Let \( x \) be the amount of food Kendall gives to her cat during the rest of the day. According to the problem, the total amount of food she feeds her cat daily is \( \frac{4}{5} \) cup, and she gives \( \frac{1}{2} \) cup in the evening.
We can set up the equation:
\[ x + \frac{1}{2} = \frac{4}{5} \]
To solve for \( x \), we need to isolate \( x \). Start by subtracting \( \frac{1}{2} \) from both sides:
\[ x = \frac{4}{5} - \frac{1}{2} \]
Next, we need a common denominator to subtract the fractions. The least common multiple of 5 and 2 is 10. We can convert each fraction:
\[ \frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10} \] \[ \frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10} \]
Now substitute these back into the equation:
\[ x = \frac{8}{10} - \frac{5}{10} \]
Now perform the subtraction:
\[ x = \frac{8 - 5}{10} = \frac{3}{10} \]
Thus, the amount of food Kendall gives to her cat during the rest of the day is
\[ \boxed{\frac{3}{10}} \]