Tukuk-OS"Per Hundred" Is the Whole Idea
A percentage is just a fraction with a fixed denominator of 100. "35%" means "35 out of every 100," which is the number 35 ÷ 100 = 0.35. That single reframe unlocks everything:
- Percent → decimal: divide by 100 (move the decimal two places left). 7% = 0.07, 150% = 1.5.
- Decimal → percent: multiply by 100. 0.62 = 62%.
- Fraction → percent: divide then × 100. 3/8 = 0.375 = 37.5%.
To take P% of a number, multiply the number by P/100. Finding 15% of 240 is 240 × 0.15 = 36. The most useful mental shortcut is that 10% is the number divided by 10, and you build from there: 15% = 10% + half of 10%; 20% = double 10%. So 15% of 240 = 24 + 12 = 36, done in your head.
Discounts: Multiply, Don't Subtract
A sale of "30% off" means you pay 70%. The fast move is to multiply the price by (100 − 30)% = 0.70 rather than working out the discount and subtracting it. A €80 item at 30% off costs 80 × 0.70 = €56 in one step.
The trap is successive discounts. "20% off, then an extra 10% off" is not 30% off. The second cut only applies to the already-reduced price: 0.80 × 0.90 = 0.72, so you pay 72% — a total discount of 28%, not 30%. Store the rule: percentage changes multiply; they never simply add.
Percentage Change: It Matters What You Divide By
To find how much something changed, divide the change by the starting value, then × 100:
> change % = (new − old) / old × 100
A price from €50 to €65 rose by 15/50 = 30%. Reversed — €65 back to €50 — falls by 15/65 ≈ 23%, not 30%. The base is different, so the percentages differ. This is the single most abused idea in headlines. If a stock rises 50% then falls 50%, you have lost money: 1.5 × 0.5 = 0.75, so you end at 75% of the start, down 25%.
Ratios and Proportions: Scaling a Recipe or a Map
A ratio compares parts: a 2:3 mix of rice to water, or a map scale of 1:50,000. Ratios are fractions in disguise — 2:3 means 2 parts out of 5 total (40%) and 3 out of 5 (60%). To split a quantity in a ratio, add the parts and divide by that total. Split 250 in the ratio 2:3 → 250 ÷ 5 = 50 → 100 and 150.
A proportion says two ratios are equal, and it is the workhorse for scaling: if 3 apples cost €2, then 12 apples cost? 2/3 = x/12, so x = 12 × 2/3 = €8. Cross-multiply and solve — that one pattern handles recipes, map distances, unit conversions and speed problems alike.
Percent, Percentage Point, and the Confusion That Fooled Everyone
There is a difference between a percentage and a percentage point. If a tax rate goes from 20% to 25%, it rose by 5 percentage points but by 25% in relative terms (5 is a quarter of 20). News reports blur these constantly, and the gap between the two readings can be large. When a figure that is itself a percentage changes, always ask: points or percent?
Why This Pays Off
Every one of these is a two-line calculation, but strung together they cover shopping, tipping, loans, statistics, exam scores, battery percentages, sports stats and cooking. Learn the four moves — convert to decimal, multiply for discounts, divide-the-change-by-the-start, cross-multiply proportions — and mental arithmetic stops being a parlor trick and becomes an everyday defence against being mislead by a number.
You can check any of these in seconds with our percentage calculator, which handles "what is X% of Y," "X is what percent of Y," increases and decreases, and percentage change, and the loan calculator shows the same percentages working at scale when interest is spread over years of payments.