Inverse Variation Calculator
Enter any two of the three values (x, y, or k) and this calculator finds the third using the inverse variation formula y = k/x. Choose what you want to solve for, enter the two known values, and the answer appears instantly with a step-by-step breakdown and a live chart of the full curve.
What is inverse variation?
Inverse variation (also called inverse proportion) describes a relationship between two variables x and y where their product is always a constant: x * y = k. Equivalently, y = k/x. When x increases, y decreases by the same factor, and vice versa. For example, if you double x, y is cut in half. This is fundamentally different from direct variation (y = k*x), where both variables move in the same direction. The graph of y = k/x is a hyperbola with two branches that never touch either axis, because neither x nor y can be zero in this relationship.
How to use this calculator
Select what you want to solve for using the 'Solve for' dropdown: y (the dependent variable), x (the independent variable), or k (the constant of variation). Then enter the two known values in the fields that appear. The unknown is computed instantly using the rearranged form of y = k/x. To solve for y, enter x and k. To solve for x, enter y and k. To find k from a known (x, y) data point, select k and enter both coordinates. The 'Show your work' panel displays every algebraic step, and the chart plots the full hyperbola for your k value so you can see how the entire curve behaves.
The three forms of the inverse variation formula
The base equation y = k/x can be rearranged to isolate any of the three values. To find y: y = k/x. To find x: multiply both sides by x and divide by y, giving x = k/y. To find k: multiply both sides by x, giving k = x*y. This last form is particularly useful for identifying inverse variation in a data set: if you multiply each x by its corresponding y and always get the same number, the data follows an inverse variation with that number as k. If the products differ, the data is not an inverse variation.
Inverse variation vs. direct variation
In direct variation (y = k*x), doubling x also doubles y, and the graph is a straight line through the origin. In inverse variation (y = k/x), doubling x halves y, and the graph is a hyperbola that never reaches either axis. A quick test: for direct variation, the ratio y/x is constant. For inverse variation, the product x*y is constant. Both are special cases of a broader power-law relationship y = k * x^n, where direct is n = 1 and inverse is n = -1. Real-world examples of inverse variation include Boyle's Law (pressure and volume of a gas at constant temperature), Ohm's Law when voltage is fixed (current and resistance), and the relationship between speed and travel time for a fixed distance.
Common real-world inverse variation relationships
| Relationship | x variable | y variable | What k represents | Example |
|---|---|---|---|---|
| Speed and time | Speed (mph or km/h) | Travel time (hours) | Fixed distance | 100 km trip: t = 100 / s |
| Boyle's Law | Volume (L) | Pressure (atm) | n * R * T (gas constant) | PV = constant at fixed T |
| Electrical resistance and current | Current (A) | Resistance (ohm) | Fixed voltage (V) | R = V / I (Ohm's Law) |
| Gravitational force vs. distance | Distance squared (m2) | Force (N) | G * m1 * m2 | F = G*m1*m2 / r^2 |
| Worker hours to complete a job | Number of workers | Hours to finish | Total man-hours required | 10 workers, 8 h job: 80/w |
| Wavelength and frequency | Frequency (Hz) | Wavelength (m) | Speed of light (c) | lambda = c / f |
Each relationship follows y = k/x where k is determined by a fixed physical or situational constant.
Frequently asked questions
What is the formula for inverse variation?
The formula is y = k/x, where k is called the constant of variation (or constant of proportionality). It can also be written as x*y = k. Rearranging: to find y, use y = k/x; to find x, use x = k/y; to find k, use k = x*y. The constant k is always the product of any corresponding x and y pair on the curve.
How do I find the constant of variation k?
Multiply any known (x, y) pair: k = x * y. For example, if a car traveling at 60 mph takes 2 hours to cover a route, the constant is k = 60 * 2 = 120 (miles). You can then predict travel time at any other speed: at 80 mph the trip takes 120/80 = 1.5 hours.
How do I know if a data table shows inverse variation?
Multiply each x value by its corresponding y value. If all products are equal (or very close, allowing for rounding), the data shows inverse variation and that product is k. If the products differ, it is not an inverse variation. For direct variation, check if y/x is constant instead.
Can x or y be zero in inverse variation?
No. Because y = k/x, dividing by zero is undefined, so x can never be zero. Similarly, if k is non-zero, y can never be zero either (you would need x to be infinite). This is why the graph of y = k/x is a hyperbola with two branches that approach but never reach either axis.
What is the difference between inverse variation and inverse proportion?
They are the same concept. 'Inverse proportion' is the more common term in British English, while 'inverse variation' is standard in American algebra courses. Both describe y = k/x where increasing one variable decreases the other by the same factor. 'Directly proportional' and 'direct variation' are similarly equivalent.
What does the graph of inverse variation look like?
The graph is a hyperbola with two branches. When k is positive, the branches sit in quadrants I (both x and y positive) and III (both negative). When k is negative, the branches are in quadrants II and IV. The curve approaches both axes but never crosses them. As x grows large, y shrinks toward zero; as x shrinks toward zero, y grows without bound.