Percent Increase and Percent Decrease

Every percent change question is really just one fraction: change divided by original, times 100. The change is new minus old. The original is whatever value you started with, before anything happened. That original always sits in the denominator, no matter whether the value went up or down. Price going from 40 to 50 is a change of 10, and 10/40 = 0.25, so a 25 percent increase. The same jump read backward, from 50 down to 40, is still a change of 10, but now 10 is divided by 50, giving 20 percent. Same ten dollars, different base, different percent.

This is why a 50 percent rise followed by a 50 percent fall never gets you back to where you started. Each percent change uses a new, different original: the second change is based on the already-inflated price, not the first starting price. A rise of p percent multiplies the price by (1 + p/100); a later fall of q percent multiplies the new price by (1 - q/100). You chain the two factors together and compare the result to 1 to see the net change, and that product is almost never exactly 1 unless p and q are both zero.

This idea of a growth factor is what connects percent change to the rest of the unit: ratios, scaling, and distributions all lean on the same multiply-by-a-factor logic. Once percent change is read as multiplying by (1 plus or minus a rate) rather than just adding or subtracting a number, compounding, scatterplot trends, and growth models all become the same calculation repeated.