Constant of Proportionality: Definition & Examples

Definitionnoun phrase

What is the constant of proportionality?

The constant of proportionality is the constant ratio between two proportional quantities, k = yx. It is the k in y = kx and stays the same for every pair of values, which makes the graph a straight line through the origin. If 3 notebooks cost 9 dollars, k = 3 dollars per notebook.

In Spanish: constante de proporcionalidad
Also called: k value, proportional constant, constant of variation, unit rate
Grade level: 6-8

Grade 7Grade 8Algebra 1CCSS 7.RP.A.2.bCCSS 7.RP.A.2.cCCSS 8.EE.B.5GEDSATACT

Constant of Proportionality in a picture

The constant of proportionality shows up three ways: as the same ratio yx in every row of the table, as k in y = kx, and as the height of the line at x = 1.

Formal definition and notation

Two quantities x and y are in a proportional relationship when yx has the same value k for every pair with x ≠ 0. That value is the constant of proportionality, and the relationship is y = kx. The constant equals the unit rate (the value of y when x = 1) and the slope of the graph, a line through (0, 0) and (1, k).

Symbol	Read it as	Meaning
k	k	the usual letter for the constant of proportionality
y = kx	y equals k x	the equation of a proportional relationship; y is k times x
yx = k	y over x equals k	the test for a proportional relationship: the same quotient for every pair

Key facts

In a table, divide y by x in each row. The same answer every time means proportional, and that answer is k.
On a graph, k is the y-value at x = 1. The line must be straight and pass through (0, 0).
The steeper the graph, the larger the constant of proportionality: y = 5x rises faster than y = 2x.
An equation like y = 3x + 2 is linear but not proportional, because x = 0 gives y = 2, not 0.
k can be a fraction or decimal: walking 3 miles in 2 hours gives k = 32 miles per hour.

Constant of proportionality

k = yx, so y = kx

k: the constant of proportionality (the unit rate)
x: the input quantity, such as number of notebooks or hours
y: the output quantity, such as total cost or distance

Check every row of a table, not just one. If yx changes, the relationship is not proportional and there is no constant of proportionality.

Try it: scale a ratio

Interactive

Change how many notebooks and dollars make one group. Every row keeps the same dollars per notebook, so the points lie on one line through (0, 0); that rate is the constant of proportionality.

This interactive needs JavaScript. The worked examples below show the same ideas step by step.

Constant of Proportionality examples, step by step

Example 1: Find k from a table

Warm-up

A table shows x = 2, 5, 8 and y = 7, 17.5, 28. Is the relationship proportional? If so, find k.

Row 1: 7 ÷ 2 = 3.5.
Row 2: 17.5 ÷ 5 = 3.5. Row 3: 28 ÷ 8 = 3.5.
Every ratio is 3.5, so the relationship is proportional and y = 3.5x.

Answer: Yes. The constant of proportionality is k = 3.5.

Example 2: Price per pound

Core

5 pounds of apples cost 8.75 dollars. The cost is proportional to the weight. Write an equation and find the cost of 12 pounds.

k = 8.755 = 1.75 dollars per pound.
The equation is C = 1.75p, where p is pounds and C is cost in dollars.
For 12 pounds: C = 1.75 × 12 = 21.

Answer: C = 1.75p, so 12 pounds cost 21 dollars.

Example 3: Proportional or not?

Test-style

A gym charges 5 dollars for 1 visit, 8 dollars for 2 visits and 11 dollars for 3 visits. Is the cost proportional to the number of visits?

Compute yx for each pair: 51 = 5, 82 = 4, 113 ≈ 3.67.
The ratios are not equal, so there is no constant of proportionality.
The cost rises by 3 dollars per visit from a 2-dollar start: y = 3x + 2. At 0 visits it would be 2 dollars, not 0.

Answer: No. The ratio y/x changes, so the cost is not proportional to the number of visits.

Common mistakes

✗ Mistakey = 2x + 3 is proportional with k = 2.

✓ InsteadIt is linear but not proportional: at x = 0, y = 3. Only y = kx is proportional.

A proportional relationship must have a constant ratio and pass through the origin when graphed.

✗ MistakeFor 4 tickets costing 30 dollars, k = 430 ≈ 0.13.

✓ Insteadk = yx = 304 = 7.5 dollars per ticket.

Divide the output by the input. xy gives tickets per dollar, which is a different rate.

✗ MistakeChecking only the first row of a table and declaring it proportional.

✓ InsteadCheck every row: 5/1, 8/2 and 11/3 are 5, 4 and 3.67, so that table is not proportional.

One ratio can always be computed. Proportional means every ratio is the same.

Not to be confused with

Constant of Proportionalityvs.Slope

Every non-vertical line has a slope; only lines through the origin have a constant of proportionality. For y = 4x they are the same number; y = 4x + 1 has slope 4 and no k.

Constant of Proportionalityvs.Unit rate

A unit rate is the amount per 1 unit in a situation, like 3 dollars per notebook. In a proportional relationship, the unit rate and k are the same number.

