Problem: Compute 16% of 25 in your head.
Solution: 16% of 25 is equivalent to 25% of 16, which is clearly 4. This is true for all numbers: $x\%$ of $y$ is always equal to $y\%$ of $x$. The first one is $\frac{x}{100} y$ and the second is $\frac{y}{100}x$, and because multiplication is commutative and associative, both are equal to $(x \cdot y) / 100$. You can pick the version that is easiest.
Discussion: While this doesn’t work for every problem, it gives you three quick options to try. The first two options are to see if either of the two numbers makes it an easy problem (like 25% or 10%). The second, which “falls out” from the proof, is that you can first just multiply the two numbers together and then divide by 100. This often works well when one of the numbers is a single digit (e.g., 4% of 41 = 1.64).
Viewing the mental arithmetic as a math puzzle inspires all kinds of creativity. You can compute “$x\%$ of $y$” in their head by first computing 1% of $y$ and then scaling it up by $x$. Or you can split the denominator 100 into two pieces, such as $((x / 10) \cdot (y / 10))$, which is the same as “compute 10% of x and y separately, then multiply them together,” making a problem like 40% of 70 = 28 seem less intimidating than the “multiply then divide by 100” approach. The common 20% gratuity calculation is “move the decimal place over by 1 and double it,” i.e., $\frac{x}{100} \cdot 20 = \frac{2 \cdot 10 \cdot x}{10 \cdot 10} = \frac{2 \cdot x}{10}$.