Lesson 17: Proportional Relationships and Unit Rates in Graphs

A puppy grows steadily: at 2 weeks it weighs 5 pounds, at 4 weeks 10 pounds, at 6 weeks 15 pounds. Weight and age here march in perfect step — a proportional relationship. Spot one, and you can predict any value from any other with a single multiplication. In this lesson you will recognise proportional relationships in tables and graphs, find the constant of proportionality, and write equations of the form y = kx that let you calculate forwards, backwards and sideways.

What makes a relationship proportional?

Two quantities are proportional when their ratio never changes: y ÷ x is always the same number, and the graph is a straight line through the origin (0, 0). The puppy: 5 ÷ 2 = 2.5, 10 ÷ 4 = 2.5, 15 ÷ 6 = 2.5 pounds per week. The constant ratio, 2.5, is the constant of proportionality, written k. The equation is y = 2.5x: weight = 2.5 × age in weeks.

Worked example 1. A table shows x: 2, 4, 6 and y: 5, 10, 15. Check the ratios: 5/2 = 2.5, 10/4 = 2.5, 15/6 = 2.5. Constant ratio, so the relationship is proportional with k = 2.5 and equation y = 2.5x. Use it: at x = 10, y = 25.

Worked example 2 (not proportional). A taxi charges y = x + 2 dollars for x miles. Ratios: at x = 2, y = 4 (ratio 2); at x = 4, y = 6 (ratio 1.5). The ratios differ, and the graph starts at (0, 2), not the origin — the $2 pickup fee breaks proportionality. Proportional means multiplication only: no added constants allowed.

Reading k from a graph

On a graph, k is the y-value when x = 1 — the unit rate, drawn. A climbing graph passes through (3, 150): 150 metres of height gained in 3 hours gives k = 150 ÷ 3 = 50 metres per hour, so the equation is h = 50t. The steeper the line, the bigger k is: plant B growing 3 cm per week outruns plant A at 2 cm per week, and its line is visibly steeper. Comparing two proportional graphs is just comparing their steepness.

Worked example 3. Two juice machines fill bottles. Machine 1's graph passes through (2, 10) and Machine 2's through (2, 6). k values: 10 ÷ 2 = 5 bottles per minute versus 6 ÷ 2 = 3. Machine 1 is faster — its line is steeper, and at (2, ...) it is already higher.

Solving with the equation

Once you have y = kx, any value gives the other. If a snail's distance and time obey d = 3t (3 cm per minute), then in 10 minutes it travels d = 3 × 10 = 30 cm, and to travel 45 cm it needs t = 45 ÷ 3 = 15 minutes. Divide to reverse; multiply to go forward.

In the real world: recipes, wages and exchange rates

Doubling a recipe keeps every ingredient proportional to the flour. Earning $12 per hour is the proportional relationship pay = 12 × hours (until overtime breaks it with a different rate — real life loves to interrupt proportionality, and noticing where it stops is a genuine skill). Currency exchange is proportional while the rate holds: at k = 1.27 dollars per pound, £50 becomes 50 × 1.27 = $63.50.

Common mistakes

• Calling y = x + 2 proportional. It has an added constant; its graph misses the origin. Proportional relationships are pure multiplication, y = kx.
• Computing k as x ÷ y. k = y ÷ x: for (3, 150), k = 150 ÷ 3 = 50, not 3 ÷ 150. Ask "how much y per one x?"
• Assuming every straight line is proportional. A straight line through (0, 5) is linear but not proportional. The line must pass through (0, 0).
• Checking only one pair of values. One pair always has some ratio. Proportionality requires the ratio to be the same for every pair in the table.

Exam Experience

1. A graph passes through the origin and the point (4, 10). What is the constant of proportionality k? A) 0.4 B) 2.5 C) 4 D) 40

2. Which equation describes a proportional relationship with k = 4? A) y = x + 4 B) y = 4x C) y = x ÷ 4 D) y = 4 ÷ x

3. The point (5, 20) lies on the graph of a proportional relationship. What is the value of y when x = 8? A) 23 B) 28 C) 32 D) 40

4. (Constructed response) A snail's journey is recorded in a table: after 2 minutes it has travelled 6 cm; after 4 minutes, 12 cm; after 6 minutes, 18 cm. (a) Show that the relationship between distance and time is proportional. (b) Find the constant of proportionality and write the equation connecting distance d and time t. (c) Predict the distance travelled after 10 minutes. (d) How long does the snail take to travel 45 cm?