Constant of Proportionalityvs.Constant

A constant is any fixed number. The constant of proportionality is the particular constant that multiplies x in y = kx.

Quick check: 3 questions

Question 1 of 3

In y = 6x, what is the constant of proportionality?

Show answer and explanation

Answer: B. In y = kx, k is the number multiplying x: 6. 16 is xy, 6x is the output, and 0 is the y-intercept.

Question 2 of 3

A proportional graph passes through (0, 0) and (4, 10). What is k?

Show answer and explanation

Answer: A. k = yx = 104 = 2.5. Choice B divides x by y, choice C reads only the y-value, and choice D adds the coordinates.

Question 3 of 3

Which equation shows a proportional relationship?

Show answer and explanation

Answer: D. y = 0.5x has the form y = kx with k = 0.5. Choices A and B do not pass through the origin, and y = x² has a changing ratio, so its graph is not a straight line.

Constant of Proportionality: frequently asked questions

How do you find the constant of proportionality?

Divide y by x for any pair of values: k = yx. From a table, check that every row gives the same quotient. From a graph, read the y-value where x = 1, or divide the coordinates of any point on the line. From an equation y = kx, it is the number multiplying x. From words, find the unit rate.

Is the constant of proportionality the same as the slope?

For a proportional relationship, yes: the graph of y = kx is a line through the origin with slope k. But not every slope is a constant of proportionality. The line y = 2x + 5 has slope 2 and is not proportional, because it does not pass through (0, 0).

Can the constant of proportionality be a fraction?

Yes. If a recipe uses 2 cups of flour for every 3 cups of milk, flour = 23 × milk, so k = 23. Any positive number can be a constant of proportionality, and in some contexts, like temperature drop per hour, k can even be negative.

Practice and tools

Effortless Math practicePractice on Effortless Math Constant of Proportionality CalculatorFind k from a table, a graph point or a word problem Unit Rate CalculatorThe unit rate is the constant of proportionality Ratio and Proportion CalculatorSolve proportions with the same ratio

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 7.RP.A.2.b, CCSS 7.RP.A.2.c, CCSS 8.EE.B.5

Cite this entry

Effortless Math Team. “Constant of Proportionality.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/constant-of-proportionality/

Flashcard

TermConstant of Proportionalitythe k in y = kx

DefinitionThe constant ratio yx between proportional quantities. If 3 notebooks cost 9 dollars, k = 3 dollars per notebook.

Study the full deck

Browse all terms in the Math Dictionary · More ratios and proportions terms · Random term

Proportional RelationshipA proportional relationship is a relationship between two quantities in which one is always the same multiple of the other, so y = kx for a fixed number k. The ratio yx is the same for every pair, and the graph is a straight line through the origin. Example: at 3 dollars per pound, cost = 3 × pounds.Ratios and Proportions · Grade 7, Grade 8, Algebra 1Unit RateA unit rate is a rate that compares a quantity to one unit of another quantity, such as 50 miles per hour or 4 dollars per notebook. To find it, divide the first quantity by the second: 150 miles in 3 hours is 150 ÷ 3 = 50 miles per hour. The word per means for each one.Ratios and Proportions · Grade 6, Grade 7, Grade 8SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5MileA mile (mi) is a US customary unit of length for long distances, equal to 5,280 feet or 1,760 yards. One mile is about 1.61 kilometers, so it is longer than a kilometer. Road distances and speed limits in the United States use miles and miles per hour.Measurement · Grade 4, Grade 5, Grade 6RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8ConstantA constant is a fixed value that does not change. In an expression, a constant is a number without a variable, such as 8 in x + 8 or −3 in 4a − 3. Special numbers such as π are constants too. Changing the variable changes the other terms, but never the constant.Algebra · Grade 6, Grade 7, Grade 8ProportionA proportion is an equation that says two ratios are equal, such as 12 = 36. It is read 1 is to 2 as 3 is to 6. In a true proportion the cross products are equal, which lets you find a missing value. Proportions are often used to solve scale, percent, and rate problems.Ratios and Proportions · Grade 6, Grade 7, Grade 8Linear EquationA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Algebra · Grade 7, Grade 8, Algebra 1Slope-Intercept FormSlope-intercept form is the linear equation form y = mx + b, where m is the slope and b is the y-intercept. You can read both values directly from the equation without rearranging: in y = 2x + 5, the slope is 2 and the line crosses the y-axis at (0, 5).Algebra · Grade 8, Algebra 1

Constant of Proportionality

What is the constant of proportionality?

Constant of Proportionality in a picture

Formal definition and notation

Key facts

Constant of proportionality

Try it: scale a ratio

Constant of Proportionality examples, step by step

Example 1: Find k from a table

Example 2: Price per pound

Example 3: Proportional or not?

Common mistakes

Not to be confused with

Quick check: 3 questions

Constant of Proportionality: frequently asked questions

How do you find the constant of proportionality?

Is the constant of proportionality the same as the slope?

Can the constant of proportionality be a fraction?

Related terms

Practice and tools

Sources

Flashcard

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