A constant product — and two values the relationship simply does not have.
Inverse variation
The product xⁿ × y is what stays constant.
y varies inversely as
k = 12
y = 12 / x
k = x × y = 4 × 3 = 12
Constant of variation
12
the product every pair shares
x = 6
y = 2
y = 12 ÷ 6 = 2
Doubling x
halves y
because y follows 1/x
A few values on this variation
Values of x and y on this inverse variation
x
1
2
3
4
6
y
12
6
4
3
2
Double x and y halves; triple x and y is a third of what it was. The product x × y stays at 12 for every pair on the curve, and checking that product is the fastest way to test whether data is an inverse variation.
x = 0 is excluded from the domain and y = 0 from the range. Both are asymptotes: the curve gets arbitrarily close to each axis and touches neither. This is the sharpest difference from a direct variation, which passes exactly through the origin.
The graph is a hyperbola in two separate branches, in the first and third quadrants because k is positive. The two branches never meet and the curve never crosses either axis.
What this tool shows
A constant product, two asymptotes, and two branches that never meet — every difference from direct variation follows from the first of those.
k as the constant product xⁿy, with predictions in both directions
The inverse-square case, where doubling x quarters y
x = 0 and y = 0 handled with explanations rather than errors
Joint variation z = kxy and combined variation z = kx/y
Constant product, not ratio Asymptotes as content, not errors Joint and combined variation too Free, no signup
Free, no signup — exact products, asymptotes explained.
Updated 6 September 2026 · Works in any browser, no installation
“y varies inversely as x” means the product x × y never changes. From the pair (4, 3) that product is 12, so the relationship is y = 12 / x — and at x = 6 it gives y = 2.
At a glance
Formula shown
y = k/x\u207f, where k = x\u207fy is the constant of variation. The graph is a hyperbola with asymptotes at x = 0 and y = 0.
Scenario support
One known pair, plus a value to predict from \u2014 in either direction.
Educational estimate
Planning support from the values you enter — not professional advice.
The product, not the ratio
Direct variation keeps y ÷ x fixed. Inverse variation keeps x × y fixed. That one substitution accounts for every other difference between the two topics.
k = x × y = 4 × 3 = 12. Every pair on the relationship multiplies to the same 12: (1, 12), (2, 6), (3, 4), (12, 1). Checking a table is a multiplication, not a division.
y = 120 / x is the everyday version. A journey of 120 km at 60 km/h takes 2 hours; at 40 km/h it takes 3. Speed up and the time drops in exact proportion, because the distance — the product — has not changed.
Two values that are simply not there
x = 0 is not a hard case, a special case, or an error case. It is not in the domain: k/0 has no value, and there is no point on the graph above it. The curve runs off to infinity as x approaches zero from either side.
y = 0 is excluded too, and for a related reason: k/x = 0 has no solution when k is not zero. A non-zero numerator over anything at all is never zero. So the curve approaches the x-axis without ever meeting it.
Both are ASYMPTOTES, and this is the sharpest contrast with direct variation, which passes exactly through the origin — the one point an inverse variation is furthest from having. Asking either question in the tool gets the explanation rather than an error message, because the explanation is the answer.
The hyperbola, and which quadrants it lives in
The graph is two separate branches, and the sign of k decides which pair of quadrants they occupy. The graph is a hyperbola in two separate branches, in the first and third quadrants because k is positive. The two branches never meet and the curve never crosses either axis.
y = −20 / x puts them in the other pair: The graph is a hyperbola in two separate branches, in the second and fourth quadrants because k is negative. The shape is identical, reflected.
The two branches never join, and that is a genuine feature rather than a drawing artefact — there is no x at which the curve crosses from one to the other, because it would have to pass through x = 0 to do it. Reading a hyperbola as one continuous curve is the commonest misreading of this graph.
The inverse-square law
“y varies inversely as the square of x” is y = k/x², and now it is x²y that is constant. k = x² × y = (2)² × 9 = 4 × 9 = 36, giving y = 36 / x².
Double x and y is divided by 4; triple x and y is divided by 9. That is the inverse-square law — the reason light, gravity and sound all fall off the way they do.
This is the shape of most of physics that spreads out from a point. Light, gravity, electrostatic force and sound intensity all fall off as the inverse square of distance, and the reason is geometric: the same total spreads over a sphere whose area grows as r². Twice as far away means a quarter as intense — not half, which is the intuition that has to be unlearned.
Joint and combined variation
When more than two quantities are involved the same single step still does the work. Joint variation is z = kxy — z varies directly as both. Combined variation mixes the two, z = kx/y.
Substitute the known case: 24 = k × 12, so k = 24 ÷ 12 = 2. gives z = 2xy, and at the second set of values z = 2 × 30 = 60.
z = 10x / y² is the combined case, with y in the denominator and squared: Substitute the known case: 10 = k × 1, so k = 10 ÷ 1 = 10. These are not new techniques. Substitute the known case to fix k, then substitute the new one — the only thing that changes is which variables multiply and which divide.
Telling the two apart from a table
Given a table and no wording, one calculation settles which relationship it is: divide for direct, multiply for inverse.
Every product x × y comes to 12, so this is an inverse variation with k = 12. for (1, 12), (2, 6), (3, 4) — the products are 12, 12, 12, all equal.
(1, 3), (2, 6), (4, 12) gives products 3, 12, 48, which grow — so it is not inverse. It happens to be a direct variation instead, with a constant ratio of 3, and the Direct Variation Calculator is where that test lives. The two tools ask opposite questions of the same table, and between them the answer is unambiguous.
Sources and methodology
Inverse proportionality is a definition rather than a cited result; the reference below covers the elementary algebra it sits in.
Method. Every figure on this page comes from src/lib/inverse-variation.ts over the exact rational arithmetic in src/lib/algebra/poly.ts. The product test compares exact fractions rather than rounded decimals. x = 0 and y = 0 return an explanation of why they are outside the domain and range rather than a division-by-zero result. That engine is verified on every change against 45 hand-written assertions, including that the predicted pair always reproduces the same constant product, and that both excluded values are reported rather than computed. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Direct VariationFind the constant of variation from one pair and predict any other, for y = kx or y = kx^n — and test whether a table really is a direct variation or just a straight line that misses the origin.
Percentage ChangeWork out percentage increase or decrease, reverse change, loss recovery, percentage points, multi-period change, and CAGR.
Doubling TimeHow long a quantity takes to double at a constant rate, or from two measurements — with the rule of 70 and rule of 72 measured against the exact answer.
Quadratic FormulaSolve any quadratic with exact roots — surds stay surds and a negative discriminant gives the complex pair — plus the vertex, the factored form and every step of the working.
This calculator handles inverse variation, y = k/xⁿ, along with joint and combined variation, using exact rational arithmetic. x = 0 is genuinely outside the domain and y = 0 is outside the range — both are reported with the reason rather than as errors. Direct variation, where the ratio rather than the product is constant, is a different relationship with its own page.
Published the inverse variation page: k as the constant PRODUCT rather than a ratio, with predictions in both directions and the power case.
Treats the asymptotes as content: x = 0 is not in the domain and y = 0 is not in the range, and asking for either gets an explanation rather than an error.
Covers joint and combined variation in the same place, since they are the same constant-finding step with more variables.
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