Proportional Relationships Explained: Tables, Graphs and Equations
A proportional relationship is the simplest kind of connection between two quantities: when one doubles, the other doubles too. You meet them every time you pay by the pound, earn by the hour or drive at a steady speed. This guide shows you exactly how to recognize a proportional relationship in a table, a graph, an equation and a word problem, and how to find its constant of proportionality.
Key takeaways
- Two quantities are in a proportional relationship when their ratio y/x is always the same number.
- That fixed number is the constant of proportionality, k. It is also the unit rate.
- Every proportional relationship has an equation of the form y = kx, with nothing added or subtracted.
- Its graph is a straight line through the origin, (0, 0), and it passes through the point (1, k).
- If y = kx + b with b not zero, the relationship is linear but not proportional.
Table of Contents
- What Is a Proportional Relationship?
- The Constant of Proportionality (k)
- How to Identify a Proportional Relationship in a Table
- How to Identify a Proportional Relationship on a Graph
- How to Identify a Proportional Relationship in an Equation
- How to Spot a Proportional Relationship in a Word Problem
- How to Write the Equation of a Proportional Relationship
- Comparing Two Proportional Relationships
- Proportional vs Linear Relationships
- Real World Examples of Proportional Relationships
- Where This Fits in School Math
- Common Mistakes
- Practice Problems With Answers
- Frequently Asked Questions
- Keep Learning
What Is a Proportional Relationship?
Two quantities x and y have a proportional relationship when y is always the same multiple of x. In other words, the ratio y/x never changes. If a movie ticket costs $12, then 1 ticket costs $12, 2 tickets cost $24 and 5 tickets cost $60. The cost divided by the number of tickets is always 12.
Mathematicians also call this direct variation or say that “y varies directly with x” or “y is directly proportional to x”. All of these phrases describe the same idea, and you can write it with the symbol ∝ as y ∝ x.
Every pair of values in a proportional relationship forms a proportion with every other pair. For the tickets, 2/24 = 5/60, because both equal 1/12. If you are new to proportions, start with our guide to what a proportion is.
The Constant of Proportionality (k)
The constant of proportionality is the fixed number that connects the two quantities. You find it by dividing y by x:
Formula
k = y ÷ x. In a proportional relationship, this gives the same value for every pair of numbers.
The constant of proportionality is the same thing as the unit rate: the amount of y for exactly 1 unit of x. In the ticket example, k = 12 dollars per ticket. If a car travels 150 miles in 2.5 hours at a steady speed, k = 150 ÷ 2.5 = 60 miles per hour.
On a graph, k is also the slope of the line. That is why steeper lines mean faster rates, higher prices or bigger unit rates.
How to Identify a Proportional Relationship in a Table
A table is proportional when y ÷ x gives the same answer in every row. Add a third column for y/x and check it.
| Hours worked (x) | Pay in dollars (y) | y ÷ x |
|---|---|---|
| 2 | 36 | 18 |
| 5 | 90 | 18 |
| 8 | 144 | 18 |
| 10 | 180 | 18 |
| Hours (x) | Cost in dollars (y) | y ÷ x |
|---|---|---|
| 1 | 45 | 45 |
| 2 | 75 | 37.5 |
| 3 | 105 | 35 |
| 4 | 135 | 33.75 |
Two quick tips help with tables. First, if the table includes x = 0, then y must also be 0. Second, do not test the pattern with differences. In the second table, the cost goes up by $30 each hour, which looks steady, but the ratio is not constant, so the relationship is not proportional.
Paste any two rows into our proportion calculator to see whether they form a proportion. It also builds a full ratio table from any proportion you enter.
How to Identify a Proportional Relationship on a Graph
A graph shows a proportional relationship when it passes two tests:
- The points form a straight line.
- The line goes through the origin, the point (0, 0).
Both tests must pass. The graph below shows two straight lines. Only the green one is proportional.
Reading k from a graph
Find the point on the line where x = 1. Its y value is the constant of proportionality. On the green line, the point (1, 2.5) tells you k = 2.5. You can also pick any point (x, y) on the line and divide y by x. The point (4, 10) gives 10 ÷ 4 = 2.5 as well.
