To solve the equation \( x - 4 = \frac{1}{4} (8x - 16) \), we start by simplifying the right side of the equation.
First, distribute \(\frac{1}{4}\) into the expression \(8x - 16\):
\[ \frac{1}{4} (8x - 16) = \frac{1}{4} \cdot 8x - \frac{1}{4} \cdot 16 = 2x - 4 \]
Now we can rewrite the equation:
\[ x - 4 = 2x - 4 \]
Next, we can eliminate \(-4\) from both sides of the equation:
\[ x - 4 + 4 = 2x - 4 + 4 \] \[ x = 2x \]
To isolate \(x\), subtract \(x\) from both sides:
\[ x - 2x = 0 \] \[ -x = 0 \]
Multiply both sides by \(-1\):
\[ x = 0 \]
Therefore, the solution to the equation is:
\[ \boxed{0} \]