The Complete Guide to Percentages

Last updated: 2026-06-25

The basics: percent means "per hundred"

Percent (%) comes from the Latin "per centum" — "per hundred." It expresses a ratio against a base of 100, so 50% means 50 out of 100, that is, 0.5. The simplest, most reliable way to find B% of a value is to multiply by B and divide by 100. For example, 10% of a $50,000 item is 50,000 × 10 ÷ 100 = $5,000.

People often find percentages hard not because the formula is complicated, but because what counts as the base (the whole, the 100%) changes from situation to situation. The same "20%" gives completely different results depending on which value it is 20% of, or what it is an increase on. So when working with percentages, always start by deciding what the base 100% is.

1) Finding X% of A

This is the most basic type. The formula is A × (B ÷ 100). If 35% of a class of 240 students wears glasses, that is 240 × 0.35 = 84 students. Conversely, to ask "A is what % of B," divide the part by the whole and multiply by 100: part ÷ whole × 100. Asking what percent 84 is of 240 gives 84 ÷ 240 × 100 = 35%, matching exactly. These two calculations are really the same equation read in opposite directions.

2) Percentage change: the base is the denominator

Percentage change (rate of change) expresses how much a value grew or shrank as a ratio. The formula is (new value − old value) ÷ old value × 100. The key is that the denominator is always the old (base) value. If monthly revenue rose from $1,000,000 to $1,200,000, that is (1,200,000 − 1,000,000) ÷ 1,000,000 × 100 = +20%. Going the other way, from 1,200,000 to 1,000,000, is (1,000,000 − 1,200,000) ÷ 1,200,000 × 100 ≈ −16.7%.

Here's the interesting part. Going from 100 to 120 is +20%, but coming back from 120 to 100 is not −20% — it is about −16.7%, because the denominator changed. So the intuition that "up 20% then down 20% lands you back where you started" is wrong. $100 up 20% is $120; down 20% from there is $96, lower than the original.

3) Percent (%) vs percentage points (pp)

This is the most commonly confused point in the news. When a value already measured in percent (interest rate, unemployment, approval rating, market share) changes, you must use "percentage points (pp)" to be precise. If the base rate rises from 3% to 5%, that is a 2 pp increase. But the same change expressed as a percentage change is (5 − 3) ÷ 3 × 100 ≈ 66.7%. Both are correct but mean very different things. "Rates went up 66%" is dramatic but misleading, so changes in percent-unit values should be stated in percentage points.

4) Discounts and markups: 1 ± B/100

Discounts and markups follow one principle — multiply by a ratio. A B% discount multiplies the original by (1 − B/100); a B% markup multiplies by (1 + B/100). A $300 item at 25% off is 300 × (1 − 0.25) = 300 × 0.75 = $225. A 25% markup is 300 × 1.25 = $375. This "multiplier" way of thinking is the key to understanding stacked discounts next.

5) Stacked discounts: multiply, don't add

"20% off plus an extra 10% off" looks like 30% off, but that is a common trap. Stacked discounts multiply the multipliers, not add the rates. 20% off is ×0.8, 10% off is ×0.9, so 0.8 × 0.9 = 0.72 — the final price is 72% of the original and the real discount is 28%, less than 30%. Reverse the order and 0.9 × 0.8 = 0.72 gives the same result, because multiplication is commutative.

6) Reversing sales tax: ÷1.1 and ÷11

The reverse calculation people use most in daily life is sales tax. When the tax rate is 10%, the tax-inclusive total is 1.1 times the net amount. So to get the net amount from a tax-inclusive total, use total ÷ 1.1, and for the tax alone, use total ÷ 11. If a receipt total is $33,000, the net amount is 33,000 ÷ 1.1 = $30,000 and the tax is 33,000 ÷ 11 = $3,000. Mistakenly computing "10% of the total is the tax" gives $3,300, which is off from the real figure, so be careful.

Three habits that reduce confusion

• Define the base (100%) first — Decide what is the whole and what is the part, and which value is the denominator, before anything else.
• Check the direction — Increase and decrease are not symmetric because the denominator differs. Remember that "up then down by the same %" does not return you to the start.
• Distinguish the units — Express changes of rates in percentage points (pp) and changes of ordinary values in percent (%).

Once you grasp the principles, percentages become a powerful tool across discounts, taxes, statistics and investment returns. To test these types with real numbers, switch modes in the percentage calculator. If you need to work with fractions alongside ratios, try the fraction calculator, and to explore the mean and spread of data, the average & standard deviation calculator.

Last updated: 2026-06-25