Direct & Inverse Variation Calculator
Find the constant of variation from a known pair, then predict y for a new x under direct (y = kx) or inverse (y = k/x) variation.
How it works
Direct: k = y ÷ x, then y = k·x · Inverse: k = y·x, then y = k ÷ x
Two quantities vary directly when their ratio is constant (y = kx): double x and y doubles. They vary inversely when their product is constant (y = k/x): double x and y halves. In both cases you first find the constant of variation k from one known (x, y) pair, then use it to predict y for any new x. The constant k is what stays fixed across the whole relationship.
Worked example
Suppose y varies directly with x and y = 12 when x = 4. Then k = 12 ÷ 4 = 3, so the rule is y = 3x. At x = 6, y = 3 · 6 = 18. If instead the variation were inverse, k = 12 · 4 = 48 and y = 48 ÷ 6 = 8.
Frequently asked questions
How do I tell direct from inverse variation?
In direct variation the ratio y ÷ x stays constant, so as one grows the other grows proportionally. In inverse variation the product x · y stays constant, so as one grows the other shrinks. If doubling x doubles y it is direct; if doubling x halves y it is inverse.
What is the constant of variation?
It is the fixed number k that links the two quantities: k = y ÷ x for direct variation, or k = x · y for inverse variation. Once you know k from any single pair, you can find y for any x, which is the whole point of the model.
Why can't x be zero in these formulas?
For direct variation, finding k needs y ÷ x, so the known x must be non-zero. For inverse variation, predicting y needs k ÷ x, so the new x must be non-zero. The calculator guards both cases instead of dividing by zero.
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This calculator is for educational purposes. Double-check important results before acting on them.