Inverse Variation Calculator (y = k/x: constant of variation & missing value)

Understanding the inverse variation calculator — concept and principle

This calculator works with inverse variation y = k ÷ xⁿ. From one known pair it finds the constant of variation k, writes the equation, and gives the missing y for a new x or the missing x for a new y. Use it for algebra word problems on speed and travel time, workers and job time, or inverse-square laws.

x times y stays at 60, so a bigger x means a proportionally smaller y.

How it is calculated

k = y₁ × x₁ⁿ

New y = k ÷ xⁿ; new x = (k ÷ y)^(1/n)

Factor of change in y = new y ÷ y₁

n = 1 is plain inverse variation; n = 2 is an inverse-square law.
Check: y × xⁿ stays equal to k. x = 0 is not allowed.

The principle

In an inverse variation the product y·xⁿ never changes. If one quantity goes up, the other must go down to keep that product fixed. With n = 1, doubling x halves y: twice the speed means half the time for the same distance. With n = 2, doubling x divides y by 4, as with light or sound intensity spreading out from a source.

Worked example

Default inputs: x₁ = 4, y₁ = 15, n = 1, find y for x = 6.

Constant of variation k = 15 × 4 = 60.
Equation: y = 60 / x; known pair (4, 15).
New y = 60 ÷ 6 = 10 (what was found: y for the new x).
Factor of change in y = 10 ÷ 15 = 0.6666666667, the reciprocal of 6 ÷ 4.
Check: y × x = 10 × 6 = 60 = k.

Each point is the calculator re-run with only “New x or new y” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

New x or new y	Answer
1.5	40
3	20
4.5	13.33333333
6	10
7.5	8
9	6.666666667
12	5

How to read the result

If 4 workers need 15 days for a job, 6 workers need 10 days, assuming everyone works at the same rate and the job can be shared. The factor of change in y is the reciprocal of the factor of change in x when n = 1, and its square when n = 2.

Common mistakes

Dividing y by x to find k, which is direct variation.
Forgetting the power in inverse-square problems.
Applying inverse variation when there is a fixed overhead, such as setup time.
Using x = 0.

Keep in mind: Real situations are rarely perfectly inverse; more workers can get in each other's way. Use the model within a sensible range.

Frequently asked questions

What is inverse variation?

A relationship y = k ÷ xⁿ in which the product y·xⁿ stays constant.

How do I find k?

Multiply y by xⁿ for any known pair.

What is an example of inverse variation?

Travel time and speed over a fixed distance.

Sources: OpenStax, College Algebra 2e (2021), §5.8 Modeling Using Variation

Checked 9 Nov 2026. OpenStax, College Algebra 2e (2021), §5.8 Modeling Using Variation (inverse variation y = k/xⁿ; k is the constant of variation).

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