Introduction
Direct and indirect proportion describe how things change in relation to each other. Understanding these help you solve real-world problems and exam questions involving rates, conversions between $ and £, metres and miles, and mathematical relationships.
In direct proportion, as one quantity doubles or triples, another quantity doubles or triples. Think of buying apples – the more apples you buy, the more you pay, always at the same rate per apple. In indirect (or inverse) proportion, as one quantity increases, another decreases at the same rate. Consider driving to a destination – double the speed you drive, and it takes half the time to arrive.
In the exam you will see questions about cooking recipes and map scales, to speed calculations and financial planning.
Exam Boards
This topic applies to: AQA (Higher & Foundation), OCR (Higher & Foundation), Edexcel (Higher & Foundation), International Edexcel (Higher & Foundation)
Note: Some advanced proportion concepts (such as inverse square relationships) are Higher tier only across all boards.
TLDR (Too Long; Didn’t Read)
Direct Proportion:
- When one quantity doubles, the other doubles
- Relationship: y = kx (where k is constant)
- Graph: straight line through origin
- General equation with symbol: y ∝ x
Indirect Proportion:
- When one quantity doubles, the other halves
- Relationship: y = k/x (where k is constant)
- Graph: hyperbola (curved)
- General equation with symbol: y ∝ 1/x
Key methods:
- Find the constant k first
- Use k to find unknown values
- Check the answers by putting the numbers in and making sure both sides of the equations are equal
What You Need to Know (From the Specification)
- Solve problems involving direct and inverse proportion, including graphical and algebraic representations
- Express direct and inverse proportion in algebraic terms
- Use the relationship between ratio and linear functions
- Interpret equations that describe direct and inverse proportion (Higher tier only)
- Construct equations that describe direct and inverse proportion (Higher tier only)
- Formulate equations involving proportion to powers or roots of quantities (Higher tier only)
Core Content Sections
Understanding Direct Proportion
Key Facts
🔑 Key Fact: In direct proportion, the ratio between corresponding values is always constant. If y is directly proportional to x, then y = kx where k is called the constant of proportionality.
🔑 Key Fact: Direct proportion graphs always pass through the origin (0,0) and form a straight line.
Explanation
Direct proportion occurs when two quantities increase or decrease at the same rate. If one quantity doubles, the other doubles. If one triples, the other triples.
We write "y is directly proportional to x" as:
y ∝ x
This means: y = kx, where k is a constant.
To solve direct proportion problems:
- Find the constant k using given values
- Use k to find unknown values
- Check your answer makes sense
Example 1: Basic Direct Proportion
If y is directly proportional to x, and y = 15 when x = 3, find y when x = 7.
Step 1: Find the constant k
y = kx
15 = k × 3
k = 15 ÷ 3 = 5
Step 2: Use k to find the unknown value
y = kx = 5 × 7 = 35
Therefore, y = 35 when x = 7.
Step 3: Check
When x = 3, y = 15, so y/x = 5
When x = 7, y = 35, so y/x = 5 ✓
Example 2: Real-World Direct Proportion
The cost of petrol is directly proportional to the number of litres bought. If 8 litres cost £11.20, how much will 15 litres cost?
Step 1: Find the constant (cost per litre)
Cost = k × litres
£11.20 = k × 8
k = £11.20 ÷ 8 = £1.40 per litre
Step 2: Calculate cost for 15 litres
Cost = £1.40 × 15 = £21.00
Therefore, 15 litres will cost £21.00.
Practice Questions
Give yourself a proper go at these before checking the solution – you’ll learn much more that way.
Practice Question 1: If p is directly proportional to q, and p = 24 when q = 8, find p when q = 12.
Solution 1: p ∝ q, so p = kq
Find k: 24 = k × 8, so k = 3
When q = 12: p = 3 × 12 = 36
Practice Question 2: The distance travelled at constant speed is directly proportional to time. A car travels 120 miles in 2 hours. How far will it travel in 3.5 hours?
Solution 2: Distance ∝ time, so Distance = k × time
Find k: 120 = k × 2, so k = 60 mph
For 3.5 hours: Distance = 60 × 3.5 = 210 miles
Learning Block 2: Understanding Indirect (Inverse) Proportion
Key Facts
🔑 Key Fact: In indirect proportion, when one quantity increases, the other decreases such that their product remains constant. If y is inversely proportional to x, then xy = k or y = k/x.
🔑 Key Fact: Indirect proportion graphs form a hyperbola – a curved line that never touches the axes.
Explanation
Indirect (or inverse) proportion occurs when one quantity increases while the other decreases, but their product stays the same. If one quantity doubles, the other halves.
We write "y is inversely proportional to x" as:
y ∝ 1/x
This means: y = k/x, where k is a constant, or xy = k.
Example 3: Basic Indirect Proportion
y is inversely proportional to x. When x = 4, y = 12. Find y when x = 6.
Step 1: Find the constant k
y = k/x, so k = xy
k = 4 × 12 = 48
Step 2: Use k to find the unknown value
y = k/x = 48/6 = 8
Therefore, y = 8 when x = 6.
