How this percentage calculator works
This page combines four tools: a basic triple (percent, base, result — any two find the third), common phrase layouts for typical word problems, percentage difference for symmetric comparisons, and percentage change for before-and-after situations with an increase/decrease mode when you apply a known percent to a known starting value.
What is a percentage?
A percentage expresses a ratio out of 100 — the symbol % means “per hundred.” Percentages relate to decimals (divide by 100) and fractions (e.g. 35% = 0.35 = 35/100). They are useful because they scale different-sized quantities onto a common scale for comparison.
Example: 25 out of 50 students is the ratio 25/50 = 0.5, and 0.5 × 100 = 50%.
The percentage formula (three values)
In decimal form, P × V₁ = V₂ where P is the percentage as a decimal, V₁ is the base, and V₂ is the result. The basic calculator lets you type percents in the usual way (e.g. 5 for 5%); it converts to a decimal internally. If you solve for the percentage, the answer is shown as a percent, not as 0.05.
Example: What percent is 1.5 of 30? P = 1.5 ÷ 30 = 0.05 → 5%.
Percentage difference vs percentage change
Difference (symmetric): |V₁ − V₂| ÷ ((V₁ + V₂)/2) × 100. Use when neither value is clearly the “original” — e.g. comparing two products or two measurements.
Example: 10 and 6 → |4| / 8 × 100 = 50% difference.
Change (directional): compare new to a chosen baseline (original). A common form is ((New − Old) ÷ Old) × 100. To apply a change: multiply by (1 ± p/100) for increase or decrease.
Examples: 500 increased by 10% → 500 × 1.1 = 550. Decreased by 10% → 500 × 0.9 = 450.
Everyday uses
- Tips: 10% = move the decimal one place left; 15% ≈ 10% + half of 10%; 20% = double 10%.
- Discounts: Sale price = original × (1 − discount/100), or subtract the discount amount from the original.
- Sales tax: Total = pretax × (1 + tax rate/100).
- Grades: Score % = (points earned ÷ points possible) × 100.
Percentage points vs percent change
Moving from 4% to 6% is a rise of 2 percentage points, but it can also be described as a 50% relative increase in the rate ((6 − 4) ÷ 4). Both are valid; they answer different questions.
Mental math symmetry
X% of Y = Y% of X — e.g. 8% of 25 = 25% of 8 = 2. Pick whichever is easier.
Common pitfalls
- Chaining +50% then −50% does not return to the start — percentages apply to the current value.
- Stacked “20% off then 15% off” is not the same as 35% off; multiply the remaining factors: 0.80 × 0.85.
- Reversing a price increase requires a smaller percent decrease than the original increase (different base).
More math tools: Scientific calculator · Math calculators.