The inverse variation calculator solves problems where one quantity goes down as another goes up, at a fixed rate. Enter one known pair (x₁, y₁) and either a new x₂ or a new y₂, and the calculator finds the constant of variation k, writes the equation, and solves for the missing value. It handles ordinary inverse variation y = k/x, inverse square variation y = k/x², inverse square-root variation y = k/√x, and any power y = k/xⁿ, and it lists a table of values so you can check your homework.
Inverse Variation Calculator
Result
Solved y = 36/x, x₂ = 9
y₂ = 4
Constant k
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Equation
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The Variation Curve
Both known and solved points sit on the same curve. Hover to slide a point along it and watch the shaded product stay equal to k.
x Up, y Down
As x steps up, the bars for y shrink, while the line for xⁿ·y stays perfectly flat at k.
x y xⁿ·y = k
What is Inverse Variation?
Two quantities vary inversely when their product is constant: x·y = k, or y = k/x, with k ≠ 0. The number k is called the constant of variation or constant of proportionality.
Because the product is fixed, a change in one variable forces the opposite change in the other. If x doubles, y halves. If x is divided by 5, y is multiplied by 5.
- Constant product: x₁·y₁ = x₂·y₂ = k for every pair.
- Graph: a hyperbola that never touches either axis.
- Area picture: every rectangle from the origin to a point on y = k/x has the same area, k.
- Language: "y varies inversely as x", "y is inversely proportional to x", and "y ∝ 1/x" all mean the same thing.
Interactive: Same Area, Different Shapes
Drag x. The rectangle stretches and squashes, but its area x·y never changes.
How to Solve Inverse Variation Problems?
Find k from the known pair, write the equation, then substitute the new value.
- Write the model. "y varies inversely as x" gives y = k/x.
- Find k. With x = 3 and y = 12, k = 3 × 12 = 36.
- Write the specific equation. y = 36/x.
- Substitute. At x = 9, y = 36/9 = 4.
A shortcut skips k entirely: x₁·y₁ = x₂·y₂, so y₂ = x₁·y₁/x₂. For inverse square variation use x₁²·y₁ = x₂²·y₂, which means y changes by the square of the scale factor.
Interactive: Scale x, Watch y
Multiply x by any factor. For y = k/xⁿ, y is divided by that factor raised to the power n.
How to Use This Inverse Variation Calculator?
Pick the type of variation, enter the known pair, enter the new value, and press solve.
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Choose the model
k/x for "inversely", k/x² for "inversely as the square", k/√x for "inversely as the square root", or k/xⁿ for any power.
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Enter x₁ and y₁
One known pair from the problem. The calculator computes k = x₁ⁿ·y₁ and shows the equation.
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Find y₂ or x₂
Use the tabs to solve for the new y or the new x. The table below the result lists more points on the same curve.
For square-root and fractional powers, x must be positive. Negative x works for whole-number powers.
Direct Variation vs Inverse Variation
Direct variation keeps the ratio y/x constant; inverse variation keeps the product x·y constant.
| Direct | Inverse | |
|---|---|---|
| Equation | y = kx | y = k/x |
| Constant | k = y/x | k = x·y |
| x doubles | y doubles | y halves |
| Graph | line through origin | hyperbola |
| Example | cost vs. items bought | time vs. speed |
To tell them apart from a table, compute both y/x and x·y for every row. Whichever stays the same identifies the type. If neither does, the data shows neither kind of variation.
Interactive: Spot the Difference
Both curves pass through the same point. Move it to see how the line and the hyperbola respond.
direct y = kxinverse y = k/x
Inverse Variation Examples Table
Classic textbook problems with their constants. Click a row to load it into the calculator.
| Problem | Model | k | Answer |
|---|---|---|---|
| y = 12 when x = 3; find y at x = 9 | y = k/x | 36 | y = 4 |
| 2 h at 60 km/h; time at 80 km/h | t = k/v | 120 | t = 1.5 h |
| 6 workers take 10 days; 4 workers | d = k/w | 60 | d = 15 days |
| Gas at 100 kPa in 2 L; pressure at 5 L | P = k/V | 200 | P = 40 kPa |
| 440 Hz has wavelength 0.78 m; 330 Hz | λ = k/f | 343.2 | λ = 1.04 m |
| 100 lux at 2 m; light at 4 m | I = k/d² | 400 | I = 25 lux |
| g = 9.81 at Earth's radius; at 2 radii | g = k/r² | ≈ 3.98 × 10⁸ | g ≈ 2.45 m/s² |
| y = 10 when x = 4; y at x = 16 | y = k/√x | 20 | y = 5 |
| y = 5 when x = 2; y at x = 4 | y = k/x³ | 40 | y = 0.625 |
Real-World Inverse Variation: Boyle's Law, Speed and Work
Inverse variation appears whenever a fixed total is shared or spread out.
