proportionalities

Proportionalities describe relationships between two quantities in which one quantity changes in a consistent, multiplicative way with respect to the other. There are two common forms: direct proportionality, where the ratio of the quantities remains constant, and inverse proportionality, where their product remains constant. When a quantity y is directly proportional to x, there is a constant k such that y = kx. When y is inversely proportional to x, there is a constant k such that xy = k, equivalently y = k/x.

Direct proportionality means the constant of proportionality k determines how steeply y grows with x. The graph is a straight line through the origin with slope k. Inverse proportionality produces a hyperbolic relationship, where increasing x reduces y and their product remains k. The constant k is often called the proportionality constant and may carry units, reflecting the physical quantities involved.

Examples illustrate the two forms. Direct proportionality: doubling the amount of material while keeping other factors fixed doubles the resulting quantity. Inverse proportionality: doubling speed while traveling a fixed distance halves the travel time. Proportionalities underpin scaling and dimensional analysis in science and engineering, enabling simple predictions from changes in one variable. They also support proportional reasoning in everyday contexts, such as recipe adjustments or budgeting, while reminding practitioners to track units and ensure clarity about which quantity is held constant.