Convert 2.`bar(46102)` in the `p/(q)` form of rational number.

Answer

To convert the repeating decimal \( 2.\overline{46102} \) into the form \( \frac{p}{q} \), we can follow these steps:

### Step 1: Define the repeating decimal
Let \( x = 2.\overline{46102} \). This means that the digits 46102 repeat indefinitely.

### Step 2: Multiply by a power of 10
Since the repeating part has 5 digits, we multiply \( x \) by \( 10^5 \) (which is 100000) to shift the decimal point 5 places to the right:
\[ 100000x = 246102.\overline{46102} \]

### Step 3: Set up the equations
Now we have two equations:
1. \( x = 2.\overline{46102} \) (Equation 1)
2. \( 100000x = 246102.\overline{46102} \) (Equation 2)

### Step 4: Subtract the two equations
Subtract Equation 1 from Equation 2:
\[ 100000x - x = 246102.\overline{46102} - 2.\overline{46102} \]
This simplifies to:
\[ 99999x = 246102 - 2 \]
\[ 99999x = 246100 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 99999 to solve for \( x \):
\[ x = \frac{246100}{99999} \]

### Step 6: Simplify the fraction
Now, we need to check if \( \frac{246100}{99999} \) can be simplified. We can find the greatest common divisor (GCD) of 246100 and 99999 and divide both the numerator and denominator by this GCD.

### Final Answer
Thus, the repeating decimal \( 2.\overline{46102} \) can be expressed in the form \( \frac{p}{q} \) as:
\[ x = \frac{246100}{99999} \]

---