Math
Inverse Variation Calculator
If y varies inversely with x, then y = k/x and the product x·y is the constant k.
Give the inverse variation calculator one known pair and it finds k exactly, writes the equation, and finds y for a new x or x for a new y, with the hyperbola graphed. Switch modes for joint variation (z = kxy) and combined variation (z = kx/y).
Part of a known pair (or triple). Cannot be 0.
The y that goes with x₁.
The x value to evaluate at.
Try an example
Result
Answer
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- Constant of variation k
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- Equation
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- Model
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- New values
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- Check
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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What the inverse variation calculator solves
Inverse variation (inverse proportion) means one quantity goes down as the other goes up so that their product stays fixed. Speed and travel time for a fixed distance, or pressure and volume of a gas at constant temperature, are classic examples. Joint variation makes z proportional to a product of variables; combined variation mixes direct and inverse variation.
Example: y = 6 when x = 4
- k = 4 × 6 = 24.
- Equation: y = 24/x.
- When x = 8: y = 24/8 = 3. Doubling x halved y.
- When y = 3: x = 24/3 = 8.
| x | 1 | 2 | 3 | 4 | 6 | 8 | 12 | 24 |
|---|---|---|---|---|---|---|---|---|
| y = 24/x | 24 | 12 | 8 | 6 | 4 | 3 | 2 | 1 |
How to use it
- Pick the type: inverse (y = k/x), joint (z = kxy), or combined (z = kx/y).
- Enter the known values from the problem to fix k.
- Enter the new value(s) and read the answer, k, and the equation. The check line shows the product or ratio is the same for both sets of values.
How to read the answer
k is the fixed product x·y (for inverse variation). The missing value is shown as an exact fraction with its decimal. A negative k puts the hyperbola in quadrants II and IV. For joint and combined variation the answer is the new z.
Common mistakes and edge cases
- Multiplying when you should divide: for inverse variation k = x·y, but y = k/x.
- Using x = 0 or y = 0: neither is possible in y = k/x.
- Confusing inverse variation with a negative How steep a line is: how much y changes for each step right in x.. y = -2x is direct variation with negative k; y = 2/x is inverse variation.
- In combined variation, put the direct variables in the The top number of a fraction; it counts how many parts you have. and the inverse ones in the The bottom number of a fraction; it says how many equal parts make one whole..
Worked examples
Find y for a new x
y = 6 when x = 4; find y when x = 8 → 3
Answer
3
Find x for a new y
y = 6 when x = 4; find x when y = 3 → 8
Answer
8
Fraction answer
y = 5 when x = 3; at x = 7, y = 15/7
Answer
15/7 (2.142857)
Negative constant
y = -2 when x = 6 gives k = -12; at x = 4, y = -3
Answer
-3
Joint variation
z = 24 when x = 4, y = 6 gives z = xy; at x = 8, y = 3, z = 24
Answer
24
Combined variation
z = 24 when x = 4, y = 6 gives z = 36x/y; at x = 8, y = 3, z = 96
Answer
96
x = 0 is not allowed
y = k/0 is undefined
Answer
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Frequently asked questions
What is inverse variation?+
A relationship y = k/x where the product of the two variables is constant. When one doubles, the other halves. It is also called inverse or indirect proportion.
How do I find the constant of variation for inverse variation?+
Multiply the known pair: k = x·y. For y = 6 when x = 4, k = 24 and the equation is y = 24/x.
What does the graph of inverse variation look like?+
A hyperbola with two branches that approach the axes but never touch them. For k > 0 the branches are in quadrants I and III; for k < 0 in quadrants II and IV.
What is joint variation?+
z varies jointly with x and y when z = kxy. The area of a rectangle varies jointly with its length and width (k = 1).
What is combined variation?+
A mix of direct and inverse variation, such as z = kx/y: z grows with x and shrinks as y grows. Gas laws and many physics formulas have this shape.
Can y be 0 in inverse variation?+
No. k/x is never 0 for k ≠ 0, and x = 0 makes the expression undefined, so neither variable can be 0.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
References
- OpenStax, College Algebra 2e, Section 5.8 “Modeling Using Variation” — Direct, inverse, joint, and combined variation.
- OpenStax, Intermediate Algebra 2e, Section 7.5 “Solve Applications with Rational Equations”
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Last updated: September 4, 2026