Proportional Relationships
A proportional relationship is y = kx: a straight line through the origin, with one constant ratio running through every pair.
A recipe feeds 3 people using 2 cups of flour. You are cooking for 9.
Nine is three times three, so you need three times the flour: 6 cups.
Look at what stayed fixed. 23, 46, 69 — all the same number, 23 of a cup each.
That unchanging ratio is the constant of proportionality, k. Here it is cups per person.
And cooking for zero people needs zero flour — which is why the graph of this goes through the origin.
That is the test. A relationship is proportional when yx is the same in every row, which happens exactly when the line passes through (0, 0). A joining fee, a delivery charge or a head start breaks it.
The three signatures
A relationship is proportional if — and only if — all three of these hold, and any one of them implies the others:
In a table: every yx gives the same number.
In a graph: it is a straight line through the origin.
In an equation: it can be written y = kx, with nothing added on.
The origin test
Zero hours worked must mean zero dollars earned. If there is a flat fee, the line misses the origin and the relationship is linear but not proportional.
Finding k from a table
Divide y by x for each row. If you get the same value every time, that value is k.
Hours 2, 5, 8 and pay $30, $75, $120: 302 = 15, 755 = 15, 1208 = 15. So k = 15 and y = 15x.
If even one row disagrees, it is not proportional.
What k means on the graph
k is the slope: how far up the line climbs for each step right.
It is also the y-value at x = 1. Find the point directly above 1 on the graph and read its height — that is k.
Steeper line means larger k means a faster rate.
Reading a proportional graph
Every point (x, y) on the line means "x of these go with y of those".
The point (1, k) is the unit rate. The point (0, 0) says nothing goes with nothing.
When two proportional relationships are graphed together, the steeper one is the better rate — if you want more y per x.
Solving with proportions
Because the ratio is constant, you can set two ratios equal and cross-multiply.
If 3 notebooks cost $4.50, then 4.503 = c7 for 7 notebooks, so 3c = 31.50 and c = $10.50.
Keep the units in the same position on both sides — dollars on top both times, or the equation lies.
Worked examples
Is it proportional?
A gym charges $25 to join plus $15 per month. Is total cost proportional to months?
- Write the rule: y = 15x + 25 A per-month rate plus a one-time fee.
- Test the origin: at x = 0, y = 25, not 0. You pay $25 before any months pass.
- Test the ratios: 401 = 40 but 552 = 27.5 Not constant.
Linear, but not proportional — the join fee breaks it.
Find k and predict
A car uses 7 gallons to drive 224 miles. Write the equation and find the distance on 11 gallons.
- k = 2247 = 32 Miles per gallon — the unit rate.
- Equation: y = 32x y is miles, x is gallons.
- y = 32 × 11 = 352 Substitute and multiply.
y = 32x, and 352 miles.
Comparing two rates
Printer A prints y = 18x pages in x minutes. Printer B prints 130 pages in 10 minutes. Which is faster, and by how much?
- A's rate is k = 18 pages per minute. Read it straight from y = kx.
- B's rate is 13010 = 13 pages per minute. Convert B to a unit rate so the two are comparable.
- 18 - 13 = 5 Compare the two constants directly.
Printer A, by 5 pages per minute.
Words to know
- Proportional
- Two quantities with a constant ratio: yx = k always.
- Constant of proportionality
- k — the unit rate, the slope, and the value of y when x = 1.
- Origin
- The point (0, 0), which every proportional graph passes through.
- Linear but not proportional
- A straight line that misses the origin, like y = 3x + 5.
Mistakes to watch for
Calling any straight line proportional. It must pass through (0,0). y = 3x + 5 is perfectly linear, but at x = 0 it gives 5, so yx is not constant and doubling x does not double y.
Computing xy instead of yx. You get the reciprocal of k — often a tidy-looking number that then gives wrong answers everywhere. Check it against a row: does k times x really give y?
Setting up a proportion with the units crossed. Dollars over items on one side must face dollars over items on the other. Flipping one turns it into a different equation that still solves neatly.
Assuming every doubling is proportional. Doubling the side of a square does not double its area — it quadruples it. Proportional means y = kx, and nothing else.
The Parthenon, a credit card and a sheet of A4 paper are all built on fixed ratios — but different ones.
A4 is special: its sides are in the ratio 1 : √2, so folding it in half gives A5 with exactly the same shape. No other ratio does that.
The system was proposed by the German scientist Georg Lichtenberg in 1786 and became a standard in 1922.