Inverse Variation
Free Inverse Variation calculator with clear step-by-step results.
About Inverse Variation
Two quantities vary inversely when their product stays constant: x x y = k. Here the constant is recovered by multiplying the known pair, and the prediction follows from y = k / x. Doubling x therefore halves y, and tripling x cuts y to a third — the exact opposite of direct variation. The curve is a hyperbola that never touches either axis, so both the known and the target x must be non-zero; as x approaches zero, y grows without bound.
Formulas
Constant product
k = x x y (from the known pair)
- k
- Constant product
- x
- Known input
- y
- Known output
Prediction
y_new = k / x_new
- x_new
- The x you want a prediction for
Worked example
With the default values below, Inverse Variation returns Predicted y of 2.0000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Known x
- 4
- Known y
- 3
- Predict y at x =
- 6
Results
- Predicted y
- 2.0000
- Constant product kk = x x y, identical for every valid pair.
- 12.0000
- Change in y
- -33.33%
- Variation Equation
- y = 12 / x
Step by step
- 1Values used
- 2Constant product
- 3Prediction
- 4Predicted y= 2.0000
- 5Constant product k= 12.0000
- 6Change in y= -33.33
- 7Variation Equation= y = 12 / x
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
How is inverse variation different from direct variation?
In direct variation the ratio y/x is constant, so x and y rise together. In inverse variation the product x times y is constant, so raising x lowers y. Direct variation graphs as a straight line through the origin; inverse variation graphs as a hyperbola.
How do I test data for inverse variation?
Multiply x by y for each pair. If every product lands on the same number, the data varies inversely and that number is k. Constant speed over a fixed distance is a classic example: travel time times speed always equals the distance.
Why is x = 0 rejected?
y = k / x divides by x, so x = 0 has no defined output. Mathematically the hyperbola has a vertical asymptote there: as x shrinks toward zero, y races off toward infinity.