What the points mean
- (0, 0) means zero of one quantity matches zero of the other, like 0 hours worked and $0 earned.
- (1, k) shows the unit rate, like $18 for 1 hour.
- Any other point (x, y) is a matching pair, like 8 hours and $144.
How to Identify a Proportional Relationship in an Equation
An equation shows a proportional relationship when it can be written as y = kx. The y value equals a constant times x, and nothing is added or subtracted.
| Equation | Proportional? | Why |
|---|---|---|
| y = 7x | Yes | k = 7 |
| y = x/4 | Yes | Same as y = 0.25x, so k = 0.25 |
| y = 2.5x | Yes | k = 2.5 |
| d = 60t | Yes | Distance equals 60 times time, k = 60 |
| y = 3x + 1 | No | Adding 1 moves the line off the origin |
| y = x² | No | The ratio y/x changes, and the graph curves |
| xy = 12 | No | This is an inverse relationship, y = 12/x |
The last row is a common trap. In xy = 12, the product stays constant, not the ratio. That makes it an inverse proportion, where y goes down as x goes up.
How to Spot a Proportional Relationship in a Word Problem
Look for a constant rate and a starting value of zero. Phrases like “per”, “each”, “for every” and “at a rate of” often signal a proportional relationship. Then ask whether zero of one quantity means zero of the other.
- Proportional: “Apples cost $1.80 per pound.” Cost = 1.80 × pounds, and 0 pounds cost $0.
- Proportional: “A pump moves 15 gallons each minute.” Gallons = 15 × minutes.
- Not proportional: “A gym charges a $50 sign up fee plus $25 per month.” The fee means 0 months still costs $50.
- Not proportional: “A candle is 12 inches tall and burns 1 inch per hour.” The height goes down from a starting value that is not zero.
How to Write the Equation of a Proportional Relationship
- Pick any pair of values (x, y) from the table, graph or story. Avoid (0, 0).
- Find k = y ÷ x.
- Write y = kx with your value of k.
- Check the equation with a second pair of values.
For example, if 4 pounds of grapes cost $10, then k = 10 ÷ 4 = 2.5, and the equation is y = 2.5x. Check: 6 pounds should cost 2.5 × 6 = $15. You can then use the equation to answer any question, like the cost of 9 pounds: 2.5 × 9 = $22.50.
Check any pair of values
Enter two pairs and our calculator tells you whether they are proportional, shows the unit rate and draws the graph.
Comparing Two Proportional Relationships
When two relationships are both proportional, compare their constants of proportionality. The larger k means a faster rate or a higher price per unit.
Say Maya earns $90 for 5 hours of tutoring, and Leo earns according to the equation y = 17x. Maya’s k is 90 ÷ 5 = 18, and Leo’s k is 17. Maya earns $1 more per hour. On a graph, Maya’s line would be slightly steeper.
Proportional vs Linear Relationships
Every proportional relationship is linear, but not every linear relationship is proportional. The difference is the starting value.
| Feature | Proportional | Linear but not proportional |
|---|---|---|
| Equation | y = kx | y = kx + b, with b not 0 |
| Graph | Straight line through (0, 0) | Straight line that misses (0, 0) |
| Ratio y/x | Constant | Changes |
| Starting value | 0 | b, such as a fee or a head start |
| Example | Pay of $18 per hour | A $15 fee plus $30 per hour |
Real World Examples of Proportional Relationships
| Situation | Equation | Constant of proportionality |
|---|---|---|
| Hourly pay at $18 per hour | pay = 18 × hours | $18 per hour |
| Driving at a steady 60 mph | distance = 60 × time | 60 miles per hour |
| Gas at $3.50 per gallon | cost = 3.50 × gallons | $3.50 per gallon |
| Converting inches to centimeters | cm = 2.54 × inches | 2.54 cm per inch |
| Sales tax of 8% | tax = 0.08 × price | 0.08 |
| Map scale of 1 inch to 20 miles | miles = 20 × inches | 20 miles per inch |
| Circle circumference | C = π × d | π, about 3.1416 |
Unit conversions are a great example. The relationship between inches and centimeters is exactly proportional, because 1 inch is defined as 2.54 centimeters, and 0 inches is 0 centimeters.
