Inverse Proportion Calculator

Solve inverse variation (y = k/x) problems instantly. Find constant of variation, missing values, and visualize the hyperbola.

Inverse proportion formula: y = k / x   or   x·y = k (constant of variation)

If two variables are inversely proportional, their product is constant: x₁·y₁ = x₂·y₂

Range of x-values to display on the graph
Calculating...

Understanding Inverse Proportion

Two variables are inversely proportional if their product is constant. As one variable increases, the other decreases proportionally. The graph is a hyperbola.

Mathematical Definition:

If a non‑zero constant k exists such that y = k / x (equivalently x·y = k), then y is inversely proportional to x. k is called the constant of variation.

y ∝ 1/x ⇔ x·y = k

Comparison: Direct vs Inverse Proportion

Property Direct (y = kx) Inverse (y = k/x)
Graph Straight line through origin Hyperbola (two branches)
Change pattern y increases when x increases y decreases when x increases
Constant relation y/x = k (ratio constant) x·y = k (product constant)
Asymptotes None x = 0 and y = 0
Example Distance = speed·time (fixed speed) Time = distance/speed (fixed distance)

Key Properties

  • Product rule: For any two points (x₁,y₁) and (x₂,y₂) on the curve, x₁·y₁ = x₂·y₂ = k.
  • Symmetry: If (a,b) lies on the graph, then (b,a) also lies on it (because a·b = b·a). The curve is symmetric about the line y = x when k>0.
  • Asymptotes: The coordinate axes are asymptotes; the function is undefined at x = 0.
  • Negative constant: If k < 0, the branches lie in the second and fourth quadrants (x and y have opposite signs).

Real‑World Examples

  • Physics: Boyle’s law: P·V = constant (pressure inversely proportional to volume for a fixed amount of gas at constant temperature).
  • Motion: For a fixed distance, speed and travel time are inversely proportional: t = d / v.
  • Optics: Light intensity I ∝ 1/d² (inverse square law).
  • Economics: With a fixed budget, the quantities of two goods you can buy are inversely proportional.
  • Work rate: Number of workers and time to complete a job (assuming same work rate) satisfy workers × time = constant.
  • Geometry: For a rectangle of fixed area, length and width are inversely proportional.

Clearing Common Misconceptions

  • Misconception: “Inverse proportion just means y decreases when x increases.” — Not exactly; the product must stay constant. For example, y = 1/x + 1 decreases but is not inversely proportional.
  • Misconception: “The graph always lies in the first and third quadrants.” — Only when k > 0. If k is negative, the graph lies in the second and fourth quadrants.
  • Misconception: “x and y can be zero.” — No, because k ≠ 0 and the expression k/x is undefined at x = 0; zero would force k = 0, which is a degenerate case.
  • Misconception: “Every rational function of the form 1/x is inverse proportion.” — Only functions exactly of the form y = k/x, not translations like y = 1/(x+1).

Deriving the Equation from a Point

Given a point (x₁, y₁), the constant is k = x₁·y₁. Therefore the inverse proportion equation is:

y = (x₁·y₁) / x

If you know another x₂, then y₂ = k / x₂; if you know y₂, then x₂ = k / y₂. This is the basis of our calculator.

Why “Inverse”?

When x is multiplied by a factor t, y is multiplied by 1/t — the opposite (inverse) factor. Formally: if x becomes t·x, then y becomes k/(t·x) = (1/t)·(k/x) = (1/t)·y.

Frequently Asked Questions

The calculator will check if the product x₂·y₂ equals k (within a small tolerance). If not, it will show that the points are not inversely proportional. If they match, it confirms consistency.

Yes! If k is negative, one variable is positive while the other is negative. The graph lies in the second and fourth quadrants. Our calculator supports negative k.

In inverse proportion, neither variable can be zero because the product k would be zero, but then the relationship is trivial. Also, the function y = k/x is undefined at x=0. The calculator will warn if you try to use zero.

The graph plots the function y = k/x over the chosen x-range, skipping x=0. It accurately shows the hyperbolic shape and the two points you entered.