This direct and inverse proportion calculator handles the two classic variation problems. In a direct proportion, y = kx: doubling x doubles y. In an inverse proportion, y = k / x: doubling x halves y. Enter one complete pair (x₁, y₁), choose the type, and give one value of the second pair; the calculator finds the constant of variation k, writes the equation, solves the missing value, and verifies the answer by substitution.

All arithmetic is exact. Fractions, mixed numbers, and decimals stay rational throughout, so k = 5/2 is reported as 5/2, not 2.4999….

02Input

How to Use the Direct and Inverse Proportion Calculator

Choose the type of proportion. Direct means the quantities rise and fall together; inverse means one rises as the other falls. The type comes from the situation, not from the numbers—see the guide below.
Enter the known pair x₁ and y₁. Both values are required, and x₁ cannot be zero, because the constant is calculated from this pair.
In the second pair, enter either x₂ or y₂ and leave the other blank. The blank field is the one that gets solved.
Read the constant k, the equation, the substitution, and the verification line. If the situation has no answer—for example, asking which x makes y = 0 in an inverse proportion—the calculator says so explicitly instead of returning a number.

Inputs accept integers, decimals, fractions such as 3/4, and mixed numbers such as 1 1/2.

03Calculation

Direct and Inverse Proportion Formulas

A direct proportion keeps the ratio of the quantities constant:

y = kx, with k = y₁ / x₁

An inverse proportion keeps the product of the quantities constant:

y = k / x, with k = x₁ · y₁

Both problems follow the same three moves: find k from the known pair, write the equation, and substitute the known value of the second pair. For a direct proportion this is equivalent to the rule of three y₂ = y₁ · x₂ / x₁, which is the same cross-multiplication the proportion calculator performs on a / b = c / d. For an inverse proportion the rule of three does not apply—the working identity is x₁ y₁ = x₂ y₂.

The testy/x vs x·y

To identify the type from data, check what stays constant: if y / x is the same for every pair, the proportion is direct; if x · y is the same, it is inverse. If neither is constant, neither model fits and this calculator’s equations do not apply.

04Recognize

How to Tell Direct from Inverse

Choosing the wrong type is the main source of wrong answers in variation problems. The three checks below identify the model before any algebra.

ADirecty = kx

Direct: More of One, More of the Other

Cost against quantity, distance against time at fixed speed, ingredient amounts in a scaled recipe. The graph is a straight line through the origin, and y / x never changes.

BInversex · y = k

Inverse: More of One, Less of the Other

Speed against travel time for a fixed distance, workers against days for a fixed job, pressure against volume at fixed temperature. The graph is a hyperbola, and x · y never changes.

CNeitherno k

Neither: Check Before Calculating

Many pairs rise together without being proportional—a line y = kx + b with b ≠ 0, for instance. If the ratio and the product both drift, no variation constant exists, and forcing one produces a confident wrong answer.

05Worked example

Worked Examples for Both Types

Direct proportion. Four notebooks cost $10. What do six cost?

Constant: k = y₁ / x₁ = 10 / 4 = 5/2 dollars per notebook.
Equation: y = (5/2)x.
Substitute: y₂ = 5/2 × 6 = 15. Six notebooks cost $15.
Verify: 15 = 5/2 × 6 holds, so the pair fits the variation.

Inverse proportion. Four painters finish a job in 10 hours. How long do eight painters take?

Constant: k = x₁ · y₁ = 4 × 10 = 40 painter-hours.
Equation: y = 40 / x.
Substitute: y₂ = 40 / 8 = 5. Eight painters take 5 hours.
Verify: 8 × 5 = 40 matches k, so the answer is consistent.

The two problems use the same numbers and opposite structures: the first keeps the ratio 10 : 4 fixed, the second keeps the product 40 fixed. Applying the direct method to the painters would give 20 hours—more painters taking longer—which the situation itself rules out.

Direct variation keeps y / x fixed and grows with x; inverse variation keeps x · y fixed and shrinks.

06Validity

Common Variation Mistakes

Applying the rule of three to inverse data: cross-multiplication assumes a direct proportion. For inverse problems the identity is x₁ y₁ = x₂ y₂; mixing them up inverts the answer’s direction.
Zero where the model forbids it: k = y / x is undefined at x = 0, and y = k / x has no point at x = 0 at all. The calculator rejects these inputs rather than dividing by zero.
Assuming every relationship is proportional: a taxi fare with a base charge rises with distance but is not direct variation. Test the ratio or product before trusting the model.
Unit mismatches: if x₁ is in hours and x₂ in minutes, k absorbs the inconsistency and the answer is silently wrong. Convert units before entering values.
Expecting a number when none exists: in an inverse proportion with k ≠ 0, no x produces y = 0. The calculator reports “no solution” for such questions, which is the mathematically correct answer.

07Interpretation

Reading Variation Results

What does the constant k actually mean?

k is the rate that defines the relationship. In a direct proportion it is the amount of y per unit of x—dollars per notebook, kilometers per hour. In an inverse proportion it is the fixed total—painter-hours of work, distance of the trip—that the two quantities share.

Is direct proportion the same as the rule of three?

The rule of three is the shortcut form of a direct proportion: it jumps straight from three known values to the fourth. This page shows the fuller k-first method because k carries meaning and because the same structure extends to inverse variation, where the shortcut fails.

Can k be negative or a fraction?

Yes to both. A fractional k such as 5/2 is routine, and a negative k describes quantities moving in opposite signed directions along a line through the origin. The calculator keeps either exact.

What if my data has more than two pairs?

Two pairs determine at most one variation model; several pairs can test whether the model actually holds. Use the proportional relationship calculator to check a whole table for a constant ratio and get k from all rows at once.

08Method

Method, Limitations, and Sources

The calculator computes k from the known pair with exact rational arithmetic, solves the single missing value, and verifies the completed pair against the defining identity—y₂ = k·x₂ for direct, x₂·y₂ = k for inverse. It handles one variation at a time; it does not fit joint or combined variation (z = kxy), power models (y = k/x²), or noisy data where k should be estimated rather than derived.

The definitions and worked structure follow OpenStax Intermediate Algebra’s chapter on direct, inverse, and joint variation and the Common Core State Standards for Mathematics, standard 7.RP.A.2 (CCSSO). Those sources cover the mathematics; this page adds exact arithmetic, explicit verification, and honest no-solution reporting.