Enter two known values — x₁, y₁, and a new x₂ (or y₂) — and the Inverse Variation Calculator solves for the missing variable using the relationship y = k/x. You also get the proportionality constant k so you can verify or extend the relationship to any other pair of values.
Unknown Variable Result
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Constant of Proportionality (k)
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Product Verification (x × y = k)
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Inverse Variation Equation
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Inverse variation (also called inverse proportion) describes a relationship where two variables change in opposite directions at a proportional rate. When one variable increases, the other decreases such that their product always equals a constant k. This is expressed as y = k/x or equivalently xy = k.
The proportionality constant k is found by multiplying the two known values: k = x₁ × y₁. For example, if x = 4 and y = 10, then k = 40. Once you know k, you can find any unknown value using y = k/x or x = k/y.
A relationship is an inverse variation if the product of the two variables (x × y) is always the same constant. Key phrases to watch for include 'varies inversely', 'inversely proportional', 'varies indirectly', or 'indirectly proportional'. The graph of an inverse variation is a hyperbola, not a straight line.
Substituting x = 8 into y = 40/x gives y = 40/8 = 5. The constant of proportionality is k = 40, and any pair of values (x, y) along this curve will always multiply to 40.
The graph of inverse proportionality (a hyperbola) never crosses either axis. As x approaches zero, y approaches infinity, and as x approaches infinity, y approaches zero — so the curve gets infinitely close to both axes but never touches them. This is why x = 0 is not allowed in inverse variation.
In direct variation, y = kx — both variables increase or decrease together, and their ratio y/x is constant. In inverse variation, y = k/x — when one variable increases the other decreases, and their product x×y is constant. Direct variation graphs are straight lines through the origin; inverse variation graphs are hyperbolas.
Yes. Inverse variation can extend beyond simple linear variables. For example, y might vary inversely with x², giving y = k/x². In such cases the same principle applies — the product of y and the expression in x equals the constant k — but the curve and calculations differ from the basic hyperbola.
Division by zero is mathematically undefined. Since y = k/x, setting x = 0 would require dividing k by zero, which has no defined value. Geometrically, this is why the hyperbola approaches but never touches the y-axis (the line x = 0).