Directly proportional describes two quantities whose ratio never changes: y = kx for a fixed nonzero number k, the constant of proportionality. If x doubles, y doubles; if x is cut in half, so is y. The graph is a straight line through the origin, and yx = k for every pair.
In Spanish
directamente proporcional
Also called
direct variation, varies directly, proportional to, direct proportion
Directly proportional quantities lie on a line through the origin, so yx is constant. The dashed line is straight but starts at 2, so yx changes.
Formal definition and notation
A variable y is directly proportional to x (also, y varies directly with x) if there is a constant k ≠ 0 such that y = kx for all values in the relationship. Equivalently, yx = k for every pair with x ≠ 0, and the pair (0, 0) belongs to the relationship. More generally, y is directly proportional to the nth power of x when y = kxn.
Test with division: yx is the same for every pair.
The graph is a straight line through the origin. A line that misses the origin, like y = x + 2, is not directly proportional.
Multiplying x by any number multiplies y by the same number.
The constant k is the unit rate and the slope of the graph.
Everyday examples: cost and number of identical items, distance and time at a steady speed, pay and hours at a fixed wage.
Direct proportion
Direct proportion
y = kx
k = yx
the constant of proportionality, found from any one pair
y1x1 = y2x2
the proportion you can solve when one value is missing
To test a table, divide y by x in every row. Directly proportional means every quotient is the same.
Try it: keep the ratio constant
Interactive
Each row multiplies both columns by the same number, so cups of flour and batches stay directly proportional. Divide any row to check that the ratio never changes.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Directly Proportional examples, step by step
Example 1: Find k, then predict
Warm-up
y is directly proportional to x, and y = 12 when x = 4. Find y when x = 10.
Is y directly proportional to x? x: 2, 5, 8. y: 9, 22.5, 36.
Divide in each row: 92 = 4.5, 22.55 = 4.5, 368 = 4.5.
Every quotient is 4.5, so the relationship is y = 4.5x.
Answer: Yes, y is directly proportional to x, with k = 4.5.
Example 3: Weight on the Moon
Test-style
Weight on the Moon is directly proportional to weight on Earth. An astronaut who weighs 150 pounds on Earth weighs 25 pounds on the Moon. How much does a 96-pound rover weigh on the Moon?
Set up the proportion: 25150 = m96.
k = 25150 = 16.
m = 16 × 96 = 16.
Answer: The rover weighs 16 pounds on the Moon.
Common mistakes
✗ MistakeA taxi charges 3 dollars plus 2 dollars per mile, so the cost is directly proportional to the miles.
✓ InsteadCost = 2m + 3 is linear, not proportional: 1 mile costs 5 dollars but 2 miles cost 7, not 10.
A fixed starting amount moves the line off the origin. Directly proportional needs y = kx with nothing added.
✗ MistakeIn the table x: 1, 2, 3 and y: 4, 6, 8, y goes up by 2 each time, so it is proportional.
✓ InsteadCheck the ratios: 41 = 4, 62 = 3, 83 ≈ 2.7. Not proportional.
Equal steps show a linear pattern. Proportional means equal ratios, which requires the line to pass through (0, 0).
✗ MistakeIf y ∝ x and y = 10 when x = 2, then y = 13 when x = 5 (add 3 to both).
✓ Insteadk = 5, so y = 5 × 5 = 25.
Proportional quantities change by multiplying, not by adding the same amount.
Directly proportional: y = kx, the ratio yx is constant and both grow together. Inversely proportional: y = kx, the product xy is constant and one shrinks as the other grows.
Every direct proportion is a linear function with b = 0. But y = 2x + 1 is linear and not proportional.
Quick check: 3 questions
Question 1 of 3
Which equation shows y directly proportional to x?
Show answer and explanation
Answer: C.y = 7x has the form y = kx with k = 7. A and D add a constant, so their lines miss the origin, and B is inverse variation.
Question 2 of 3
d is directly proportional to t, and d = 18 when t = 3. What is d when t = 7?
Show answer and explanation
Answer: A.k = 183 = 6, so d = 6 × 7 = 42. 22 adds 4, 126 multiplies 18 by 7, and 7.7 divides the wrong way.
Question 3 of 3
5 pounds of apples cost 7.50 dollars. The cost is directly proportional to the weight. What do 8 pounds cost?
Show answer and explanation
Answer: B. The unit rate is 7.50 ÷ 5 = 1.50 dollars per pound, so 8 pounds cost 12.00 dollars. 10.50 adds 3 dollars, 5.33 inverts the ratio, and 15.50 adds 8.
Directly Proportional: frequently asked questions
How do you know if two quantities are directly proportional?
Divide one by the other for every pair. If yx gives the same number each time, they are directly proportional. On a graph, look for a straight line that passes through the origin. In an equation, look for the form y = kx with no number added or subtracted.
What is the constant of proportionality?
It is the fixed number k in y = kx, found by dividing y by x for any pair. It tells how much y changes for each 1-unit change in x, so it is the unit rate and the slope of the line. If 4 tickets cost 34 dollars, k = 8.5 dollars per ticket.
Is directly proportional the same as direct variation?
Yes. y varies directly with x, y is directly proportional to x, and y ∝ x all mean y = kx for a nonzero constant k. Algebra books often say direct variation, while grade 7 books say proportional relationship. All of these name the same idea.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 7.RP.A.2.a, CCSS 7.RP.A.2.c
Cite this entry
Effortless Math Team. “Directly Proportional.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/directly-proportional/
Flashcard
TermDirectly Proportionaly = kx
DefinitionThe ratio yx stays the same; double one and the other doubles. If 3 pens cost 6 dollars, 9 pens cost 18 dollars.
PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7Constant of ProportionalityThe 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.Ratios and Proportions · Grade 7, Grade 8, Algebra 1DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.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 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 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 1SubstituteSubstitute 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 8QuotientA 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 8Inversely ProportionalInversely proportional describes two quantities whose product never changes: y = kx, or xy = k, for a fixed nonzero number k. When one quantity doubles, the other is cut in half. On a 120-mile trip, speed and time are inversely proportional: 60 mph takes 2 hours and 30 mph takes 4 hours.Ratios and Proportions · Grade 8, Algebra 1, Algebra 2Linear FunctionA linear function is a function whose graph is a straight line. You can write it as f(x) = mx + b, where m is the slope, or constant rate of change, and b is the y-intercept. Each time x increases by 1, the output changes by the same amount, m.Functions · Grade 8, Algebra 1Proportional 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 1ProportionA 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 8Cross MultiplicationCross multiplication is a shortcut for solving a proportion: if ab = cd, then ad = bc. Multiply each numerator by the denominator on the other side and set the two products equal. To solve x6 = 1015, write 15x = 60, so x = 4. It works only when one fraction equals one fraction.Ratios and Proportions · Grade 7, Grade 8, Algebra 1