What this calculator does
When two quantities are in direct proportion, one is always a fixed multiple of the other: y = kx. That fixed multiple, k, is called the constant of proportionality, and it can be found from any single matching pair of x and y values by dividing y by x.
The constant of proportionality only holds if the relationship really is a direct proportion, meaning it passes through zero and scales evenly. Entering a second x-y pair lets this calculator check that: if both pairs give the same k, the two points genuinely lie on the same proportional line; if they give different values of k, the data is not a direct proportion after all.
The formula
Divide y by x for the first pair to get k. If a second pair is entered, the same division is applied to it and the two results are compared; matching values confirm a genuine direct proportion, while differing values show the relationship is not proportional across both points.
| Term | Meaning |
|---|---|
| Constant of proportionality (k) | The fixed multiplier in a direct proportion: k = y ÷ x. |
| Direct proportion | A relationship of the form y = kx, where y is always the same multiple of x and the line passes through the origin. |
| Proportionality constant | Another name for the same value, k, used interchangeably with constant of proportionality. |
The inputs explained
| Field | What to enter |
|---|---|
| x (first pair) | The x value from a known matching pair. |
| y (first pair) | The y value from the same pair. |
| x (second pair, optional check - set to 0 to skip) | An optional second x value, used only to check consistency; set to 0 to skip this check. |
| y (second pair, optional check) | The matching y value for the second pair, if used. |
When to use it
Finding k from a word problem
A question stating that y is directly proportional to x, and giving one matching pair of values, is solved directly by dividing that pair to get k, and this calculator also writes out the resulting y = kx equation.
Checking whether a data set is really proportional
Given two supposedly matching pairs, working out k from each and comparing them is the standard way to confirm, or rule out, that a direct proportion actually applies across the whole data set.
Predicting a new y value from a known constant
Once k is known, the same y = kx equation this calculator produces can be used to work out y for any new x value without recalculating k each time.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the constant from the first pair compares against a fixed second pair
The first pair held at x=4 with y varying, checked against a fixed second pair that implies k=4.
| y (first pair) | Constant of proportionality (k) from first pair | Do the two pairs agree on the same constant? |
|---|---|---|
| 8 | 2.000 | No - the first pair gives k = 2.000 and the second gives k = 4.000, so these two points do not lie on the same direct proportion line |
| 12 | 3.000 | No - the first pair gives k = 3.000 and the second gives k = 4.000, so these two points do not lie on the same direct proportion line |
| 16 | 4.000 | Yes - both pairs give k = 4.000, consistent with a direct proportion |
| 20 | 5.000 | No - the first pair gives k = 5.000 and the second gives k = 4.000, so these two points do not lie on the same direct proportion line |
| 24 | 6.000 | No - the first pair gives k = 6.000 and the second gives k = 4.000, so these two points do not lie on the same direct proportion line |
How the second pair's constant changes as x2 varies
The second pair's x value varying against a fixed y2 of 24, checked against the fixed first pair.
| x (second pair) | k from second pair | Do the two pairs agree on the same constant? |
|---|---|---|
| 3 | 8.000 | No - the first pair gives k = 4.000 and the second gives k = 8.000, so these two points do not lie on the same direct proportion line |
| 4 | 6.000 | No - the first pair gives k = 4.000 and the second gives k = 6.000, so these two points do not lie on the same direct proportion line |
| 6 | 4.000 | Yes - both pairs give k = 4.000, consistent with a direct proportion |
| 8 | 3.000 | No - the first pair gives k = 4.000 and the second gives k = 3.000, so these two points do not lie on the same direct proportion line |
| 12 | 2.000 | No - the first pair gives k = 4.000 and the second gives k = 2.000, so these two points do not lie on the same direct proportion line |
Questions
What is the constant of proportionality?
It is the fixed number k in a direct proportion y = kx, found by dividing any matching y value by its x value. It stays the same for every matching pair on a genuine direct proportion.
How is this different from a general rate or gradient?
A direct proportion is a specific case of a straight-line relationship that passes through the origin, so its gradient and its constant of proportionality are the same number. A line that does not pass through the origin has a gradient but no single constant of proportionality, since y is not simply a multiple of x.
What does it mean if my two pairs give different values of k?
It means the data does not represent a single direct proportion. Either one of the values was recorded incorrectly, or the underlying relationship is not actually proportional (for example it might have a fixed starting value added on, rather than scaling purely from zero).
Can the constant of proportionality be negative?
Yes. If y decreases as x increases in a straight-line relationship through the origin, k is negative, and the same formula, k = y ÷ x, still applies.
For a related everyday ratio calculation built from two related figures, see the percent change calculator.