To find out how much water leaked from the bucket, subtract the amount of water remaining from the amount initially filled.

Donald initially filled the bucket with $\frac{15}{16}$ of a gallon of water, and after some time, $\frac{11}{16}$ of a gallon remained.

The amount of water that leaked out is:

$\frac{15}{16} - \frac{11}{16} = \frac{15 - 11}{16} = \frac{4}{16}$

Now, simplify $\frac{4}{16}$:

$\frac{4}{16} = \frac{1}{4}$

So, $\frac{1}{4}$ of a gallon of water leaked out.

Final Answer: $\frac{1}{4}$ of a gallon leaked out.

Would you like further details or clarifications on this solution?

Here are some related questions for further practice:

1. If Donald had started with $\frac{3}{4}$ of a gallon, how much would have leaked out if $\frac{2}{4}$ remained?
2. How much more water would have leaked if $\frac{10}{16}$ of a gallon had remained instead of $\frac{11}{16}$?
3. What fraction of the initial water remained in the bucket after the leak?
4. If the bucket initially had $2$ gallons of water, and $\frac{3}{8}$ leaked, how much remained?
5. If Donald added another $\frac{1}{16}$ gallon after the leak, how much water would be in the bucket?

Tip: When subtracting fractions with the same denominator, just subtract the numerators and keep the denominator the same.