One idea, four disguises
Every percentage problem is the same relationship — part = whole × rate — asked from a different direction. Know which of the three pieces you are missing and the formula picks itself:
- Missing the part: "What is 15% of 80?" → 80 × 15 ÷ 100 = 12. This is the shape of tips, commissions, and deposit calculations.
- Missing the rate: "38 out of 50 — what percent?" → 38 ÷ 50 × 100 = 76%. Test scores, conversion rates, savings rates.
- Comparing two values: "Rent went from $1,400 to $1,540" → (1,540 − 1,400) ÷ 1,400 × 100 = a 10% increase. Note the old value is always the divisor.
- Missing the whole: the reverse-percentage case, the one most people get wrong — covered below.
The calculator's four tabs map to exactly these shapes, and it always prints the formula next to the answer so you can redo the same calculation on the back of a receipt.
Mental math that actually works
The single most useful trick is that 10% is the number with its decimal point moved one place left, and everything else is built from that. An 18% tip on a $62 bill: 10% is $6.20, 5% is half of that ($3.10), 1% is $0.62. So 18% = 6.20 + 3.10 + 0.62 + 0.62 + 0.62 = $11.16 — which is exactly what 62 × 0.18 gives.
The second trick is that percentages commute: X% of Y always equals Y% of X. 4% of 75 feels awkward; 75% of 4 is obviously 3. Same answer, because both are just 4 × 75 ÷ 100.
The three classic percentage traps
Trap 1 — subtracting instead of dividing (reverse percentages). A price tag reads $89.99 including 8.25% sales tax, and you want the pre-tax price. Taking 8.25% off $89.99 gives $82.57 — wrong. The tax was charged on the original price, so you divide: 89.99 ÷ 1.0825 = $83.13, with $6.86 of tax. Whenever a percentage has already been applied, undo it with division.
Trap 2 — adding stacked percentages. "20% off, plus an extra 15% off at checkout" is not 35% off. The second discount applies to the already-reduced price: 0.80 × 0.85 = 0.68, so you pay 68% — a 32% total discount. On $250, that's $170 rather than the $162.50 a 35%-off deal would give. Retailers phrase promotions this way precisely because it sounds bigger than it is.
Trap 3 — percentage points vs percent change. A savings rate moving from 4% to 6% rose two percentage points, but the percent change is (6 − 4) ÷ 4 = 50%. Headlines exploit the ambiguity in both directions: "rates jumped 50%" and "rates edged up 2 points" can describe the same event. The % change tab computes the relative version; just remember which one a headline means before reacting to it.
Why gains and losses aren't symmetric
Percent changes use whatever number they start from as the base, and that base shifts after every step. Start with 160, lose 25%, and you're at 120. Gain 25% back and you reach 150 — still 6.25% below where you began. The deficit grows fast with size: a 50% fall needs a 100% rise to recover. This is why an investment that "went down 30% then up 30%" is quietly still 9% underwater, and why comparing two changes only makes sense when you know their bases.
The same base-shifting explains compound growth: 5% per year for three years is not 15% but 1.05 × 1.05 × 1.05 − 1 = 15.76%. For money questions where that difference compounds over decades, the compound interest calculator does the repeated multiplication for you; for a one-off price change, the increase/decrease tab here is enough.