The percent change formula
The formula is: (new minus old) divided by old, times 100. Three steps: subtract to get the change, divide by the starting value to get the relative change, multiply by 100 to express it as a percent.
Walk through a raise from 60,000 to 63,000 dollars. Subtract: 63,000 minus 60,000 equals 3,000. Divide: 3,000 divided by 60,000 equals 0.05. Multiply: 0.05 times 100 equals 5 percent. A 5 percent raise.
Now a decrease: an electric bill falling from 200 to 160 dollars. Subtract: 160 minus 200 equals negative 40. Divide: negative 40 divided by 200 equals negative 0.20. Multiply: negative 0.20 times 100 equals negative 20 percent. The negative sign means a decrease, so the bill dropped 20 percent.
The golden rule: always divide by the old number
The denominator is the starting point, not the end point and not an average. This is the entire game. Every percent-change error is a base error.
Price example: rent goes from 1,500 to 1,650 dollars. The change is 150. Divide by the old number, 1,500: 150 divided by 1,500 equals 0.10, a 10 percent increase. If you mistakenly divided by the new number, 1,650, you would get 9.1 percent, which is wrong. Ask yourself "percent of what?" and the answer is always the original value.
Worked examples
Example 1, a price hike: coffee goes from 4.50 to 5.25 dollars. Change: 0.75. Relative change: 0.75 divided by 4.50 equals 0.1667. Times 100 gives 16.7 percent. A 16.7 percent increase.
Example 2, business growth: monthly revenue goes from 40,000 to 55,000 dollars. Change: 15,000. Relative change: 15,000 divided by 40,000 equals 0.375. Times 100 gives 37.5 percent. Revenue grew 37.5 percent.
Example 3, a pay cut: salary goes from 72,000 to 68,400 dollars. Change: negative 3,600. Relative change: negative 3,600 divided by 72,000 equals negative 0.05. That is a 5 percent decrease.
Example 4, weight loss: 210 pounds down to 189. Change: negative 21. Relative change: negative 21 divided by 210 equals negative 0.10. A 10 percent decrease.
The trap: up 50%, then down 50%
A 50 percent increase followed by a 50 percent decrease does not return you to the start, because the two percentages use different bases. Start at 100. Up 50 percent adds 50, landing at 150. Down 50 percent removes half of 150, which is 75, landing at 75. You end 25 percent below where you began.
This asymmetry matters in investing. A stock that falls 50 percent must rise 100 percent to recover, because the gain is measured from the smaller base. If a 10,000 dollar portfolio drops to 5,000, the 5,000 dollar climb back is 100 percent of the new base.
Increase vs. decrease vs. difference
Percent change has a direction: it starts at the old number and moves to the new one. Percent difference is directionless and compares two numbers symmetrically, dividing by their average. If you want "how much did it change from X to Y," use percent change. If you want "how far apart are these two values," use percent difference.
Going from 100 to 150 is a 50 percent increase in percent-change terms. The symmetric percent difference is 50 divided by 125 (the average), times 100, which is 40 percent. Different questions, different answers, so pick the one that matches what you are asking.