What makes a relationship proportional?
Two quantities are in a proportional relationship when their ratio never changes. If you double one, the other doubles. If you triple one, the other triples.
Example: Apples cost $2 per kilogram.
| Mass (kg) x | 1 | 2 | 3 | 5 |
|---|---|---|---|---|
| Cost ($) y | 2 | 4 | 6 | 10 |
The ratio y/x is always 2. That fixed number is the constant of proportionality, k. It’s also the unit rate: $2 per kg.
The equation
Every proportional relationship can be written as
y = kx
For the apples: y = 2x.
Three tests
| Test | Proportional if… |
|---|---|
| Ratio table | y ÷ x is the same for every pair |
| Graph | It’s a straight line through the origin (0, 0) |
| Equation | It has the form y = kx (no "+ something") |
A non-example
A taxi charges $4 to start plus $2 per km.
| km | 1 | 2 | 3 |
|---|---|---|---|
| Cost ($) | 6 | 8 | 10 |
Ratios: 6, 4, 3.33 — not constant. The equation is y = 2x + 4. The graph is a straight line, but it starts at (0, 4), not the origin. So it’s not proportional. (It’s linear, but that’s different.)
Reading k from a graph
On the graph of a proportional relationship, the point (1, k) shows the unit rate. Any point (x, y) on the line gives k = y/x.
Examples of proportional relationships
- Distance travelled at a constant speed: d = 60t
- Pay at an hourly rate: P = 17h
- Cost of gas: C = 1.6L
- Perimeter of a square: P = 4s
Examples that are not proportional
- Age and height (not a constant ratio)
- Phone plan with a monthly fee plus per-gigabyte charge
- Area of a square vs side (A = s², a curve)
Worked examples
Example 1: Is this proportional? x: 2, 4, 6; y: 5, 10, 15.
5/2 = 10/4 = 15/6 = 2.5. Yes, k = 2.5, y = 2.5x.
Example 2: A recipe uses 3 eggs for 12 muffins. Write the equation for eggs (e) and muffins (m).
k = 3/12 = 0.25 → e = 0.25m.
Example 3: A graph is a straight line through (0, 3) and (2, 7). Proportional?
It doesn’t pass through the origin: no.
Common mistakes
- Thinking every straight line is proportional. It must pass through (0, 0).
- Checking only one pair of values.
- Dividing x by y in one row and y by x in another.
Try it yourself
- Is x: 1, 3, 5 and y: 4, 12, 20 proportional? Find k.
- Is y = 3x + 1 proportional?
- You earn $90 for 6 hours. Write an equation for pay p and hours h.
- Using your equation, what do you earn in 15 hours?
Answers: 1) yes, k = 4 2) no 3) p = 15h 4) $225
Quick recap
- A proportional relationship has a constant ratio y/x = k.
- Its equation is y = kx and its graph is a straight line through the origin.
- k is the unit rate.