Proportional Relationships — Print

A proportional relationship is y = kx: a straight line through the origin, with one constant ratio running through every pair.

What the practice questions cover

These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

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.

a recipe for 3 people needs 2 cups of flour 3 people 2 6 people 4 9 people 6 double the people, double the flour — the ratio never budges

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.

A proportional graph: a straight line through the origin (0,0) (1, k) y = kx every point has the same ratio y ÷ x = k

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?

  1. Write the rule: y = 15x + 25 A per-month rate plus a one-time fee.
  2. Test the origin: at x = 0, y = 25, not 0. You pay $25 before any months pass.
  3. 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.

  1. k = 2247 = 32 Miles per gallon — the unit rate.
  2. Equation: y = 32x y is miles, x is gallons.
  3. 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?

  1. A's rate is k = 18 pages per minute. Read it straight from y = kx.
  2. B's rate is 13010 = 13 pages per minute. Convert B to a unit rate so the two are comparable.
  3. 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.

Problem 1

Which equation represents a proportional relationship?

  1. y = 4x
  2. y = 4x + 3
  3. y = 4x
  4. y = x2
Worked answer

y = 4x

y = 4x has the form y = kx with k = 4.

The others add a constant, divide, or square — none is proportional.

Problem 2

A table shows x = 2, 4, 6 with y = 10, 20, 30. Find the constant of proportionality k.

Worked answer

5

102 = 5, 204 = 5, 306 = 5

k = 5

Problem 3

A proportional relationship passes through (3, 21). Find k.

Worked answer

7

k = 213 = 7

So y = 7x.

Problem 4

Every proportional graph must pass through which point?

  1. (1, 1)
  2. (0, 0)
  3. (1, 0)
  4. (0, 1)
Worked answer

(0, 0)

Zero of one means zero of the other, so the line goes through the origin (0, 0).

Problem 5

Five pencils cost $2.00. What do 12 pencils cost?

Worked answer

4.80

2.00 ÷ 5 = $0.40 each

0.40 × 12 = $4.80

Problem 6

A table shows x = 3, 5, 8 with y = 12, 20, 32. Find y when x = 11.

Worked answer

44

123 = 4, and the other rows agree, so k = 4

y = 4x

y = 4 × 11 = 44

Problem 7

Which relationship is NOT proportional?

  1. 8 apples for $4; 12 apples for $6
  2. A car travelling 50 miles every hour
  3. A gym charging $25 to join plus $15 a month
  4. 3 cups of flour per dozen cookies
Worked answer

A gym charging $25 to join plus $15 a month

With the gym, zero months still costs $25, so the graph misses (0,0).

It is linear (y = 15x + 25) but not proportional.

Problem 8

A car travels 224 miles on 7 gallons. How far on 11 gallons?

Worked answer

352

k = 2247 = 32 mpg

y = 32x

32 × 11 = 352 miles

Problem 9

A proportional graph passes through (4, 10). Find k as a decimal.

Worked answer

2.5

k = 104 = 2.5

Problem 10

Eight pounds of rice cost $20. How many pounds can you buy with $32.50?

Worked answer

13

20 ÷ 8 = $2.50 per pound

32.50 ÷ 2.50 = 13 pounds

Problem 11

A recipe uses 3 cups of flour for 24 cookies. How much flour for 60 cookies?

Worked answer

7.5

324 = 18 cup per cookie

18 × 60 = 7.5 cups

Problem 12

Printer A follows y = 18x pages in x minutes. Printer B prints 130 pages in 10 minutes. How many more pages per minute does the faster one print?

Worked answer

5

A: 18 pages per minute

B: 130 ÷ 10 = 13 pages per minute

18 - 13 = 5

Problem 13

In a proportional relationship, y = 45 when x = 12. Find x when y = 75.

Worked answer

20

k = 4512 = 3.75

75 = 3.75x

x = 75 ÷ 3.75 = 20

Problem 14

Line A passes through (2, 5) and line B through (3, 7). Both are proportional. Which is steeper?

  1. Line A
  2. Line B
  3. They have the same steepness
Worked answer

Line A

A: k = 52 = 2.5

B: k = 73 ≈ 2.33

2.5 > 2.33, so A is steeper.

Problem 15

On a map, 2.5 cm represents 15 km. How many km does 7 cm represent?

Worked answer

42

15 ÷ 2.5 = 6 km per cm

6 × 7 = 42 km

Problem 16

A table shows x = 2, 4, 7 with y = 7, 14, 24. Is it proportional?

  1. Yes, with k = 3.5
  2. No — the last pair breaks the pattern
  3. No — it does not pass through the origin
Worked answer

No — the last pair breaks the pattern

72 = 3.5 and 144 = 3.5 — so far so good

But 247 ≈ 3.43, which does not match

One disagreeing row is enough to rule it out.

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