The Constant of Proportionality

If apples cost 3 pounds a kilo, cost is always weight times 3. That 3 never changes.

That fixed multiplier has a name.

Constant of proportionality is the number one quantity is always multiplied by to get the other.

Written as an equation

The relationship is y = kx, where k is the constant. Here it is y = 3x.

Finding it from a table

Divide any output by its input. Every row gives the same answer, which is the test.

The number you have been multiplying by

In a proportional relationship, y = kx, where k is the constant of proportionality. It is the unit rate: the amount of y for each 1 of x. Every ratio table you built in Grade 6 had this number hiding in it.

Finding it from a table

Divide any y by its matching x. If the relationship really is proportional, every pair gives the same answer — and that answer is k. If different pairs give different values, the relationship is not proportional.

It usually means something

For cost and weight, k is the price per kilogram. For distance and time, it is the speed. The constant is rarely just a number: naming what it represents is usually the most informative thing you can say about the relationship.

There are two constants, one each way

If y = 3x then x = y/3, so the constant depends on which variable you treat as the input. Both are correct and both are useful; state which direction you mean, especially when the units make the two very different numbers.