Express `4.bar(163)` in the form of `(p)/(q)`.

Explanation

To express \( 4.\overline{163} \) in the form of \( \frac{p}{q} \), we can follow these steps:

### Step 1: Define the variable
Let \( x = 4.\overline{163} \).

### Step 2: Identify the repeating part
The repeating part is \( 163 \), which has 3 digits.

### Step 3: Multiply by a power of 10
Since the repeating part has 3 digits, multiply both sides of the equation by \( 1000 \):
\[ 1000x = 4163.\overline{163} \]

### Step 4: Set up the equation
Now we have two equations:
1. \( x = 4.\overline{163} \) (Equation 1)
2. \( 1000x = 4163.\overline{163} \) (Equation 2)

### Step 5: Subtract the first equation from the second
Subtract Equation 1 from Equation 2:
\[ 1000x - x = 4163.\overline{163} - 4.\overline{163} \]
This simplifies to:
\[ 999x = 4163 - 4 \]

### Step 6: Simplify the right side
Calculate \( 4163 - 4 \):
\[ 999x = 4159 \]

### Step 7: Solve for \( x \)
Now, divide both sides by \( 999 \):
\[ x = \frac{4159}{999} \]

### Final Result
Thus, \( 4.\overline{163} \) can be expressed as:
\[ \frac{4159}{999} \]