- Boyle's law: at constant temperature, pressure × volume is constant. Squeeze a gas to half its volume and its pressure doubles.
- Speed and time: for a fixed distance d, time = d/speed. Going 25% faster cuts the time by 20%, not 25%.
- Workers and days: a job needing 60 worker-days takes 10 days with 6 people or 15 days with 4.
- Gears: the number of teeth times the rotation speed is the same for two meshing gears.
Interactive: Squeeze the Gas
Push the piston. The molecules bump the walls more often as the space shrinks, so pressure rises as volume falls.
Joint and Combined Variation
Combined variation mixes direct and inverse variation in one formula, such as y = k·z/x: y grows with z and shrinks with x.
Newton's law of gravitation is the classic example: F = G·m₁·m₂/r². The force varies jointly with both masses and inversely with the square of the distance.
- Joint variation: y = k·x·z (both factors push y up).
- Combined variation: y = k·z/x or y = k·z/x².
- Solving: plug one complete set of values in to find k, then use k for the new values, exactly as in simple inverse variation.
Example: if y = 10 when z = 4 and x = 2, then k = y·x/z = 5. When z = 9 and x = 3, y = 5·9/3 = 15.
Interactive: Gravity Between Two Masses
Change the masses and the distance. Force grows with each mass and falls with the square of the distance.
Other Inverse Calculators
Related tools that solve problems in reverse.
Inverse Laplace Calculator
Turns F(s) back into f(t) with partial fractions, a pole-zero map, and a live time-response graph.
(3s+5)/(s²+3s+2) → 2e⁻ᵗ + e⁻²ᵗ Open calculator Inverse Modulo Calculator
Finds a⁻¹ mod m with the extended Euclidean algorithm, a modular clock, and a congruence solver.
17⁻¹ mod 43 = 38 Open calculator Inverse Normal Calculator
Returns the x value or z-score for any area under a normal curve, with left, right, and central tails.
invNorm(0.975) = 1.9600 Open calculator Inverse Square Calculator
Solves I₁d₁² = I₂d₂² for light, sound, gravity, and radiation, plus the reciprocal 1/x².
2× distance → 1/4 intensity Open calculator Inverse Function Calculator
Finds f⁻¹(x) by swapping x and y and solving, with the y = x mirror graph and a domain check.
f(x) = 2x + 3 → (x − 3)/2 Open calculator Inverse Matrix Calculator
Inverts 2×2 to 6×6 matrices by Gauss-Jordan elimination with exact fractions and every row step.
A · A⁻¹ = I Open calculator Inverse Logarithm Calculator
Takes the antilog in base 10, e, 2, or any base, with characteristic, mantissa, and a log ruler.
antilog(2.5) = 316.23 Open calculator Inverse Poisson Calculator
Finds the count k for a target probability, or the rate λ, with PMF bars and a CDF staircase.
qpois(0.95, 10) = 15 Open calculator Frequently Asked Questions
Answers to common questions about inverse and inverse square variation
y = 4. First find the constant: k = x·y = 3 × 12 = 36. Then y = 36/9 = 4. Tripling x divided y by 3, which is the signature of inverse variation.
Multiply a known pair: k = x·y for y = k/x. For inverse square variation y = k/x², use k = x²·y, and for y = k/xⁿ use k = xⁿ·y. Any one data point is enough, because k is the same for every pair.
A hyperbola that falls as x grows and approaches both axes without touching them. For k > 0 the curve sits in the first and third quadrants. It never crosses y = 0 because k/x is never zero, and it is undefined at x = 0.
In direct variation y = kx, so y/x is constant and doubling x doubles y. In inverse variation y = k/x, so x·y is constant and doubling x halves y. The direct graph is a straight line through the origin, while the inverse graph is a hyperbola.
No. The +2 breaks the rule because x·y is no longer constant. At x = 1, x·y = 7, but at x = 5, x·y = 15. A true inverse variation has the form y = k/x with no added constant, so the product x·y must give the same k for every point.
Travel time and speed: for a fixed distance, time = distance/speed. A 120 km trip takes 2 hours at 60 km/h and 1.5 hours at 80 km/h. Other examples are Boyle's law (P·V = k for a gas at constant temperature) and workers versus days to finish a job.
y = k/x², where y is divided by 4 whenever x doubles. Light intensity, gravity, and electrostatic force all follow it. If a light gives 100 lux at 2 m, k = 400, and at 4 m it gives 400/16 = 25 lux.