Formula and Method for Inverse Variation
Two quantities x and y show inverse variation (also called inverse proportion) when their product stays constant: y = k / x, where k is the nonzero constant of variation. As one variable grows, the other shrinks by the same factor, so x·y = k for every matching pair on the relationship. This calculator generalizes the idea to y = k / xⁿ, where n = 1 is basic inverse variation and n = 2 is inverse-square variation, so you can model both.
How the calculation works
Enter one known pair (x₁, y₁) that satisfies the relationship, along with the power n (use 1 unless the problem specifies otherwise, such as an inverse-square law). The calculator finds the constant of variation with k = x₁ⁿ × y₁. Once k is known, the full equation y = k / xⁿ is fixed, so entering a new x-value (x₂) lets the calculator solve for the corresponding y₂ with y₂ = k / x₂ⁿ. The verification result multiplies x₂ⁿ by y₂ to confirm it returns the same k, which is a quick way to check that the relationship holds.
Common mistakes
- Confusing inverse with direct variation: in direct variation y = kx, y grows as x grows. In inverse variation y = k/x, y shrinks as x grows — the two are opposites.
- Using x = 0: inverse variation is undefined at x = 0 because division by zero has no value; the graph of y = k/x has a vertical asymptote there.
- Forgetting the power n: not every inverse relationship is 1/x — inverse-square (n = 2) and inverse-cube (n = 3) laws are common in physics and require raising x to that power, not just dividing once.
Real-world applications
- Speed and travel time: at a fixed distance, time = distance / speed, so doubling speed halves travel time.
- Physics inverse-square laws: light intensity, gravitational force, and electric field strength all weaken with the square of distance (n = 2).
- Boyle's law in chemistry: at constant temperature, gas pressure and volume vary inversely (P = k / V).
- Work-rate problems: the number of workers needed to finish a job in a given time varies inversely with the time allowed.