Where This Fits in School Math
Proportional relationships are a major focus of 7th grade math in the United States. The Common Core standard 7.RP.A.2 asks students to recognize and represent proportional relationships between quantities, including in tables, graphs, equations and diagrams. In algebra, the same idea returns as direct variation. The free OpenStax Elementary Algebra lesson on direct and inverse variation has extra worked examples.
Common Mistakes
- Thinking every straight line is proportional. The line must also pass through (0, 0).
- Checking differences instead of ratios. Steady increases can still come from a non zero start. Always divide y by x.
- Dividing x by y. The constant of proportionality is y ÷ x. Dividing the other way gives the reciprocal, which answers a different question.
- Ignoring the fee. A fixed fee, a starting amount or a head start breaks proportionality.
- Confusing constant product with constant ratio. If x × y stays the same, the relationship is inverse, not proportional.
Practice Problems With Answers
1. Is this table proportional? x: 3, 6, 9 and y: 12, 24, 36.
Show answer
Yes. 12 ÷ 3 = 4, 24 ÷ 6 = 4 and 36 ÷ 9 = 4, so k = 4 and y = 4x.
2. Is y = 5x + 2 a proportional relationship?
Show answer
No. The + 2 means the line crosses the y axis at 2, not at the origin.
3. A line passes through (0, 0) and (5, 35). Find k and the equation.
Show answer
k = 35 ÷ 5 = 7, so y = 7x.
4. Pears cost $6 for 4 pounds. Write the equation and find the cost of 10 pounds.
Show answer
k = 6 ÷ 4 = 1.5, so y = 1.5x. For 10 pounds, y = 1.5 × 10 = $15.
5. Car A goes 210 miles in 3.5 hours. Car B follows d = 62t. Which is faster?
Show answer
Car A: 210 ÷ 3.5 = 60 mph. Car B: 62 mph. Car B is faster.
6. Is xy = 20 a proportional relationship?
Show answer
No. The product is constant, so y = 20/x is an inverse relationship.
Frequently Asked Questions
What is a proportional relationship?
A proportional relationship is one where two quantities always have the same ratio. If y is always k times x, the relationship is proportional and its equation is y = kx.
How do you tell if a relationship is proportional?
Divide y by x for every pair of values. If you always get the same number, the relationship is proportional. On a graph, look for a straight line through the origin.
What is the constant of proportionality?
It is the fixed number k in y = kx. You find it with k = y ÷ x. It is the same as the unit rate and the slope of the graph.
What does a proportional relationship look like on a graph?
It is a straight line that passes through the origin, (0, 0). The steeper the line, the larger the constant of proportionality.
Why must a proportional graph go through the origin?
Because y = kx gives y = 0 when x = 0. Zero of one quantity always matches zero of the other, like 0 hours worked and $0 earned.
Is every linear relationship proportional?
No. A linear relationship y = kx + b is proportional only when b = 0. If there is a starting value, such as a fee, the line is straight but not proportional.
How do you find the constant of proportionality from a table?
Divide each y value by its x value. If every result matches, that number is k. For x = 2 and y = 36, k = 18.
Is a proportional relationship the same as direct variation?
Yes. Direct variation, direct proportion and proportional relationship all describe y = kx, where y changes by the same factor as x.
Is the unit rate the same as the constant of proportionality?
Yes. Both tell you how much y there is for exactly 1 unit of x. If 5 pounds cost $10, the unit rate and the constant of proportionality are both $2 per pound.
What is an example of a relationship that is not proportional?
A taxi that charges a $3 base fare plus $2 per mile is not proportional. Its equation is y = 2x + 3, so the line does not pass through the origin.
Keep Learning
- Start with the basics: what a proportion is, with its parts and properties.
- Learn the opposite idea in our guide to inverse proportion.
- Check any two pairs of values with the proportion calculator.
- Simplify or compare rates and ratios with the ratio calculator.