The four percentage formulas
A percent is just a fraction with 100 on the bottom, so 15% means 15/100 or 0.15. Every percentage problem is one of four shapes. (1) What is X% of Y: multiply Y by X/100. A 15% tip on an $80 dinner is 80 x 0.15 = $12. (2) X is what percent of Y: divide X by Y and multiply by 100. Scoring 25 out of 200 is 25 / 200 x 100 = 12.5%. (3) Percent change from X to Y: subtract, divide by the starting value, multiply by 100. A raise from $80,000 to $100,000 is (100,000 - 80,000) / 80,000 = 25%. (4) Increase or decrease Y by X%: multiply by (1 + X/100) or (1 - X/100). A $60 jacket at 20% off is 60 x 0.8 = $48, and a $60 item with 8.25% sales tax is 60 x 1.0825 = $64.95.
The most common mistake is using the wrong denominator in a percent change. The base is always the starting value: going from 80 to 100 is a 25% increase, but going from 100 back to 80 is a 20% decrease. Gains and losses are not symmetric, which is also why a stock that drops 50% needs a 100% gain to recover.
Percent vs. percentage points
When a rate moves from 3% to 4%, it has risen by one percentage point but by 33.3% in relative terms ((4 - 3) / 3). News reports and mortgage quotes use both, and mixing them up changes the story dramatically: a tax rate that goes from 20% to 22% is up 2 points, or 10%. If you need the relative change between two percentages, put them into calculator 3 as ordinary numbers.
Other traps: two successive 10% increases are not 20% but 21% (1.1 x 1.1 = 1.21); a 20% increase followed by a 20% decrease leaves you 4% below where you started; and "30% off" means you pay 70%, so a $50 item costs $35. To back out sales tax from a receipt, divide by 1 plus the rate rather than subtracting the rate: a $64.95 total at 8.25% tax was $60 before tax, not $59.59.
Everyday uses
Tipping: 18% of $46.50 is $8.37; a quick way is 10% ($4.65) plus half of that ($2.33) plus a bit more. Grades: 42 correct out of 50 is 84%. Shopping: an extra 25% off an already 40% off price is 0.6 x 0.75 = 0.45, so 55% off the original, not 65%. Salary: a 4% raise on $55,000 is $2,200, bringing you to $57,200. Investing: a fund that went from $10,000 to $12,500 gained 25%.
All four sections calculate at once as you type, and each row shows the arithmetic so you can copy the working into a spreadsheet or an email. Results are rounded to four decimals with trailing zeros removed.