To find out how much water is left in the bucket after Katy pours out 1/3 of a gallon, let's break the problem down step by step:

1. Initial amount of water: Katy initially filled the bucket with $\frac{1}{2}$ gallon of water.
2. Amount poured out: She poured out $\frac{1}{3}$ gallon of water.

Now, we subtract the amount poured out from the initial amount:

$\text{Amount left} = \frac{1}{2} - \frac{1}{3}$

To subtract these fractions, we need a common denominator. The least common denominator (LCD) between 2 and 3 is 6.

$\frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}$

Now, subtract the fractions:

$\frac{3}{6} - \frac{2}{6} = \frac{1}{6}$

So, after pouring out the water, $\frac{1}{6}$ gallon of water is left in the bucket.

Would you like more details on this process, or do you have any questions?

Here are 5 related questions you can explore:

1. What is the process to subtract fractions with different denominators?
2. How do you find the least common denominator (LCD) of two numbers?
3. How much water would be left if she poured out $\frac{1}{4}$ gallon instead of $\frac{1}{3}$?
4. What happens if Katy initially filled the bucket with 1 gallon instead of $\frac{1}{2}$?
5. Can you convert the amount of water left into ounces, knowing there are 128 ounces in a gallon?

Tip: When subtracting fractions, always find a common denominator before performing the operation.