Nobody gets through school without learning percentages, and almost everybody gets a few of them wrong as an adult. The errors are consistent, which makes them easy to list. Each puzzle below has a tempting answer and a correct one. Try to answer before reading on.
A fund worth 100 falls 50% and then climbs 50%. The tempting answer is "back where it started". It is not. After the fall it is worth 50. A 50% gain on 50 is 25, so it ends at 75. The gain was applied to a smaller base.
The general rule: to undo a loss of x%, you need a gain of x / (100 - x), expressed as a percentage. A 10% loss needs an 11.1% gain. A 30% loss needs 42.9%. A 50% loss needs 100%. An 80% loss needs 400%. This is why large drawdowns are so hard to recover from, and why someone who says "I lost 40% but I'm making it back at 20% a year" is looking at a long road. A 40% loss needs a 66.7% gain, which takes about 2.8 years at 20% a year.
A store advertises 30% off, then gives you a coupon for an extra 20% off the sale price. Start with 100. After the first discount you pay 70. The coupon takes 20% of 70, which is 14, leaving 56. The total discount is 44%, not 50%.
Order does not matter here, because multiplication is commutative: 0.7 x 0.8 and 0.8 x 0.7 both equal 0.56. It does matter if the coupon applies to the original price instead, which retailers sometimes do and which will be stated in the fine print. Three successive 10% discounts leave you paying 72.9, an effective 27.1% off, not 30%. Each extra discount shaves off a little less than the one before.
Both "one percent" and "22%" are correct, and that is the trap. It rose by 1 percentage point. It also rose by 22.2% in relative terms, since 5.5 is 22.2% higher than 4.5. Headlines pick whichever sounds more dramatic.
When someone quotes a change in a rate, ask which one they mean. A drop in unemployment from 5% to 4% is one percentage point, and also a 20% fall in the rate. A mortgage rate moving from 6% to 7% is one point, a 16.7% increase in the rate, and, more usefully, roughly an 11% increase in your monthly payment on the same loan (1,798.65 to 1,995.91 on 300,000 over 30 years). The payment change is the number that touches your wallet.
The quick answer is 100, and it is correct here. The trap runs the other way. If you take the 80 price and add 20% back, you get 96, not 100. Going backward, you must divide by 0.8, not multiply by 1.2. The rule is that a percentage change applies to the number it was calculated on, not to the number you now hold.
This comes up constantly with tax. A receipt shows a total of 119 including 19% VAT. The tax is not 19% of 119 (which would be 22.61). The pre-tax price is 119 / 1.19 = 100, so the tax is 19. The VAT calculator and sales tax calculator both handle the reverse direction, which saves you from doing it by hand on a pile of invoices.
Yes, and it is a handy mental shortcut. 8% of 50 is 4, and 50% of 8 is also 4, because both are 8 x 50 / 100. If a calculation looks awkward one way, flip it. 16% of 25 is awkward, but 25% of 16 is 4. Easy.
The related mistake is confusing "X is what percent of Y" with "X is what percent more than Y". If a salary goes from 40,000 to 50,000, the new salary is 125% of the old one, and it is 25% more. Mixing those up leads to statements like "we grew by 125%", which would mean the figure more than doubled.
It depends which number you use as the base, which is why textbooks specify it. Percent error is normally calculated against the true (accepted) value: |90 - 100| / 100 = 10%. If you instead divide by your estimate, you get 11.1%. Neither is wrong, but you must be consistent and say which one you used. The percent error calculator uses the standard definition, with the true value as the base.
For everyday work, the percentage calculator covers the three standard questions (what is X% of Y, X is what percent of Y, and percent change) in one place, and the discount calculator handles sale prices and stacked offers. Both are quicker than redoing the arithmetic on a phone calculator, and they will not forget which number is the base.
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