]

### Step 6: Isolate \(x\)
Now, we can express \(x\):
\[ x = \frac{13594.7425742574...}{9000} \]

### Step 7: Convert to a fraction
To eliminate the decimal, we can multiply the numerator and denominator by 10000 (since we have four decimal places):
\[ x = \frac{135947425.7425742574...}{9000000} \]

### Step 8: Simplify the fraction
To simplify, we can express the numerator as a whole number:
Let \(p = 135947425\) and \(q = 9000000\). We can reduce this fraction if possible.

### Final Result
Thus, the repeating decimal \(0.137\overline{3206}\) can be expressed as:
\[ x = \frac{135947425}{9000000} \]