Proportional Relationship
In short: Proportional relationship describes two quantities whose ratio never changes, so a table gives the same y divided by x and the graph is a line through the origin.
7th GradeRatios and Percents7.RP.A.2a
What is Proportional Relationship?
In a proportional relationship, doubling one quantity doubles the other, tripling one triples the other, and zero pairs with zero. A farm stand that sells apples at a flat price per pound works this way.
Precisely, two quantities \(x\) and \(y\) are proportional when \(y = kx\) for a fixed nonzero number \(k\). Every pair has \(\frac{y}{x} = k\), and its graph is a line that starts at \((0, 0)\).
How to spot it
- A table shows pairs where y divided by x gives the same number in every row.
- A graph is a ruler-straight stretch starting exactly at the corner where the axes meet.
- A story charges a flat rate per item with no starting fee, no deposit and no bonus.
- An equation has the form y = kx with nothing added or subtracted.
The method
- Compute \(\frac{y}{x}\) for each pair in the table.
- If every quotient is equal, the relationship is proportional; one different quotient breaks it.
- For a graph, check that the points are collinear and that the line starts at \((0, 0)\).
- Write the equation \(y = kx\) using the common quotient.
Proportional Relationship: solved example
Problem
A farm stand in Vermont sells apples at $2.25 per pound. Is the cost proportional to the weight? Test it with 2, 4 and 6 pounds, which cost $4.50, $9.00 and $13.50.
Solution
Ratios of cost to weight: \(\frac{4.50}{2} = 2.25\), \(\frac{9.00}{4} = 2.25\) and \(\frac{13.50}{6} = 2.25\).
All three match, so the cost is proportional to weight, with \(y = 2.25x\).
Try it
Question 1
A climbing wall costs $35 to join, then $8 for each visit. Is the total proportional to the visits?
Show the answer
No. Zero visits still cost $35, so the line misses the origin and the quotients differ.
Question 2
Pairs (3, 12), (5, 20), (8, 32). Is y proportional to x?
Show the answer
Yes. \(\frac{12}{3} = \frac{20}{5} = \frac{32}{8} = 4\).
Proportional Relationship vs. Linear Function
Every proportional relationship is linear, but not every linear one is proportional. A line like \(y = 3x + 5\) has a starting amount, so it misses the origin and the ratio of y to x keeps changing.
Keep going
Lessons
Worksheets and quizzes
- Proportional Relationships and y = kx – Worksheet
- Proportional Relationships and y = kx – Quiz
- Proportional Relationships and Slope – Worksheet
- Proportional Relationships and Slope – Quiz
Related words
Whole dictionary: A–Z · 7th Grade · Ratios and Percents