Proportion, Equations and Graphs: Direct and Inverse
Ratio, proportion and rates • Ratio and proportion
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Key concepts
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Proportion as equality of ratios
Proportion means two ratios are equal, written as a:b = c:d or a/b = c/d. Ratios convert directly into fractions so cross-multiplication provides a reliable algebraic test: a·d = b·c. Units must match across corresponding quantities for the proportion to hold; mismatched units cause incorrect setup.
Direct proportion and its equation
Direct proportion means one quantity is a constant multiple of another and is written y = kx, where k is the constant of proportionality. The constant k equals the ratio y/x for any corresponding pair of values. A graph of direct proportion is a straight line through the origin with gradient k; the line passes through (x, kx) for all x in the domain.
Inverse proportion and its equation
Inverse proportion means one quantity is proportional to the reciprocal of another and is written y = k/x, often described as ‘y is inversely proportional to x’. The product xy equals the constant k for all corresponding pairs. The graph of inverse proportion is a rectangular hyperbola with x and y axes as asymptotes; x and y cannot equal zero simultaneously, so domain restrictions apply (x ≠ 0, y ≠ 0).
Constructing equations from context
Identification of direct or inverse relationship comes from language and units in the problem. Known pairs of values supply the constant k by substitution into y = kx or y = k/x. The equation then models the situation and allows prediction, rearrangement to find unknowns, and algebraic checks for consistency. Clear specification of units and domain limits prevents invalid solutions.
Relating ratios, fractions and linear functions
A ratio a:b converts to the fraction a/b and can represent the gradient of a straight line as rise/run. A linear function y = mx + c reduces to a proportional relationship only when c = 0; then the gradient m equals the constant of proportionality. Interpretation of ratio as fraction, slope and multiplier unifies arithmetic and algebraic approaches.
Gradient as rate of change
The gradient of a straight line equals the change in the vertical variable per unit change in the horizontal variable (rise/run). In direct proportion graphs, the gradient equals the constant of proportionality and therefore represents a constant rate of change. The gradient calculation requires consistent units on both axes to produce meaningful rates.
Interpreting and solving from graphs
Direct proportion graphs appear as straight lines through the origin; the gradient indicates the multiplier. Inverse proportion graphs appear as hyperbolas with horizontal and vertical asymptotes, and points satisfy xy = k. Graphical solutions include reading coordinates, using gradients for linear models, finding intersections to solve simultaneous relationships, and estimating values by interpolation; graphical answers carry approximation error and require statement of units and accuracy.
Key notes
Important points to keep in mind