Constant of proportionality

The constant of proportionality, k, is the number you multiply x by to get y. Find it with k = y ÷ x, from a table, graph, equation or story.

The idea: the multiplier

In a proportional relationship, y is always the same number times x. That number is the constant of proportionality, k. The equation is y = kx.

k is also the unit rate: the value of y when x is 1. If 4 pounds of apples cost $10, then k = 10 ÷ 4 = 2.5 dollars per pound.

See it: pictures to draw with your child

Copy these onto paper or a whiteboard. A quick sketch makes the idea concrete before any written method.

The point at x = 1 shows k. The steeper line has the bigger k.

Divide y by x, or find the row for x = 1: k = 3.5.

The unit rate is k: $1.75 per notebook, so c = 1.75n.

Finding k from a table

Pick any row and divide y by x. Check one more row to be sure. For x: 2, 4, 6 and y: 7, 14, 21, k = 7 ÷ 2 = 3.5.

If the table has a row with x = 1, you can read k straight off. That row is the unit rate.

Finding k from a graph

Find a point on the line with easy numbers, like (2, 6). Divide y by x: 6 ÷ 2 = 3, so k = 3.

Or look at the point where x = 1. The y value there is k. A steeper line has a bigger k.

Finding k from an equation or a story

In an equation like y = 4.5x, k is the number in front of x: 4.5. In c = 12h, the pay per hour is $12, so k = 12.

In a story, k is the unit rate. “5 notebooks cost $8.75” gives k = 8.75 ÷ 5 = $1.75 per notebook, so c = 1.75n.

Find k as a unit rate

Type a y amount and its x amount. The unit rate is the constant of proportionality.

k with fractions (grade 7)

Grade 7 finds unit rates from fractions. Kai walks 1/2 mile in 1/4 hour. k = 1/2 ÷ 1/4 = 1/2 × 4 = 2 miles per hour.

The method is the same: divide y by x. The fraction pages show how to divide by a fraction if that step is new.

Quiz: find k

Find the constant of proportionality from a table, an equation or a story.

Where this fits: ratios step by step

What is a ratio
Part-to-part and part-to-whole ratios
Equivalent ratios
Simplifying ratios
Ratio tables
Double number lines
Tape diagrams
Comparing ratios
Unit rates
Unit price and best buy
Speed, distance and time
Solving proportions
Proportional relationships
Constant of proportionality (this page)
Scale drawings and maps
Ratio word problems

What to say to your child

A short script for teaching it the first time. Pause after each line and let your child do the next step.

“4 pounds of apples cost $10. How much is 1 pound?” (2.50 dollars.)
“That number is k, the constant of proportionality.”
“Write the equation: cost = …” (2.5 times pounds.)
“What do 6 pounds cost?” (15 dollars.)
“On the graph, what is y when x is 1?” (2.5, which is k.)

More worked examples

Example 1: table

x: 3, 6, 9. y: 12, 24, 36. 12 ÷ 3 = 4, so k = 4 and y = 4x.

Example 2: graph

The line passes through (5, 2). k = 2 ÷ 5 = 0.4, so y = 0.4x.

Example 3: equation

d = 65t. k = 65 miles per hour.

Example 4: fractions

3/4 cup of rice makes 1/2 a batch. k = 3/4 ÷ 1/2 = 1.5 cups per batch.

Practice questions (answers below)

Ask these out loud or write them down. Check each answer, and ask “how did you get that?” for one or two.

y = 6x. k?
x: 2, 4. y: 10, 20. k?
x: 4, 8. y: 2, 4. k?
A line through (3, 12). k?
$45 for 3 hours. k?
7 bags hold 84 marbles. k?
y = 0.25x. k?
1/2 mile in 1/6 hour. k?
k = 8. What is y when x = 5?
Stretch: k = 2.5 and y = 20. What is x?

Show the answers

6
5
0.5
4
$15 per hour
12 per bag
0.25
3 miles per hour
40
8

Common mistakes

Dividing x by yGets k = 0.4 instead of 2.5 for $10 for 4 pounds.Fix: k = y ÷ x, the amount of y for each 1 of x.

Using a point not on the lineReads a point near the line.Fix: choose a point where the line crosses grid lines exactly.

Finding k for a non-proportional tableDivides one row and calls it k.Fix: check every row gives the same ratio first.

How to teach it at home

Pay day

Work out k for an hourly wage and write the equation.

Car trip

Find k in miles per hour from a trip, then predict a longer trip.

Recipe k

Find cups of flour per batch and write c = k × b.

By grade

Grade 7 (7.RP.A.2b): identify the constant of proportionality in tables, graphs, equations and diagrams.
Grade 7 (7.RP.A.1): unit rates with fractions.
Grade 8: k becomes the slope of a line.
In India, direct proportion in class 8 uses the same idea. Next: scale drawings.

More for each grade: Grade 7 math and the CBSE class 7 maths guide.