Let's break down the information given:
- There was half of the pie left after Lance's mother made it: \( \frac{1}{2} \) of the whole pie.
- After Lance eats a piece of pie, \( \frac{1}{6} \) of the whole pie remains.
So, if we denote the amount of pie Lance ate as \( x \), we can set up the equation:
\[ \frac{1}{2} - x = \frac{1}{6} \]
To solve for \( x \), we first need to find a common denominator for \( \frac{1}{2} \) and \( \frac{1}{6} \). The common denominator is 6.
Convert \( \frac{1}{2} \) into sixths:
\[ \frac{1}{2} = \frac{3}{6} \]
Now we can rewrite the equation:
\[ \frac{3}{6} - x = \frac{1}{6} \]
To isolate \( x \), we can rearrange the equation:
\[ x = \frac{3}{6} - \frac{1}{6} \]
Simplifying this gives:
\[ x = \frac{2}{6} = \frac{1}{3} \]
So, Lance ate \( \frac{1}{3} \) of the whole pie.
Now looking at the answer choices:
- A. \( \frac{1}{3} \) of the whole pie
- B. \( \frac{1}{4} \) of the whole pie
- C. \( \frac{1}{5} \) of the whole pie
- D. \( \frac{1}{6} \) of the whole pie
None of the provided options match \( \frac{1}{3} \). It appears that either the options provided do not have the correct answer, or there is a mistake in interpreting the problem or the expected options.
But based on the calculations, the fraction of pie that Lance ate is \( \frac{1}{3} \).