Shivani served \( \frac{1}{2} \) gallon of hot chocolate and \( \frac{1}{4} \) gallon of apple cider. To find out how much more hot chocolate than apple cider she served, we subtract the amount of apple cider from the amount of hot chocolate:
\[ \frac{1}{2} - \frac{1}{4} \]
To perform this subtraction, we need a common denominator. The denominators 2 and 4 have a common denominator of 4. We can convert \( \frac{1}{2} \) to a fraction with a denominator of 4:
\[ \frac{1}{2} = \frac{2}{4} \]
Now we can subtract:
\[ \frac{2}{4} - \frac{1}{4} = \frac{2 - 1}{4} = \frac{1}{4} \]
Therefore, Shivani served \( \frac{1}{4} \) more hot chocolate than apple cider.
Thus, the answer is
\[ \frac{1}{4} \]
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