⚖️ Direct Proportion

Cambridge Lower Secondary · Grade 7 · Ratio & Proportion

The Key Relationship
y ∝ x  ⟹  y = kx
Double x → Double y. Triple x → Triple y.
Constant of Proportionality
k = y ÷ x   (always the same!)
If k is constant, x and y are directly proportional.
Unitary Method
Find value of 1 unit, then scale
5 apples cost £2 → 1 apple costs £0.40 → 8 cost £3.20

Watch direct proportion in action! 🍬

Bags of sweets: price goes up proportionally with quantity

1 bag
£0.50
2 bags
£1.00
3 bags
£1.50
4 bags
£2.00
5 bags
£2.50

k = £0.50 per bag — always constant! ✓

What you'll learn:

  • What direct proportion means: y ∝ x, written as y = kx
  • Identifying direct proportion from tables (constant ratio y/x)
  • Finding the constant of proportionality k
  • Unitary method: find value of 1, then scale up
  • Direct proportion graphs: straight line through the origin
  • Real-world contexts: exchange rates, speed, recipes, unit pricing
  • Setting up and solving proportion equations
Created by Mr Josh

📖 Learn: Direct Proportion

Part 1: What is Direct Proportion?

Two quantities are in direct proportion when one increases at the same rate as the other.

📌 If y is directly proportional to x, we write: y ∝ x
This means: y = kx for some constant k
Key property: if x doubles, y doubles. If x triples, y triples. If x halves, y halves.
💡 Real-life examples: total cost of identical items, distance at constant speed, currency exchange.

Part 2: Identifying Direct Proportion from Tables

To check if a table shows direct proportion, calculate y ÷ x for each pair. If the ratio is always the same, they are directly proportional.

x (items)y (cost £)y ÷ x
252.50
4102.50
6152.50
10252.50
✅ y ÷ x = 2.50 always → Direct proportion! k = 2.50

Now a non-example — check the ratios:

xyy ÷ x
133
273.5
3124
❌ y ÷ x varies → NOT direct proportion
💡 Also check: does the relationship pass through the origin (0, 0)? Direct proportion always does.

Part 3: Finding the Constant of Proportionality k

k = y ÷ x    →    y = kx
k is called the constant of proportionality (or unit rate)
Example: 8 metres of rope costs £3.60. Find k.
k = 3.60 ÷ 8 = £0.45 per metre
Equation: y = 0.45x where x = metres, y = cost in £
Use this to find cost of 15 m: y = 0.45 × 15 = £6.75
💡 k has units! If y is in £ and x is in metres, then k is in £/m.

Part 4: Unitary Method

The unitary method finds the value of 1 unit first, then scales up (or down).

Step 1: Find the value of 1 unit by dividing
Step 2: Multiply to find the target amount
🎂 Example: 4 cakes need 600 g of flour. How much for 7 cakes?
÷4:   1 cake → 600 ÷ 4 = 150 g
×7:   7 cakes → 150 × 7 = 1050 g
💡 Always write the ÷ step and the × step clearly. It's the most reliable method in exams!

Part 5: Direct Proportion Graphs

The graph of a direct proportion is always a straight line through the origin (0, 0).

Equation: y = kx → gradient = k
If k is large: steep line. If k is small: shallow line.
Three key features: ① Straight line   ② Through origin   ③ Positive gradient
⚠️ A straight line that does NOT pass through the origin is NOT direct proportion.
💡 The gradient of the line equals k. You can read k directly from the graph: k = rise ÷ run.
Created by Mr Josh

💡 Worked Examples

Example 1: Identifying Direct Proportion from a Table

Determine whether the table shows direct proportion.

xy
312
520
832
1144
Step 1: Calculate y ÷ x for each row:
12 ÷ 3 = 4  ·  20 ÷ 5 = 4  ·  32 ÷ 8 = 4  ·  44 ÷ 11 = 4
Step 2: The ratio y/x = 4 for every pair ✅
Conclusion: Yes — directly proportional with k = 4. Equation: y = 4x
✅ Check: when x = 0, y = 0 — always passes through origin.

Example 2: Finding k and Using the Equation

A car uses petrol at a constant rate. 5 litres cover 65 km. Find k and the distance covered by 12 litres.

Step 1: Let y = distance (km), x = petrol (litres)
Step 2: k = y ÷ x = 65 ÷ 5 = 13 km/litre
Step 3: Equation: y = 13x
Step 4: When x = 12:   y = 13 × 12 = 156 km
⛽ The constant k = 13 means the car travels 13 km on every litre of petrol.