Answer key and explanations

• 1. B — k = y ÷ x = 10 ÷ 4 = 2.5.
• 2. B — y = kx with k = 4 is y = 4x.
• 3. C — k = 20 ÷ 5 = 4, so y = 4 × 8 = 32.
• 4. (a) Ratios: 6 ÷ 2 = 3, 12 ÷ 4 = 3, 18 ÷ 6 = 3 — the ratio is constant at 3, so the relationship is proportional. (b) k = 3 cm per minute; equation d = 3t. (c) d = 3 × 10 = 30 cm. (d) t = 45 ÷ 3 = 15 minutes.

Scoring guide for question 4 (4 points)

• 1 point for checking at least two (preferably all three) ratios and finding them equal.
• 1 point for k = 3 with units (cm per minute) and the equation d = 3t.
• 1 point for the prediction, 30 cm.
• 1 point for the reversed calculation, 15 minutes, found by dividing 45 by 3.

Going further: steeper lines and sneaky tables

Worked example 4. Two lines through the origin: line A passes through (2, 8), so k = 4; line B passes through (3, 9), so k = 3. Line A is steeper and represents the faster rate — 4 units of y for every 1 of x beats 3. When graphs share an axis, the steeper line always has the larger k, no calculation needed once you trust the picture.

Worked example 5. The relationship y = 6x: find x when y = 42. Divide: x = 42 ÷ 6 = 7. Equations of the form y = kx solve backwards by division just as two-step equations did — find the forward rule first, then undo it.

Worked example 6 (the sneaky table). x: 1, 2, 3; y: 4, 8, 13. Ratios: 4/1 = 4, 8/2 = 4, 13/3 ≈ 4.33. The first two pairs agree, the third breaks ranks — so the relationship is not proportional. Proportionality is a promise kept by every pair, and checking only two pairs is how this trap catches people. (The differences, 4 then 5, also betray it: not even linear.)

Worked example 7. A photograph 10 cm wide and 6 cm tall is enlarged proportionally to 25 cm wide. The scale factor is 25 ÷ 10 = 2.5, so the new height is 6 × 2.5 = 15 cm. Enlargements, reductions and photocopier settings are proportional relationships wearing a percent costume (250% here — lesson 5 again).

Quick practice

• 1. Find k for a proportional graph through (4, 22).
• 2. For y = 0.5x, find y when x = 14.
• 3. Is y = 3x + 0 a proportional relationship? Why?
• 4. A point (6, 15) lies on a proportional graph. Find k and the value of y at x = 10.

Answers. 1. k = 22 ÷ 4 = 5.5. 2. y = 0.5 × 14 = 7. 3. Yes — it is y = kx with k = 3; the "+0" adds nothing and the graph passes through the origin. 4. k = 15 ÷ 6 = 2.5, so y = 2.5 × 10 = 25.

Investigation: two taps, one tank

Tap A fills a tank steadily; its graph passes through (5, 20), so k = 20 ÷ 5 = 4 litres per minute and its rule is V = 4t. Tap B's graph passes through (4, 10): k = 10 ÷ 4 = 2.5 litres per minute, rule V = 2.5t. Open both taps together and the rates add: 4 + 2.5 = 6.5 litres per minute (each minute genuinely delivers 4 litres from A and 2.5 from B). A 130-litre tank fills in 130 ÷ 6.5 = 20 minutes. Adding rates is one of the most useful moves in applied mathematics — it runs work crews, printers and internet downloads too ("two workers paint a fence" problems are this, word for word). But watch the boundary of the model: the moment the tank is full, proportionality stops. And a car park charging $3 entry plus $2 per hour is not a proportional system at all — its graph starts at (0, 3) — which is why "both taps" logic would fail there. Rates add only when each process is truly proportional, from zero.

Challenge corner

A proportional table with decimals: x: 2, 3, 5 and y: 3, 4.5, 7.5. Check the ratios: 3 ÷ 2 = 1.5; 4.5 ÷ 3 = 1.5; 7.5 ÷ 5 = 1.5 — constant, so the relationship is proportional with k = 1.5 and rule y = 1.5x. Predict y at x = 9: 1.5 × 9 = 13.5. Decimal constants frighten students into suspecting the relationship is broken; it is not — proportionality cares about the ratio being constant, not about it being whole. Real unit rates are usually decimals ($3.75 per hour of parking, 1.8°F per °C from the next lesson): the mathematics of this course does not live in a whole-number world, and the sooner the decimals feel ordinary, the better. A steady habit: when a table mixes integers and decimals, convert everything to decimals first, then test the ratios.