Percentage change calculator

Definition

The percentage change from an old value to a new value is the ratio of the increment to a reference scale, expressed in percent. In this calculator the primary formula is p = ((new − old) / |old|) · 100, provided old ≠ 0. The denominator uses the absolute value |old|, so the direction of change (the sign of p) comes from the difference new − old, while the scale stays positive even when old is negative. The absolute difference is Δ = new − old.

This model is standard in finance and descriptive statistics when reporting rise/fall against a base. MathWorld and business-math texts often write percent change as (new − old)/old · 100; the |old| variant agrees whenever old > 0, and when old < 0 it avoids flipping the scale through a negative denominator. Computation: diff / |oldV| * 100.

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

Historical and conceptual context

Percent (Latin per centum — per hundred) solidified in trade and tax; percentage change compares an increment to a base, not to the new value. YoY/MoM indicators and returns use this logic. Continuous exponential growth e^{rt} − 1 is a compounding limit — a separate model from a one-shot percent change.

Conceptually the calculator does not compute percentage points between two rates (pp) unless you choose the helper mode. It is also not a VAT or margin calculator — only the relation old → new.

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

Formal properties

Domain: old ≠ 0, new ∈ ℝ (change mode). Result p ∈ ℝ. p > 0 means an increase in the sense of a positive difference against the positive scale |old|; p = 0 ⇔ new = old; p = −50 means Δ = −0.5·|old|.

Identities when old > 0: new = old · (1 + p/100); p = (new/old − 1)·100. When old < 0 the calculator still divides by |old|, so new = old + (p/100)·|old| in “old + p% → new” mode.

The reverse change from new to old is ((old − new)/|new|)·100 when new ≠ 0 — the panel may show it as a separate readout.

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

Full derivation

Step 1. Increment Δ = new − old.

Step 2. Reference scale in this tool: S = |old| > 0.

Step 3. Relative fraction r = Δ/S; percent p = 100·r.

Step 4 (recovery). Given p and old: new = old + (p/100)·|old|. Check: ((new − old)/|old|)·100 = p. When old > 0 this coincides with the classical (new − old)/old·100.

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

Special cases

Examples: 100 → 120 gives +20%; 80 → 60 gives −25%; 50 → 50 gives 0%; −10 → −5 gives +50% with denominator |−10| = 10 and Δ = 5. Doubling old > 0: +100%.

old = 0 with nonzero new is an error (division by zero). Very small |old| yields huge |p| — a feature of the formula, not a defect of the calculator.

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

Numerical and computational notes

Computation: pct = (diff / |oldV|) * 100. The calculator accepts a comma, a period, and grouping spaces, depending on the page language. Modes “old + p% → new” and the inverse use the same |old| scale.

The calculator does not annualize a rate (APR/APY) automatically.

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

Related concepts and authorities

Related: exponential function (continuous compounding), linear interpolation (a different interpolation between points), min–max. For VAT and margin use the percentage calculators on this site.

Authorities: business-mathematics texts; MathWorld — Percentage; Wikipedia — Relative change and difference; NIST uncertainty guides (relative changes). Educational tool.

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