Constant of Proportionality Calculator (k = y/x from a table; direct or inverse)

Understanding the constant of proportionality calculator — concept and principle

This calculator finds the constant of proportionality k from a table of x and y values, for a direct (y = kx) or inverse (y = k ÷ x) relationship. It computes k for every pair, checks whether they agree and writes the equation. Use it for middle-school ratio problems, recipe scaling, unit prices and science data.

Direct proportion is a straight line through the origin with slope k.

How it is calculated

Direct proportion: y = kx, so k = y ÷ x for each pair

Inverse proportion: y = k ÷ x, so k = x × y for each pair

If every pair gives the same k (to rounding), the relationship is proportional and the equation is shown.
x must not be 0 for a direct relationship, because y ÷ x would be undefined.

The principle

Two quantities are directly proportional when doubling one doubles the other, so their ratio never changes. That fixed ratio is k, the unit rate: here, how much y you get per one unit of x. On a graph a direct proportion is a straight line through the origin with slope k. In an inverse proportion, doubling one halves the other, so their product stays fixed. A table where y ÷ x changes, such as y = 2x + 1, is linear but not proportional.

Worked example

Default inputs: pairs (2, 7), (4, 14), (6, 21), (10, 35), direct.

k from each pair: 7 ÷ 2, 14 ÷ 4, 21 ÷ 6, 35 ÷ 10 = 3.5, 3.5, 3.5, 3.5.
Smallest and largest k: 3.5 … 3.5, from 4 pairs.
Proportional? yes — every pair gives the same k.
Constant of proportionality k = 3.5; equation y = 3.5x.

How to read the result

k is the amount of y per unit of x: if x is kilograms and y is euros, k is the price per kilogram. Use the equation to predict new values: at x = 8, y = 3.5 × 8 = 28. If the k values differ, check the data for a typo or a fixed starting amount; a relationship with a fixed fee is linear but not proportional.

Common mistakes

Dividing x by y instead of y by x.
Calling any straight line proportional, even one that misses the origin.
Using the ratio from only one pair without checking the others.
Mixing direct and inverse relationships.

Keep in mind: Measured data rarely give exactly equal ratios. Small differences may be measurement error; large ones mean the relationship is not proportional.

Frequently asked questions

How do you find the constant of proportionality from a table?

Divide each y by its x; if all the results are the same, that value is k.

What does the constant of proportionality mean?

It is the unit rate: how much y changes for one unit of x.

How do I know a relationship is inversely proportional?

The product x × y is the same for every pair.

Sources: OpenStax, Elementary Algebra, §8.9 Use Direct and Inverse Variation

Checked 9 Nov 2026. OpenStax, Elementary Algebra (Marecek & Anthony-Smith), §8.9 Use Direct and Inverse Variation: y varies directly with x if y = kx (k ≠ 0, the constant of variation), inversely if y = k/x; find k by substituting a known pair (e.g. y = 20 when x = 8 gives k = 2.5).

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