Math: increase then decrease by same percentage, or vice-versa

Tutoring math, you run into curiosities. The tutor mentions one with percentage.

If an amount is increased, then decreased, by the same percentage, the net effect is shrinkage, regardless of which happens first.

Let the amount be 100. Furthermore, for an example, let the percent change be 8%, which equals 0.08. Then the final value, after 8% increase, then 8% decrease, will be

100(1+0.08)(1-0.08)

=100(1.08)(0.92)=99.36

Since order of multiplying doesn’t matter, reversing the increase and decrease yields the same result.

Let’s assume the percent change is not 0. If you imagine x as being its decimal equivalent, then the effect on an amount A is

A(1+x)(1-x)=A(1-x²).

Since 1-x² < 1, the net effect will be shrinkage.

Source:

Travers, Kenneth J. et al. Using Advanced Algebra. Toronto: Doubleday, 1977.