Eddie's Snack Shack used \( \frac{1}{2} \) of a bottle of ketchup. They used \( \frac{1}{10} \) as much mustard as ketchup.
To find out how much mustard they used, we can calculate:
\[ \text{Mustard used} = \frac{1}{10} \times \text{Ketchup used} = \frac{1}{10} \times \frac{1}{2} \]
Now we multiply the fractions:
\[ \text{Mustard used} = \frac{1 \times 1}{10 \times 2} = \frac{1}{20} \]
Thus, Eddie's Snack Shack went through \( \frac{1}{20} \) of a bottle of mustard.
The final answer is:
\[ \frac{1}{20} \]