Direct and Inverse Proportion

Direct and Inverse Proportion

Two quantities are in direct proportion if doubling one doubles the other — their ratio stays constant. They are in inverse proportion if doubling one halves the other — their product stays constant. Both types appear numerically (tables, word problems), graphically (linear and hyperbolic curves), and algebraically (equations with a constant of proportionality).

Direct Proportion

\( y \propto x \) means \( y = kx \) for some constant \( k \). Graph: straight line through origin. Key tests: \( y/x \) is constant; both quantities equal zero together.

  • Find \( k \): use any known pair \( (x_1, y_1) \): \( k = y_1/x_1 \).
  • Predict: \( y_2 = k \times x_2 \).

Inverse Proportion

\( y \propto \frac{1}{x} \) means \( y = \frac{k}{x} \), equivalently \( xy = k \). Graph: a hyperbola — decreasing curve that approaches both axes but never touches them. Key test: the product \( xy \) is constant.

  • Find \( k \): \( k = x_1 y_1 \) (the constant product).
  • Predict: \( y_2 = k/x_2 \).
y=kx Direct x y Inverse y=k/x x y

Other Proportional Relationships

More complex proportions also arise:

  • \( y \propto x^2 \): \( y = kx^2 \) — graph is a parabola through the origin.
  • \( y \propto \sqrt{x} \): \( y = k\sqrt{x} \) — graph is a half-parabola through the origin.
  • \( y \propto \frac{1}{x^2} \): \( y = \frac{k}{x^2} \) — inverse square law.

Worked Examples

\( y \propto x \). When \( x = 4 \), \( y = 10 \). Find \( y \) when \( x = 14 \).
\[ k = \frac{10}{4} = 2.5 \qquad y = 2.5 \times 14 = 35 \]
\( y \propto \frac{1}{x} \). When \( x = 6 \), \( y = 4 \). Find \( y \) when \( x = 8 \). Find \( x \) when \( y = 12 \).
\[ k = 6 \times 4 = 24 \qquad y = \frac{24}{8} = 3 \qquad x = \frac{24}{12} = 2 \]
\( y \propto x^2 \). When \( x = 3 \), \( y = 36 \). Find \( y \) when \( x = 5 \).
\[ k = \frac{36}{9} = 4 \qquad y = 4 \times 25 = 100 \]
\( y \propto \frac{1}{x^2} \). When \( x = 2 \), \( y = 9 \). Find \( y \) when \( x = 6 \).
\[ k = 9 \times 4 = 36 \qquad y = \frac{36}{36} = 1 \]

 Key Takeaways

  • Direct: \( y = kx \), ratio \( y/x \) is constant, graph is a straight line through origin.
  • Inverse: \( y = k/x \), product \( xy \) is constant, graph is a hyperbola.
  • Find \( k \) using any known pair of values; then use \( k \) to find unknowns.
  • For \( y \propto x^n \): \( k = y/x^n \). For \( y \propto 1/x^n \): \( k = y \cdot x^n \).
  • Distinguish type from the table: constant \( y/x \) → direct; constant \( xy \) → inverse.