Inverse Variation Calculator

Created by Krishna Nelaturu
Reviewed by Anna Szczepanek, PhD and Steven Wooding
Last updated: Dec 19, 2022

Our inverse variation calculator is here to help you observe two variables inversely proportional to each other. If you have two parameters that depend on each other and wish to check whether they vary inversely, you've come to the right place!

In this companion article, we shall briefly look at calculating inverse proportionality, including:

  • What is an inverse variation?
  • Inverse variation equation.
  • Inverse variation examples.
  • Some FAQs.

You can also assess the direct variation between two variables with our direct variation calculator.

Inverse variation equation of two variables

We say two variables are inversely varying (or inversely proportional) if an increase in one variable results in a proportional decrease in the other.

We can write the inverse proportion between two variables as:

y1x,y \propto \frac{1}{x},

where:

  • yy – A dependant variable; and
  • xx – An independent variable.

Note that xx must be non-zero. Using a constant of proportionality kk, we can get the formula for inverse variation as:

y=kxy = \frac{k}{x}

Naturally, inverse proportionality graphs reflect a reciprocal function.

Plot of a typical inverse variation.
Inverse variation graph.

How do you calculate the proportionality constant of inverse variation?

Once you've determined that two variables, x and y, are inversely proportional, calculating their proportionality constant is easy:

  1. Multiply the variables x and y to get k = x · y.
  2. Verify your answer with Omni's inverse variation calculator.

Examples of inverse variation

Let's look at a few examples of inverse variation in real life:

  • Speed and time: The speed of an object is inversely proportional to the time it needs to cover a given distance. A car moving at 60 mph will take 100 minutes to cover 100 miles, whereas a vehicle moving at 50 mph will need 120 minutes to cross the same distance.
Speed1Time\qquad \rm{Speed} \propto \frac{1}{Time}
  • Newton's law of gravitational force: The gravitational force FGF_G between two masses is inversely proportional to the square of the distance rr between them. So if an Earth-sized planet orbits the Sun at a larger distance than us, it would feel a weaker gravitational pull than we do.
FG1r2\qquad F_G \propto \frac{1}{r^2}

For more on gravitational force, head to our gravitational force calculator.

  • Coulomb's law: Similar to the gravitational force, the electromagnetic force FF between two charged particles is inversely proportional to the square of the distance rr between them.
F1r2\qquad F \propto \frac{1}{r^2}

You can learn more about this force with our Coulomb's law calculator.

How to use this inverse variation calculator

This inverse variation calculator requires only two inputs from you:

  1. The independent variable xx; and
  2. The constant of inverse variation kk.

From these two values, this inverse variation equation calculator will give you the following outputs:

  1. The dependent variable yy; and
  2. If x>0x>0, the calculator will also show you a graph of the inverse variation, so you can visually assess the variation.

But you don't need to hold back – You can enter any two knowns into this inverse proportion calculator to find the remaining unknown.

FAQ

How do you recognize inverse variation?

Recognizing inverse variation between two variables is possible once you have multiple values:

  1. Collect the values of these variables at multiple data points to get: (x1, y1), (x2, y2), (x3, y3),....

  2. Multiply the x-term of each data point with its corresponding y-term, to get: (x1 · y1), (x2 · y2), (x3 · y3),....

  3. To tell if the variables vary inversely or not:

    • If the result is constant, i.e., if (x1 · y1) = (x2 · y2) = (x3 · y3)...., the variables vary inversely.

    • If the result keeps varying, the variables do not vary inversely.

What is y in the inverse variation y = 14/x, at x = 4?

y = 3.5. These steps will help you arrive at the same conclusion:

  1. Use x = 4 in the inverse variation equation y = 14/x to get y = 14/4.
  2. Solve the inverse variation equation to get y = 3.5.
  3. Verify the result with our inverse variation calculator.

Which axis does the graph of inverse proportionality cross?

The graph of inverse proportionality cannot cross or intersect any axis. Looking at the inverse variation equation y = k/x, we can see that neither x nor y can be zero.

Krishna Nelaturu
y = k/x
Inverse proportionality constant (k)
Independent variable (x)
Dependent variable (y)
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