A 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.
In Spanish
relación proporcional
Also called
direct proportion, direct variation, directly proportional, y = kx
Both graphs are straight lines, but only y = 2x starts at the origin and keeps the ratio yx constant. The line y = x + 3 is linear but not proportional.
Formal definition and notation
Two quantities x and y are in a proportional relationship when there is a nonzero constant k with y = kx for every pair of values. Then yx = k for every pair with x ≠ 0, and x = 0 goes with y = 0. The numberk is the constant of proportionality, equal to the unit rate and to the slope of the graph.
Symbol
Read it as
Meaning
y = kx
y equals k times x
the equation of every proportional relationship
k = yx
k equals y over x
the constant of proportionality, the same for every pair
y ∝ x
y is proportional to x
a shorthand used in science books
Key facts
Table test: divide each y by its x. If every quotient is the same, the relationship is proportional.
Graph test: the points lie on a straight line that passes through the origin(0, 0). A line that misses the origin is not proportional.
Equation test:y = 4x is proportional; y = 4x + 1 is not, because of the added 1.
The point (1, k) is always on the graph: k is the unit rate.
Doubling x doubles y, tripling x triples y, and so on.
The proportional relationship test
The proportional relationship test
y = kx ⇔ yx = k for every pair
x
the first quantity (input), such as hours or pounds
y
the second quantity (output), such as dollars or miles
k
the constant of proportionality: y for each 1 unit of x
Three ways to test: equal ratios in a table, a straight-line graph through (0, 0), or an equation of the form y = kx with no added constant.
Try it: build a proportional table
Interactive
Set how many pounds and dollars make one group. Every row of the table is a multiple of that group, and every point on the graph lands on one line through the origin.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Proportional Relationship examples, step by step
Example 1: Test a table
Warm-up
Is this relationship proportional? x: 2, 4, 6; y: 7, 14, 21.
Divide each y by its x: 72 = 3.5, 144 = 3.5, 216 = 3.5.
Every ratio is 3.5, so the relationship is proportional with k = 3.5.
Equation: y = 3.5x.
Answer: Yes, it is proportional: y = 3.5x.
Example 2: A table that is not proportional
Core
A gym charges this much for x visits: 1 visit, 15 dollars; 2 visits, 20 dollars; 3 visits, 25 dollars. Is cost proportional to visits?
Ratios: 151 = 15, 202 = 10, 253 ≈ 8.33.
The ratios are not equal, so the relationship is not proportional.
Find the pattern: each visit adds 5 dollars, and 0 visits would cost 15 − 5 = 10 dollars, a sign-up fee.
So y = 5x + 10: the graph is a line, but it starts at (0, 10), not the origin.
Answer: Not proportional: cost is y = 5x + 10, which has a 10 dollar starting fee.
Example 3: Use the constant to predict
Test-style
A car travels 186 miles on 6 gallons of gas. Distance is proportional to gas used. How far can it travel on 15 gallons?
The relationship is the whole rule y = kx; the constant is the single number k inside it. In y = 4x, the relationship is proportional and the constant is 4.
A proportion is one equation saying two ratios are equal, like 34 = 912. A proportional relationship is a whole pattern where every pair has the same ratio.
All proportional relationships are linear, but y = mx + b with b ≠ 0 is linear and not proportional.
Quick check: 3 questions
Question 1 of 3
Which equation shows a proportional relationship?
Show answer and explanation
Answer: C.y = 6x has the form y = kx. The first two add or subtract a constant, so they miss the origin, and y = 6x is inverse variation: y shrinks as x grows.
Question 2 of 3
In a proportional relationship, y = 12 when x = 3. What is y when x = 10?
Show answer and explanation
Answer: B.k = 123 = 4, so y = 4 × 10 = 40. 19 adds 7 to 12 because x went up by 7; 30 and 36 come from multiplying by 3 instead of 4.
Question 3 of 3
A graph of a proportional relationship must pass through which point?
Show answer and explanation
Answer: A. When x = 0, y = k × 0 = 0, so the origin is always on the graph. The point (1, k) is also on it, but (1, 1) only when k = 1.
How do you know if a graph shows a proportional relationship?
Check two things. The graph must be a straight line, and the line must pass through the origin (0, 0). A curve is never proportional, and a straight line that crosses the y-axis anywhere except 0 is not proportional either. If both checks pass, the slope of the line is the constant of proportionality.
Is a proportional relationship the same as a linear relationship?
No. Every proportional relationship is linear, but not every linear relationship is proportional. y = 2x and y = 2x + 5 both graph as straight lines with slope 2, yet only y = 2x goes through the origin and keeps yx constant. A proportional relationship is the special linear case with y-intercept 0.
What is an example of a proportional relationship in real life?
Paying by the pound or by the hour. If grapes cost 3 dollars per pound, 2 pounds cost 6 dollars and 5 pounds cost 15 dollars: cost = 3 × pounds. Other examples are miles driven at a steady speed, ingredients when you scale a recipe, and earnings at a fixed hourly wage with no bonus.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 7.RP.A.2.a, CCSS 7.RP.A.2.b, CCSS 7.RP.A.2.c
Cite this entry
Effortless Math Team. “Proportional Relationship.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/proportional-relationship/
Flashcard
TermProportional Relationshipy = kx
DefinitionTwo quantities where y is always the same multiple of x; the ratio y/x stays constant and the graph is a line through the origin. Example: y = 3x.
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When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2Unit 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 5OriginThe 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 8RatioA 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 8MileA 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 6SubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.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 1Y-AxisThe y-axis is the up-and-down number line on the coordinate plane, with positive numbers above the origin and negative numbers below it. It meets the x-axis at the origin, (0, 0). A point’s y-coordinate is read along the y-axis, and every point on the y-axis has x-coordinate 0, like (0, 4).Geometry · Grade 5, Grade 6, Grade 7Y-InterceptThe y-intercept is the point where a graph crosses the y-axis. Every point on the y-axis has x = 0, so you find the y-intercept by putting x = 0 into the equation. In y = mx + b, the y-intercept is b, written as the point (0, b).Algebra · Grade 8, Algebra 1Equivalent FractionsEquivalent fractions are fractions that name the same value, like 12, 24 and 36. To make one, multiply or divide the numerator and denominator by the same nonzero number. That multiplies the fraction by 1, so its value does not change, and the fractions land on the same point of a number line.Number System · Grade 3, Grade 4, Grade 5FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2Scale FactorA scale factor is the number used to multiply side lengths when making a similar figure larger or smaller. A scale factor of 2 doubles every length, and a scale factor of 12 makes lengths half as large. Find it by dividing a new length by the matching original length. Areas change by the square of the scale factor.Ratios and Proportions · Grade 7, Grade 8, Algebra 1Percent DecreasePercent decrease is how much a quantity went down, written as a percent of its original value. Subtract the new value from the original, divide by the original, and multiply by 100. A price that drops from 80 dollars to 60 dollars has a decrease of 20 dollars, and 2080 = 25%.Ratios and Proportions · Grade 7, Grade 8, Algebra 1