Choosing the right calculation
Most percentage mistakes come from picking the wrong formula, not from the arithmetic. The question "what is 20% of 150" and the question "30 is what percent of 150" look similar but divide different numbers. A quick test: if you know the percentage and want an amount, use the first card. If you know two amounts and want the percentage between them, use the second. If you have a before and an after value and want to describe the move, use the third. If you want to apply a markup, discount, tax or tip to a number, use the fourth.
Percentage change and the base problem
Percentage change always divides by the starting value, and that makes it asymmetric. An increase from 80 to 100 is 25%, but the decrease from 100 back to 80 is 20%. A stock that falls 50% has to rise 100% to recover. This is why "up 10% then down 10%" does not land you back where you started: 100 goes to 110, then down 10% of 110 is 99.
The same asymmetry explains why a discount followed by a tax is not the same as a tax followed by a discount in general, and why two successive 10% discounts are a 19% discount, not 20%. When you chain percentages, apply them one at a time, or multiply the factors: 0.9 x 0.9 = 0.81, which is 19% off.
Percentage points versus percent
When a rate moves from 10% to 12%, it has risen by 2 percentage points but by 20 percent. News reports mix these up constantly. If you need the relative change between two percentages, enter them as plain numbers in the "percentage change" card: from 10 to 12 gives a 20% increase. If you need the absolute difference, just subtract.
Everyday uses
- Tips and tax: use "Y plus X%" with the bill and the rate. For a 7.25% sales tax on $49.99, the result is $53.61.
- Discounts: use "Y minus X%". A jacket at $120 with 30% off is $84. To find the original price from a sale price, divide by (1 - discount): $84 / 0.7 = $120.
- Grades and surveys: use "X is what % of Y". 42 correct out of 50 is 84%.
- Growth: use percentage change for revenue, followers, weight or anything measured over time. The direction label makes the sign obvious.
- Pricing: for cost-based pricing with a markup or a target margin, the markup calculator and the profit margin calculator do the whole job, including the difference between the two.
Common mistakes
- Dividing by the wrong number. For percentage change, always divide by the starting value, not the final one and not the difference.
- Adding percentages of different bases. 10% of your rent plus 10% of your groceries is not 20% of anything useful. Convert to amounts first.
- Reversing a percentage by subtracting it. If a price rose 25% to $100, the original was $80 (100 / 1.25), not $75.
- Treating percentage points as percent. An interest rate going from 4% to 5% is a 25% increase in the rate.