Ratios & Proportional Relationships

Constant of Proportionality

Identifying the constant of proportionality from tables, graphs, and equations.

Foundational15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Identifying Proportional Relationships

The One Number That Defines the Whole Relationship

Once a relationship is confirmed proportional, every pair of values shares the same ratio. Before reading on, think about why just that single shared number might be enough to describe the entire relationship — every possible pair of values at once, not just the ones already given.

Definition — Constant of Proportionality

The fixed ratio (often written k) between two quantities in a proportional relationship, satisfying y = kx for every corresponding pair of values x and y.

The constant of proportionality is really just the unit rate from Grade 6, given a formal name and a formal role: once you know k, you can find y for any x at all, not just the values already listed in a table — which is exactly why a proportional relationship can be captured by one short equation, y = kx.

Worked Example — Finding k from a Table

A table shows 4 hours for $60 and 7 hours for $105. Find the constant of proportionality. Divide any pair: 60 ÷ 4 = 15, and 105 ÷ 7 = 15. The constant of proportionality is k = 15 dollars per hour.

Worked Example — Writing and Using the Equation

Using k = 15 from the last example, write an equation for total pay y in terms of hours worked x, then find the pay for 10 hours. The equation is y = 15x. For 10 hours: y = 15 × 10 = 150. The pay for 10 hours is $150.

Worked Example — Finding k from a Graph

A proportional relationship's graph passes through the point (5, 20). Find the constant of proportionality. Since y = kx, solve for k using this point: 20 = k × 5, so k = 4.

y = kx

Tip

The constant of proportionality can always be found from just one non-zero pair of values — divide y by x — since the ratio is guaranteed to be the same for every pair in a proportional relationship.

Common Mistakes

Dividing x by y instead of y by x when finding the constant of proportionality, inverting its meaning.
The constant of proportionality k satisfies y = kx, so k is found by dividing y by x — check which quantity plays which role before dividing.
Using the equation y = kx to make predictions for a relationship that hasn't actually been confirmed proportional.
The equation y = kx only applies once a relationship has been checked and confirmed proportional — verify with several pairs of values first.

Key Takeaways

The constant of proportionality, k, is the fixed ratio y/x shared by every pair in a proportional relationship.
Once k is known, the equation y = kx predicts y for any value of x.
The constant of proportionality can be found from a table, a graph, or directly from the relationship's own equation.

Summary

The constant of proportionality unifies a proportional relationship's table, graph, and equation into one number. The final lesson in this unit applies proportional reasoning to one of its most common real-world uses — percent change.

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