Percent is easy once you know what is the whole.

Percent means "per hundred", so a percentage is always a fraction with 100 on the bottom. To take X% of a number N you compute N × X ÷ 100. To find what percent A is of B you compute A ÷ B × 100. Those two are inverses of each other, and almost every percentage question is one of them wearing a disguise. If you can name which number is the whole and which is the part, you already know which formula to use.

Because the point of a percent change is to say how big the move was relative to where you started. The formula is (new − old) ÷ old × 100. That is why the two directions are asymmetric: 50 up to 100 is a 100% increase, but 100 back down to 50 is only a 50% decrease, even though the absolute move is 50 both times. If you divide by the new value instead you get a number that no one else will recognise.

Because the second discount is taken from a price that has already shrunk. A $100 item at −20% becomes $80, and 10% off $80 is $8, so you pay $72 — a 28% total discount, not 30%. The general rule is to multiply the factors: 0.80 × 0.90 = 0.72. The same logic applies in reverse to stacked increases, which compound above the sum: +10% then +10% is +21%, not +20%.

Divide, do not subtract. If a jacket is $63 after 30% off, the original was 63 ÷ (1 − 0.30) = 63 ÷ 0.70 = $90. Taking 30% off $63 would give $44.10, which is wrong by a wide margin, because the 30% was a share of the original price and not of the sale price. The same division pulls a pre-tax amount out of a tax-inclusive total: divide the total by 1 plus the tax rate.

A percentage point is an absolute difference between two percentages; a percent change is a relative one. If an interest rate goes from 4% to 6%, that is a rise of 2 percentage points and a rise of 50 percent. Newspapers and central banks use percentage points precisely to avoid the ambiguity, and mixing the two is one of the most common ways a correct calculation ends up telling a wrong story.

Yes, whenever the part is bigger than the reference. If revenue goes from $2M to $5M that is a 150% increase, and 300 is 150% of 200. The only case where percentages are capped at 100 is when they describe a share of a fixed whole that cannot be exceeded — the parts of a single pie, for example. Percentages can also be negative, which simply means the quantity moved down.

A markup is measured against cost, a margin against the selling price. If an item costs $80 and sells for $100, that is a 25% markup ($20 ÷ $80) and a 20% margin ($20 ÷ $100). Both describe the same $20, so quoting the markup when someone expects the margin overstates your profitability. The conversion is margin = markup ÷ (1 + markup), and markup = margin ÷ (1 − margin).

Divide every part by the greatest common divisor of all the parts. For 250 : 400 the GCD is 50, so the ratio reduces to 5 : 8. If your parts have decimals, multiply them all by the same power of ten first to get whole numbers, then reduce. A ratio is in lowest terms when the only whole number that divides all its parts is 1 — which is exactly the definition the calculator applies.

No — that is probably the single most expensive misreading in mixing, dosing and dilution. A ratio of 1 : 3 means one part of the first thing for every three of the second, so the whole is four parts and the first thing is one quarter, or 25%. One third would be written 1 : 2. Whenever a label gives a ratio, add the parts together before you convert it to a fraction or a percentage.

Only when the two quantities really are proportional, meaning their ratio stays constant across the whole range you care about. Doubling the flour doubles the number of cookies, so recipes work. Doubling the workers does not halve every project, because coordination and shared equipment break the proportionality. In a direct proportion x = B × C ÷ A; in an inverse proportion, where the product rather than the ratio is constant, x = A × B ÷ C.

Anchor on 10%, which is the number with the decimal point moved one place to the left, then build from there. 20% is double that, 5% is half of it, and 1% is the decimal point moved two places. So 15% of 240 is 24 + 12 = 36. The other handy trick is that X% of Y always equals Y% of X, so 4% of 75 is the same as 75% of 4, which is 3.

It means multiplying the two decimal forms. 50% of 20% is 0.50 × 0.20 = 0.10, that is 10%. This shows up in commission splits, conversion funnels and compound tax rules, and it is why a funnel with four 50% steps converts at 6.25% rather than 200%. In this hub, the "add or subtract a percent" branch with a second percentage does exactly this compounding for you.