Joint Variation Calculator

This calculator works out joint variation, where one quantity depends on the product of two others, using the formula z = k x y. It is common in maths and science: the area of a triangle varies jointly with its base and height, and many physics formulas tie an outcome to two driving quantities multiplied together, which is what separates joint variation from simple direct variation. To use it, enter a known set of x, y and z, values you already know fit the relationship, and the calculator divides z by the product of x and y to find the constant k. Then enter a new x and y, and it applies z = k x y to predict the matching z. You get three results: the full formula with your constant filled in, the constant k itself, and the predicted z for your new values, all updating instantly as you type so you can test several pairs without re-entering anything. It suits students checking homework, teachers preparing examples, and anyone wanting a fast, reliable answer instead of working the algebra by hand. Just make sure your known x and y are not zero, since dividing by zero has no answer, and remember results are rounded for display. A worked example and the full formula are set out below if you want to follow the method step by step.

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Known x

Known y

Known z

New x

New y

z = 2 x y

Constant k2

Predicted z40

The formula

Joint variation has the form z = k x y, where k is the constant. Find k from a known triple as k = z / (x y), then predict any z from z = k x y.

Worked example

If z = 12 when x = 2 and y = 3, then k = 12 / (2 times 3) = 2, so z = 2xy. At x = 4, y = 5, z = 2 times 4 times 5 = 40. Enter 2, 3, 12, 4, 5 to confirm.

Frequently asked questions

What is joint variation?

A relationship where one quantity varies as the product of two others, z = kxy.

How is it different from direct variation?

Direct variation has one driving variable; joint variation has two, multiplied together.

What is a real example?

The area of a triangle varies jointly with its base and height, with constant one half.

Who this calculator is for

This calculator is for students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard formula is applied.
Results are rounded for display.

Formula and sources

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Scientific Calculator: general scientific maths.
Percentage Calculator: percentages and changes.
Ratio Calculator: simplify and scale ratios.

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