Percentage change calculator

Definition

Percentage change (relative change × 100) from an old value b to a new value a is the ratio of the increment to the base, expressed in percent: p = ((a − b) / b) · 100 when b ≠ 0. The absolute increment is Δ = a − b. The relative fraction is r = Δ / b; the percent is p = 100 · r. This calculator implements exactly ((a − b) / b) * 100 in the browser’s floating-point arithmetic — without an absolute value in the denominator.

The definition is standard in business mathematics, YoY/MoM reporting, and school algebra. MathWorld (Percentage) and Wikipedia (Relative change and difference) state percent change as (new − old)/old · 100. The |old| variant matches when old > 0; this page keeps the classical denominator b, matching the page formula for procent-zmiany.

In teaching practice it helps to separate the mathematical formula from floating-point limits: this calculator shows the browser’s floating-point computation (about 15–16 significant digits), not a paper proof of symbolic identities. When inputs are extreme, check the domain first, then the floating-point range.

Historical and conceptual context

Percent (Latin per centum — per hundred) settled in trade, tax, and bookkeeping. Percentage change compares the increment with the initial base, not with the final value. Hence the asymmetry: +25% from 80 to 100 is not the mirror of −25% on the way back — on the way back you divide by 100 and get −20%. Price, revenue, and traffic indexes use the same logic.

Conceptually this calculator does not compute percentage points between two rates (pp) and does not solve “what is the new value after p%” — those questions live on the sibling percentage-change page. It is also not a VAT calculator or a percent-of-one-amount tool. It answers one question: by what percent the new value departs from the old relative to the old base.

In teaching practice it helps to separate the mathematical formula from floating-point limits: this calculator shows the browser’s floating-point computation (about 15–16 significant digits), not a paper proof of symbolic identities. When inputs are extreme, check the domain first, then the floating-point range.

Formal properties

Domain: b ≠ 0, a ∈ ℝ (in the school form usually a, b ≥ 0). Result p ∈ ℝ. p > 0 ⇔ a > b; p = 0 ⇔ a = b; p < 0 ⇔ a < b. Identities when b ≠ 0: a = b · (1 + p/100); p = (a/b − 1) · 100; r = a/b − 1.

The reverse change from a to b is ((b − a)/a) · 100 when a ≠ 0 — the panel shows it as a separate readout. Chaining two percentage changes on shifting bases is not ordinary percent addition: +10% then −10% does not return exactly to the start, because the second step has a new base.

In teaching practice it helps to separate the mathematical formula from floating-point limits: this calculator shows the browser’s floating-point computation (about 15–16 significant digits), not a paper proof of symbolic identities. When inputs are extreme, check the domain first, then the floating-point range.

Full derivation

Step 1. Increment Δ = a − b.

Step 2. Reference scale: S = b, requiring b ≠ 0.

Step 3. Relative fraction r = Δ / S = (a − b) / b.

Step 4. Percent p = 100 · r = ((a − b) / b) · 100.

Step 5 (recovery). Given p and b: a = b + (p/100) · b = b · (1 + p/100). Check: ((a − b)/b) · 100 = p.

Step 6 (example). a = 100, b = 80 → Δ = 20, r = 0.25, p = 25. a = 80, b = 100 → Δ = −20, r = −0.2, p = −20.

In teaching practice it helps to separate the mathematical formula from floating-point limits: this calculator shows the browser’s floating-point computation (about 15–16 significant digits), not a paper proof of symbolic identities. When inputs are extreme, check the domain first, then the floating-point range.

Special cases

Examples: 80 → 100 gives +25%; 100 → 80 gives −20%; 50 → 50 gives 0%; 200 → 250 gives +25%; 10 → 15 gives +50%; 15 → 10 gives (−5)/15 · 100 ≈ −33.333%. Doubling a positive base: +100%.

b = 0 with nonzero a is an error (division by zero). Very small |b| yields huge |p| — a feature of the formula. The form does not take negative amounts; the algebra would work on negatives the same way, but here we keep a non-negative domain.

In teaching practice it helps to separate the mathematical formula from floating-point limits: this calculator shows the browser’s floating-point computation (about 15–16 significant digits), not a paper proof of symbolic identities. When inputs are extreme, check the domain first, then the floating-point range.

Numerical notes

Computation: pct = ((a - b) / b) * 100. The calculator accepts a comma or a period as the decimal separator. Locale en-US / pl-PL affects display only.

Compared with the |b| variant: when b > 0 the results match. When b < 0 the classical formula flips the scale through a negative denominator; the |b| variant keeps a positive base. This page does not use |·| — matching the page formula.

In teaching practice it helps to separate the mathematical formula from floating-point limits: this calculator shows the browser’s floating-point computation (about 15–16 significant digits), not a paper proof of symbolic identities. When inputs are extreme, check the domain first, then the floating-point range.

Related notions and sources

Related: percent of a number, percentage change with |old|, rule of three (proportion), ratio of two numbers, percentage points (a separate scale). For VAT and margin use the finance calculators.

Sources: Wikipedia — Relative change and difference; MathWorld — Percentage; NIST / business-mathematics textbooks (percent change); statistical agencies’ relative-index practice (context, not a separate formula). Educational tool — not a textbook substitute.