Up 50%, then down a third, where are you?

Question

A stock rises 50\% one day and falls 33	frac{1}{3}\% the next. Solve in your head. What is the net change from the start? Aim: under 10 seconds.

Accepted Answer

Convert both moves to fractions: +50\% = 	imes	frac{3}{2}, \qquad -33	frac{1}{3}\% = 	imes	frac{2}{3}. These are exact reciprocals, so 	frac{3}{2} 	imes 	frac{2}{3} = 1.00. The net change is zero. Spotting the reciprocal pair is far faster than multiplying 1.5 	imes 0.6667 by hand. Takeaway. Memorize the clean percentage-to-fraction pairs: +50\% = 	imes\frac{3}{2} undone by -33	frac{1}{3}\% = 	imes\frac{2}{3}, just as +25\% = 	imes\frac{5}{4} is undone by -20\% = 	imes\frac{4}{5}. Reciprocals cancel exactly, which instantly tells you the gain needed to reverse any given loss.

Up 50%, then down a third, where are you?

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