Inverse variation is one of 2 variation formulas in the algebra section of this library, and it is used at high school level.
Why inverse variation works
Here it is the product that stays fixed rather than the ratio. Picture a job that always needs 24 worker-hours: more workers means proportionally fewer hours, and the product of the two never moves. Solving xy = k for y gives k divided by x.
What each symbol means
$k$ is the constant of variation.
Inverse variation: when it holds
$x\ne0$; for physical positive quantities, doubling $x$ halves $y$.
When it stops applying
The rule cannot handle x equal to zero, and it can never produce y equal to zero either. Both branches of the graph run alongside the axes forever without touching them, so any quantity that genuinely reaches zero is not modelled by inverse variation.
Inverse variation: a worked example
If $y=6$ when $x=4$, then $k=24$ and $y=24/x$.
The mistake to avoid
What people do: Setting up the same cross-multiplied proportion used for direct variation.
Why it goes wrong: With y = 6 at x = 4 the constant is 24, so at x = 8 the answer is 3. The direct-style proportion gives 12, which doubles the output when it should have halved it.
Do this instead: Multiply the pairs instead of dividing them: x1 times y1 equals x2 times y2 is the correct set-up for inverse variation.
Inverse variation: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $k$ is the constant of variation.
- Check the conditions before substituting. $x\ne0$; for physical positive quantities, doubling $x$ halves $y$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Variation
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where inverse variation comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Ratios and Proportions — the lesson behind this formula: compare quantities and scale equivalent relationships.
- Algebra Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about inverse variation
What does the graph look like?
Two separate curves called a hyperbola, one in each of two opposite quadrants. Each branch drops steeply near the vertical axis and flattens toward the horizontal axis without ever meeting it.
Is varying inversely with the square of x different?
Yes, that rule is y = k over x squared, and it drops much faster. Tripling x divides y by nine, which is how gravitational and light intensity problems behave.
How do I spot inverse variation in a word problem?
Look for one quantity rising while the other falls in the same proportion, and check that the product of matching pairs stays constant across the data given.
Can the constant be negative?
Yes. A negative constant puts the two branches in the other pair of opposite quadrants, though most physical problems use positive quantities and stay in the first quadrant.