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Instant Calculation
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In an inverse variation relationship, written xy=k (equivalently y=k/x), one quantity shrinks as the other grows, in exactly the proportion needed to keep their product constant. Double x and y is cut in half; triple x and y drops to a third of its original value — the two variables move in opposite directions, but always in a fixed, predictable ratio to each other, governed by the constant k. This relationship shows up throughout physics and everyday situations involving a fixed total being divided differently: travel speed and time for a fixed distance vary inversely (drive twice as fast and the trip takes half the time, since speed × time = distance, a constant), and gas pressure and volume at constant temperature follow the same pattern (Boyle's Law) — squeeze a gas into half the volume and its pressure doubles. This tool solves an inverse variation equation xy=k for whichever of the three values — k, x, or y — is unknown, given the other two.
How it works
Given any two of the three values {x, y, k}, the tool finds the third directly from the equation xy=k by rearranging as needed: if x and y are known, k=xy; if k and x are known, y=k/x; if k and y are known, x=k/y. Because the relationship requires the product of x and y to equal k, and division by zero is undefined, neither x nor y can ever be zero in a valid inverse variation with a nonzero k — the tool checks for and rejects such invalid inputs.
- Enter solve for.
- Enter x (leave blank if solving for x).
- Enter y (leave blank if solving for y).
- Enter k (leave blank if solving for k).
- Click Calculate to see your results.
Examples
Finding the constant of variation
If x=4 and y=3, then k=4×3=12, so y=12/x.
Finding y given k and x
If k=12 and x=6, then y=12/6=2.
Finding x given k and y
If k=20 and y=5, then x=20/5=4, since rearranging xy=k to solve for x gives x=k/y.
Who should use it
- Algebra coursework on inverse variation and proportional relationships.
- Word problems involving inversely proportional quantities (e.g. speed and time, or pressure and volume).
- Checking whether a data set represents an inverse (rather than direct) proportional relationship.
Industry applications
- Mathematics education
- Physics (e.g. Boyle's Law-style relationships)
Advantages
- Solves for any of the three unknowns (k, x, or y).
- Rejects invalid zero inputs that would make the relationship undefined.
- Applicable to common physics and real-world inverse relationships.
Limitations
- Only handles the simple xy=k form, not more complex inverse relationships.
Common mistakes to avoid
- Assuming x or y can be 0 in an inverse variation — since xy=k with k≠0, neither variable can ever equal zero.
- Confusing an inverse variation's constant PRODUCT (x×y=k) with a direct variation's constant RATIO (y/x=k) — checking the wrong one leads to a false conclusion about which relationship applies.
- Assuming inverse variation means "y decreases as x increases" always follows a straight line — it actually follows a curved hyperbola, not a straight line like direct variation.
Best practices
- Check that a real-world relationship's product (not ratio) stays constant before modeling it as inverse variation.
- Verify the constant k using more than one data point when possible, to confirm the relationship genuinely holds throughout, not just for one pair.
- Keep track of units when computing k, since it carries the combined units of x times y.
Tips
- A quick check for inverse variation in a data table: if every x×y product is the same constant, the relationship is an inverse variation.
- Remember the graph is a curve (hyperbola), not a straight line — a helpful visual distinction from direct variation.