Example 3: Unitary Method — Currency Exchange

£1 = 4.60 AED. How many AED do you get for £35?

Step 1 (find 1 unit): £1 = 4.60 AED   ← this IS the unit rate
Step 2 (scale up): £35 = 4.60 × 35 = 161 AED
What if we want to reverse? How many £ for 230 AED?
÷4.60:   1 AED = 1 ÷ 4.60 = £0.2174...
×230:   230 AED = 0.2174 × 230 ≈ £50
💱 Exchange rates are real-world constants of proportionality!

Example 4: Setting Up a Proportion Equation

y is directly proportional to x. When x = 6, y = 21. Find y when x = 10.

Step 1: Write y = kx
Step 2: Substitute known values: 21 = k × 6
Step 3: Solve for k: k = 21 ÷ 6 = 3.5
Step 4: Equation: y = 3.5x
Step 5: Find y when x = 10: y = 3.5 × 10 = 35
📝 Always show the y = kx equation clearly — it's the method examiners want to see.
Created by Mr Josh

📈 Visualizer

📉 Proportion Graph
💱 Currency Converter
🃏 Is it Proportional?
🔢 Unitary Animator

Adjust k and see the line y = kx animate through the origin!

2.0

Hover over the line to read coordinates

Live Currency Converter — k is highlighted!

GBP (£)
⇒
AED
460.00

Working shown:

Drag each table card into the correct bin!

Check y ÷ x — if it's constant: Direct Proportion ✓

✅ Direct Proportion

❌ NOT Direct Proportion

Watch the unitary method step-by-step!

Created by Mr Josh

✏️ Exercise 1: Identify Direct Proportion

For each table, calculate y ÷ x for all pairs. Type YES or NO.

Created by Mr Josh

✏️ Exercise 2: Find the Constant of Proportionality

Each table shows direct proportion. Find the value of k.

Created by Mr Josh

✏️ Exercise 3: Unitary Method

Use the unitary method: ÷ to find 1 unit, then × to scale up.

Created by Mr Josh

✏️ Exercise 4: Proportion Equations

y ∝ x. Use y = kx to find the missing values.

Created by Mr Josh

🌍 Exercise 5: Real-World Proportion Problems

Apply direct proportion to real-life contexts. Show your method!

Created by Mr Josh

📝 Practice Questions

Write full working. Reveal answers when done.

  1. A table shows: x = 2, y = 8; x = 5, y = 20; x = 9, y = 36. Is y directly proportional to x? Give a reason.
  2. y is directly proportional to x. When x = 4, y = 18. Find k and write the equation.
  3. y = kx. When x = 7, y = 42. Find y when x = 11.
  4. 6 pens cost £2.40. Find the cost of 10 pens using the unitary method.
  5. A recipe for 4 people needs 320 g of rice. How much for 7 people?
  6. A car travels 180 km on 15 litres of petrol. Find the distance on 22 litres.
  7. £1 = 1.27 USD. Convert £85 to USD.
  8. A table shows: x = 3, y = 9; x = 6, y = 19. Is this direct proportion? Explain.
  9. y ∝ x. When x = 8, y = 52. Find y when x = 3.
  10. A map has scale 1:50 000. A road is 3.4 cm on the map. Find its real length in km.
  11. 12 workers can build a wall in a directly proportional context: 12 workers produce 360 bricks per hour. How many bricks per hour for 5 workers?
  12. A graph of y against x passes through (0, 4) and (3, 10). Is this direct proportion? Why?
  13. Find k if y = kx and the graph passes through (5, 35).
  14. Water flows into a tank at a constant rate. After 4 minutes there are 28 litres. How many litres after 11 minutes?
  15. Write the equation of proportionality if k = 0.75.
  16. A taxi charges per km. 8 km costs £12.80. Find the cost of 15 km.
  17. Silver costs £14 per gram. Write the formula for cost C in terms of mass m, then find cost of 23 g.
  18. x = 5, y = 30; x = 9, y = 54; x = 12, y = 72. Find k. Find x when y = 96.
  19. Explain why a straight-line graph through (0, 0) represents direct proportion.
  20. If y = 3.5x, find the value of x when y = 49.
  1. Yes — y ÷ x = 4 for all pairs; constant ratio confirms direct proportion.
  2. k = 18 ÷ 4 = 4.5; y = 4.5x
  3. k = 42 ÷ 7 = 6; y = 6 × 11 = 66
  4. 1 pen = £2.40 ÷ 6 = £0.40; 10 pens = £0.40 × 10 = £4.00
  5. 1 person: 320 ÷ 4 = 80 g; 7 people: 80 × 7 = 560 g
  6. k = 180 ÷ 15 = 12 km/litre; distance = 12 × 22 = 264 km
  7. 85 × 1.27 = $107.95
  8. 9 ÷ 3 = 3; 19 ÷ 6 ≈ 3.17 — ratios differ → NOT direct proportion
  9. k = 52 ÷ 8 = 6.5; y = 6.5 × 3 = 19.5
  10. 3.4 cm × 50 000 = 170 000 cm = 1.7 km
  11. k = 360 ÷ 12 = 30 bricks/worker; 5 × 30 = 150 bricks/hour
  12. No — passes through (0, 4), not (0, 0); y-intercept ≠ 0 → NOT direct proportion
  13. k = 35 ÷ 5 = 7
  14. k = 28 ÷ 4 = 7 L/min; after 11 min: 7 × 11 = 77 litres
  15. y = 0.75x
  16. k = 12.80 ÷ 8 = £1.60/km; 15 km = 1.60 × 15 = £24.00
  17. C = 14m; 14 × 23 = £322
  18. k = 30 ÷ 5 = 6; when y = 96: x = 96 ÷ 6 = 16
  19. When x = 0, y = 0 (through origin); gradient = k constant → directly proportional by definition
  20. x = 49 ÷ 3.5 = 14
Created by Mr Josh

