Reverse Percentage Change

It takes little computation to realize that the magnitude of percentage change from a to b isn't the same as that from b to a. The jump from 40 to 60 is 50% increase, but from 60 to 40 is 33.3% decrease. What's the exact formula for converting the magnitude of percentage changes? Let a and b be any nonzero integers.

For convention, let's have these labels:

• a: starting number, 40 in this case
• b: ending number, 60 in this case
• x: change from a to b, calculated as (b - a) / a, 50% in this case
• y: change from b to a, calculated as (a - b) / b, -33.3% in this case

The goal is to derive y from x. Follow these steps:

• x = (b - a) / a
• -x = (a - b) / a
• -x * a = (a - b)
• - x * a / b = (a - b) / b

So given x as the percentage change from a to b, the percentage change from b to a is simply the - x * a / b. In this example, -(50%) * 40 / 60 = -33.3% indeed. This works when a is bigger than b as well: a = 20, b = 10, x = (10 - 20) / 20 = -50%, y = -50% * 20 / 10 = 100%.