Constant of Proportionality Calculator
Find the constant of proportionality k from a single (x, y) pair. Choose direct variation (y = kx) or inverse variation (y = k/x), enter your x and y values, and get k, the full equation, and a doubling check that shows how the relationship behaves.
Example: with Type of variation Direct: y = kx (y grows with x) · x value 4 · y value 12 → Constant of proportionality: k = 3.
- Equationy = 3x (y / x = 3 for every pair)
- Doubling checkAt x = 8, y = 24 — doubling x doubles y
Computed by the calculator below using its default values. Change any input to see your own numbers.
Direct variation: k = y / x, the unit rate. Inverse variation: k = x · y, the constant product. A relationship is proportional only if k comes out the same for every pair.
What k actually tells you
In direct variation, k is the unit rate — how much y you get per one unit of x. If 4 tickets cost $12, then k = 12 / 4 = 3 dollars per ticket, and the whole relationship is y = 3x. On a graph, direct proportionality is a straight line through the origin with slope k; if the line misses the origin, the relationship is linear but not proportional.
Inverse variation keeps the product constant instead of the ratio. If 6 workers finish a job in 4 days, k = 6 × 4 = 24 worker-days, so y = 24 / x: doubling the workers halves the days. The doubling check in the results is the quickest way to tell the two apart — direct doubles y, inverse halves it.
How it’s calculated
Direct variation: k = y ÷ x, giving y = kx. Inverse variation: k = x × y, giving y = k / x. Default example: x = 4, y = 12 gives k = 3 and y = 3x. Results are rounded to 6 decimal places; x = 0 is rejected because k is undefined there in both models.
A single pair determines k only if the relationship really is proportional — verify with a second pair (same y/x ratio, or same x·y product) before trusting the equation.
Direct variation with k = 3 (y = 3x)
| x | y | y / x |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 6 | 3 |
| 4 | 12 | 3 |
| 10 | 30 | 3 |
Computed with y = 3x; a constant y / x column is the signature of direct proportionality.
Common mistakes
- Computing x / y instead of y / x — with x = 4 and y = 12, k is 3, not 0.333. Check that y = kx reproduces your pair.
- Calling a linear relationship proportional when it has an intercept: y = 3x + 2 has no constant of proportionality.
- Using the direct formula for an inverse problem — if y drops as x rises, k is the product x·y, not the ratio.
- Trusting k from one pair without testing a second pair for the same ratio or product.
Frequently asked questions
What is the constant of proportionality formula?
For direct variation, k = y / x and the equation is y = kx. For inverse variation, k = x × y and the equation is y = k / x.
How do I find k from a table?
Divide y by x in every row (direct) — if you get the same number each time, that number is k. For inverse variation, multiply x by y in every row instead and look for a constant product.
Is the constant of proportionality the same as slope?
For direct variation, yes: y = kx is a line through the origin with slope k. But a line with a y-intercept, like y = 3x + 2, has a slope and no constant of proportionality.
Can k be negative or a fraction?
Yes to both. k = −2 means y runs opposite to x, and k = 1/4 means y is a quarter of x. The only forbidden value in these models is division by x = 0.