🏆 Challenge Questions

Harder problems — push your thinking! Write full working.

  1. y ∝ x. When x = 1.5, y = 10.5. Find y when x = 6.4. Give your answer to 2 decimal places.
  2. A table has values: (x, y) = (2, 5), (4, 10), (6, 15), (n, 40). Find n. If the table shows direct proportion, write the equation and find y when x = 3.6.
  3. Two quantities P and Q are directly proportional. P = 24 when Q = 9.6. Find P when Q = 14.4. Then find Q when P = 35.
  4. A graph of y = kx passes through (4, 14). A second line y = 2x + 1 is drawn on the same axes. Find the coordinates of the point where the two lines cross. Explain why only y = kx represents direct proportion.
  5. In a recipe, flour and butter are in direct proportion. 250 g flour needs 80 g butter. A baker wants to use 375 g flour. How much butter? If butter costs £2.40 per 100 g, what is the total butter cost?
  6. Exchange rate: £1 = 4.60 AED and £1 = 1.27 USD. If Ali has 690 AED, how many USD can he get? (Work via £ first.)
  7. The distance–time graph of a cyclist is a straight line through the origin. At t = 25 min the distance is 10 km. Find the speed in km/h. Write the proportion equation and find the distance at t = 1 hour 12 minutes.
  8. y is proportional to x. The equation is y = kx. If you double x and halve k, what happens to y? Generalise: if x is scaled by factor m and k by factor n, express the new y in terms of the original y.
  1. k = 10.5 ÷ 1.5 = 7; y = 7 × 6.4 = 44.80
  2. k = 5 ÷ 2 = 2.5; equation y = 2.5x; n = 40 ÷ 2.5 = 16; y at x=3.6: 2.5 × 3.6 = 9
  3. k = 24 ÷ 9.6 = 2.5; P = 2.5 × 14.4 = 36; Q = 35 ÷ 2.5 = 14
  4. k = 14 ÷ 4 = 3.5; y = 3.5x. Set 3.5x = 2x + 1 → 1.5x = 1 → x = 2/3; y = 3.5 × 2/3 = 7/3 ≈ 2.33. Only y = 3.5x passes through origin — direct proportion requires y = 0 when x = 0.
  5. k = 80 ÷ 250 = 0.32; butter = 0.32 × 375 = 120 g; cost = (120/100) × £2.40 = £2.88
  6. 690 AED ÷ 4.60 = £150; £150 × 1.27 = $190.50
  7. k = 10 ÷ 25 = 0.4 km/min = 24 km/h; d = 0.4t (min) or d = 24t (hours); t = 72 min → d = 0.4 × 72 = 28.8 km
  8. New y = (mk)(nx) — wait, new k = k/2 and new x = 2x → new y = (k/2)(2x) = kx = y. y stays the same! General: new y = (mn) × original y.
Created by Mr Josh