Intuitively, why's X% of Y = Y% of X?
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I didn't spot this trick until I read this:
LPT: X percent of Y is equal to Y percent of X.
I know that multiplication is commutative. Indubitably, $\dfrac{X}{100}Y= X\dfrac{Y}{100}$ But this algebra doesn't betray the intuition! How can this be intuited?
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Just consider that Z is Z percent of 100. Then you get:
X percent of Y is X percent of Y percent of 100, which is Y percent of X percent of 100, which is Y percent of X.
I think that X percent of Y percent of something is the same as Y percent of X percent of something is intuitive. And that Z is Z percent of 100 is quite obvious from the definition of percent.
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