The question you opened
18 percent of 50, or 50 percent of 18
One receipt shows 18 percent of 50 dollars. The next shows 50 percent of 18 dollars.
Which receipt shows the bigger number?
They are equal
A percent is a multiplication, and multiplication ignores order. Both come to 9 dollars, because 0.18 times 50 is the same as 0.50 times 18.
Write both as products and the answer appears without arithmetic. Eighteen percent of 50 is 0.18 times 50. Fifty percent of 18 is 0.50 times 18. Move the decimal point in the first factor of one and into the second factor of the other and you have the same two numbers in the other order. Both equal 9.
The general statement is that x percent of y always equals y percent of x. The proof is one line: x percent of y is x times y divided by 100, and so is y percent of x. Nothing about the numbers matters, which is why this works for any pair.
As a shortcut it is worth real time. Sixteen percent of 25 looks like work. Twenty five percent of 16 is a quarter of 16, which is 4, and the two are equal. Four percent of 75 becomes 75 percent of 4, which is 3. Thirty six percent of 50 becomes 50 percent of 36, which is 18.
The rule turns an awkward multiplication into a familiar fraction whenever one of the two numbers is 25, 50, 20, or 10. Tips are the everyday case. Eighteen percent of a 50 dollar check is half of 18, so 9 dollars. Fifteen percent of a 40 dollar check is 40 percent of 15, which is 6. The flip removes the step where people reach for a phone.
The trap in the card is that the two receipts feel different because the words feel different. Half is a big, visible act. Eighteen percent is a small, technical one. The feeling attaches to the verb rather than to the product, and the product is all that is on the receipt.
Technique: Flip the percent to the easy side