Constant of Proportionality Calculator

Enter a matching x and y pair from a direct proportion to find the constant of proportionality k = y/x, then predict any value on the line y = kx.

Quick Facts

Formula
k = y / x (from y = kx)
k is the fixed ratio between the two quantities; the same k holds for every pair on the line.

Your Results

Calculated
Constant of proportionality (k)
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k = y / x
Equation
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Direct proportion y = kx
Predicted y
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y = k × x for your chosen x

Ready

Enter a matching x and y pair and press Calculate.

What the constant of proportionality is

Two quantities are directly proportional when their ratio stays the same no matter how large or small they get. That fixed ratio is the constant of proportionality, written k. The relationship is captured by the equation y = kx: whenever x grows, y grows by the same factor, and the graph is a straight line through the origin (0, 0).

To find k from a single matching pair of values, divide the y-value by the x-value:

k = y / x

For example, if 3 notebooks cost $12, then k = 12 / 3 = 4 dollars per notebook. That k lets you fill in the whole table: 5 notebooks cost 4 × 5 = $20, and $30 buys 30 / 4 = 7.5 notebooks. The constant carries units of "y per x" — dollars per notebook, miles per hour, grams per cup — which is why it is often read as a unit rate.

Why the constant matters

The constant of proportionality is the single number that fully describes a direct proportion. Once you know k, you never need the original data pair again: you can predict any y from y = kx, or recover any x from x = y / k. It is the same idea as the slope of a line through the origin, and the same idea as a unit rate, a scale factor, or a density — different names for the ratio that stays constant.

Common real-world constants of proportionality

  • Unit price: total cost = (price per item) × quantity. k is the price per item.
  • Constant speed: distance = speed × time. k is the speed (e.g. 60 miles per hour).
  • Density: mass = density × volume. k is the density (water ≈ 1 gram per milliliter).
  • Currency & unit conversion: the exchange rate or conversion factor is a constant of proportionality (e.g. 2.54 cm per inch).
  • Map scale: real distance = scale × map distance. k is the scale factor.

How to check a relationship is truly proportional

A relationship is only a direct proportion if every pair gives the same k. Compute y / x for several rows of a table; if the quotients match, the data is proportional and that shared value is k. If the ratios drift, the relationship is not directly proportional — it may be linear with a non-zero intercept (y = kx + b), inverse (y = k / x), or something else entirely. A quick visual test: the graph of a direct proportion always passes through the origin.

Frequently Asked Questions

What is the constant of proportionality?
It is the fixed ratio k in a direct proportion y = kx. You find it by dividing any y-value by its matching x-value: k = y/x. It tells you how much y changes for each one-unit increase in x, and it is the same as the unit rate or the slope of the line through the origin.
How do I find k from a table or graph?
From a table, pick any row and compute y ÷ x; the answer should be identical for every row of a proportional table. From a graph, k is the slope — the rise over run measured from the origin, since a direct proportion always passes through (0, 0). Reading the point where x = 1 gives k directly, because y = k·1 = k.
Is the constant of proportionality the same as slope?
Yes, for a line through the origin. The equation y = kx is a straight line with slope k and y-intercept 0, so the constant of proportionality and the slope are the same number. The distinction only appears for lines that do not pass through the origin (y = kx + b), which are linear but not directly proportional.
Why can't x be zero?
k = y/x involves division by x, which is undefined when x = 0. In a genuine direct proportion the pair (0, 0) always lies on the line, but you can't use it to compute k because 0/0 gives no information. Use any other pair where x ≠ 0 instead.

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