Constant of proportionality: what it is (k = y ÷ x), how to find it from a table, a graph, an equation or a word problem, how to tell whether a relationship is proportional at all, how to graph a proportional relationship (a straight line through the origin, with (1, k) the unit rate), the inverse case (k = x × y), worked examples at every step, and a printable worksheet with answers for grades 6 to 8

"Constant of proportionality" is a grade 7 phrase in the United States (Common Core 7.RP.2 asks students to "identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions"), and it is searched forty thousand times a month because the word "constant" makes a simple idea sound hard. The idea: when two quantities are proportional, one is always the same number times the other, and that number is k. This page shows how to find k from each of the four places the test puts it, how to tell when a relationship is not proportional, how to graph one, and what changes for inverse proportion. The printable worksheet has twelve questions of the exam kind with answers.

Download the constant of proportionality worksheet (PDF) (the definition and the four methods on one page, then twelve questions from tables, graphs, equations and word problems with answers; two pages, free).

What it is

Two quantities are proportional when one is always the same multiple of the other. If y is proportional to x, then

y = k × x, and so k = y ÷ x

for every pair of values. The number k is the constant of proportionality. It is also called the unit rate (the value of y when x = 1), and on a graph it is the slope of the line.

If 3 pencils cost $1.50, then k = 1.50 ÷ 3 = 0.50: fifty cents per pencil. Ten pencils cost 0.50 × 10 = $5, and the equation is cost = 0.50 × pencils. The k carries a unit, dollars per pencil, and saying the unit aloud ("per") is the quickest way to recognise a constant of proportionality in a word problem.

How to find k from a table

Divide every y by its x. If all the answers agree, that answer is k; if they do not, the table is not proportional.

k = 50 miles per hour, and y = 50x.

Not proportional: the quotients differ (this table is y = 3x + 2, linear but with a start value). A table with a row where x = 0 and y is not 0 is never proportional.

The tables on tests are sometimes given sideways, or with x and y in the other order; the rule is still "divide the dependent quantity by the one it depends on".

How to find k from a graph

A proportional relationship always graphs as a straight line through the origin (0, 0). Three ways to read k from it:

1. Find x = 1 on the horizontal axis and read the y-value directly above it: that point (1, k) gives the constant.
2. Pick any clear point on the line, (x, y), and divide: k = y ÷ x. The point (4, 12) gives k = 3.
3. Rise over run: from one point to another, the change in y divided by the change in x.

A graph that is a straight line but misses the origin is not proportional, however neat; a graph that curves is not proportional. Both appear as distractors on the grade 7 and 8 tests. The steeper the line, the larger k: on one set of axes, the line y = 3x is above the line y = 2x for every positive x.

How to find k from an equation

In y = kx, k is the number multiplying x:

If an equation is given in another form (2y = 6x, or x = y ÷ 5), rearrange it to y = kx first: 2y = 6x gives y = 3x, so k = 3; x = y ÷ 5 gives y = 5x, so k = 5.

How to find k from a word problem

The constant is the unit rate: the amount of one thing for one unit of the other. Find the pair of numbers given, and divide.

The question then usually asks for a value the problem did not give: at 30 miles per gallon, 11 gallons takes the car 330 miles; 7 cakes need 10.5 eggs.

Graphing a proportional relationship

To graph y = kx: plot the origin (0, 0), plot the point (1, k), and draw the straight line through them (and extend it). Add one more point, (2, 2k), as a check that the line is straight. For y = 2.5x the points are (0, 0), (1, 2.5), (2, 5), (4, 10).

To compare two proportional relationships, graph both on the same axes: the one with the larger k is steeper. This is the grade 8 version of the topic (Common Core 8.EE.5): "Which runner is faster, the one in the table or the one on the graph?" is answered by finding each k.

The inverse case

In inverse (indirect) proportion, one quantity goes up as the other goes down, and it is the product that is constant:

x × y = k, so y = k ÷ x.

If 4 workers finish a job in 6 days, k = 4 × 6 = 24 worker-days, and 8 workers take 24 ÷ 8 = 3 days; at 60 miles per hour a trip takes 2 hours, so k = 120 miles and at 40 miles per hour it takes 3 hours. The graph of inverse proportion is a curve that never touches either axis, not a line through the origin.

Where it sits in school

Ratios and unit rates are grade 6 in the United States (6.RP); the constant of proportionality by name, in tables, graphs, equations and word problems, is grade 7 (7.RP.2), and comparing proportional relationships and connecting k to slope is grade 8 (8.EE.5). In England it is Key Stage 3 (direct and inverse proportion, Years 8 and 9) and GCSE; in CBSE it is Class 8 (Direct and Inverse Proportions), where the constant is written as y/x = k; in Australia, Year 8 and 9. The question that costs the most marks at every level is the non-example: a straight line that misses the origin, which students call proportional because it is straight.

Practice: twelve questions

1. A table has x = 3, 6, 9 and y = 12, 24, 36. Find k and write the equation.
2. A table has x = 1, 2, 3 and y = 4, 7, 10. Is it proportional?
3. A line passes through the origin and (5, 20). What is k?
4. A line passes through (0, 3) and (2, 7). Is it proportional?
5. What is the constant of proportionality in y = 0.6x?
6. Rewrite 3y = 15x as y = kx and give k.
7. 8 notebooks cost $20. Find k and the cost of 15 notebooks.
8. A cyclist rides 45 km in 3 hours. Write the equation and find the distance in 5 hours.
9. On a graph of a proportional relationship the point (1, 7) is on the line. What is the equation?
10. Two lines through the origin: one passes through (2, 10), the other through (4, 12). Which has the larger k?
11. 6 pumps empty a tank in 8 hours. How long do 4 pumps take?
12. A table has x = 0, 2, 4 and y = 1, 5, 9. Is it proportional?

Answers: 1) k = 4, y = 4x. 2) No: 4, 3.5, 3.33 are different. 3) k = 4. 4) No: it misses the origin. 5) 0.6. 6) y = 5x, k = 5. 7) k = $2.50 per notebook; 15 cost $37.50. 8) d = 15t; 75 km. 9) y = 7x. 10) The first (k = 5 against k = 3). 11) Inverse: 6 × 8 = 48 pump-hours, so 48 ÷ 4 = 12 hours. 12) No: at x = 0, y should be 0.

Download the constant of proportionality worksheet (PDF)

Where to get help

A 1-on-1 online maths class with a tutor who holds a master's in mathematics, for grades 6 to 8 in the US curriculum and the equivalent years in England, Australia and CBSE: ratios and unit rates with real prices and speeds on the shared screen, then the four places the test hides k, then the non-examples until the child spots them. The recording and notes after every class, the same tutor every week, US$10 to $13 an hour. The grade 6 math page has the grades; the free 30-minute first class is a real lesson, usually on the homework the child brings.

Questions parents ask

See how we teach this, 1-on-1 online →