Step 3: Check
When x = 4, xy = 4 × 12 = 48
When x = 6, xy = 6 × 8 = 48 ✓
Example 4: Speed and Time Problem
The time taken for a journey is inversely proportional to the speed. At 60 mph, a journey takes 2 hours. How long will it take at 80 mph?
Step 1: Find the constant (distance)
Time = k/Speed, so k = Time × Speed
k = 2 × 60 = 120 miles
Step 2: Calculate time at 80 mph
Time = k/Speed = 120/80 = 1.5 hours
Therefore, the journey will take 1.5 hours at 80 mph.
Practice Questions
Practice Question 3: If m is inversely proportional to n, and m = 15 when n = 4, find m when n = 10.
Solution 3:
m ∝ 1/n, so m = k/n, which means mn = k
Find k: 15 × 4 = 60, so k = 60
When n = 10: m = 60/10 = 6
Practice Question 4: The number of workers needed to complete a job is inversely proportional to the time taken. 8 workers can complete the job in 15 days. How many workers are needed to complete it in 10 days?
Solution 4:
Workers ∝ 1/time, so Workers = k/time
Find k: 8 = k/15, so k = 120
For 10 days: Workers = 120/10 = 12 workers
Learning Block 3: Advanced Proportion (Higher Tier)
Key Facts
🔑 Key Fact: Variables can be proportional to powers or roots of other variables. For example, y ∝ x² means y = kx², and y ∝ √x means y = k√x.
🔑 Key Fact: In inverse square relationships (y ∝ 1/x²), doubling x makes y one-quarter of its original value.
Explanation
Advanced proportion relationships involve powers and roots:
- y ∝ x² (y proportional to x squared)
- y ∝ x³ (y proportional to x cubed)
- y ∝ √x (y proportional to square root of x)
- y ∝ 1/x² (y inversely proportional to x squared)
Example 5: Square Proportion
The area of a circle is proportional to the square of its radius. If a circle with radius 3 cm has area 28.3 cm², find the area when the radius is 5 cm.
Step 1: Set up the relationship
A ∝ r², so A = kr²
Step 2: Find the constant k
28.3 = k × 3²
28.3 = k × 9
k = 28.3 ÷ 9 = 3.14…
Step 3: Find area when r = 5
A = kr² = 3.14 × 5² = 3.14 × 25 = 78.5 cm²
Example 6: Inverse Square Relationship
The intensity of light is inversely proportional to the square of the distance from the source. At 2 metres, the intensity is 100 units. Find the intensity at 5 metres.
Step 1: Set up the relationship
I ∝ 1/d², so I = k/d²
Step 2: Find the constant k
100 = k/2²
100 = k/4
k = 400
Step 3: Find intensity at 5 metres
I = k/d² = 400/5² = 400/25 = 16 units
Practice Questions
Practice Question 5: (Higher tier only) The volume of a sphere is proportional to the cube of its radius. If a sphere with radius 2 cm has volume 33.5 cm³, find the volume when the radius is 3 cm.
Solution 5:
Volume ∝ r³, so V = kr³
Find k: 33.5 = k × 2³, so 33.5 = k × 8, giving k = 4.1875
When r = 3: V = 4.1875 × 3³ = 4.1875 × 27 = 113.1 cm³
Practice Question 6: (Higher tier only) The gravitational force between two objects is inversely proportional to the square of the distance between them. If the force is 36 N when objects are 10 m apart, find the force when they are 15 m apart.
Solution 6:
Force ∝ 1/d², so F = k/d²
Find k: 36 = k/10², so 36 = k/100, giving k = 3600
When d = 15: F = 3600/15² = 3600/225 = 16 N
Recognising Proportion in Real Life
One thing about proportion is how often it crops up in everyday situations. After my many years of tutoring, I’ve noticed that students who think about these in real life are much more confident with the abstract mathematics.
Here are some classic examples you’ll encounter:
Direct Proportion Examples:
- Shopping: More items → higher cost
- Travel: Longer distance → more fuel needed
- Cooking: More people → more ingredients needed
- Work: More hours → higher pay
Indirect Proportion Examples:
- Speed vs Time: Faster speed → less time for journey
- Workers vs Time: More workers → less time for job
- Sharing: More people → smaller share each
- Brightness: Further from light → dimmer appearance
I always tell my students to think about whether the quantities go "the same way" (both increase together = direct) or "opposite ways" (one increases, other decreases = inverse).
Common Mistakes to Avoid
After 14 years of tutoring, I’ve seen these errors countless times. Let me help you avoid them:
| Mistake | Why It Happens | Correct Approach |
|---|---|---|
| Confusing direct and indirect proportion | Students mix up when quantities move in the same direction vs opposite directions | Always ask: "If one quantity doubles, what happens to the other?" Same direction = direct, opposite = indirect |
| Forgetting to find the constant first | Rushing to solve without establishing the relationship | Always find k first using given values, then use k to solve for unknowns |
| Setting up the wrong equation | Not being careful with fraction positions in indirect proportion | Remember: y ∝ 1/x means y = k/x, NOT y = x/k |
| Not checking answers | Assuming calculations are correct without verification | Substitute back into original conditions to verify your constant k |
| Mixing up powers in advanced proportion | Confusion between x², 1/x², √x etc. | Write out the full relationship (y = kx² not just y ∝ x²) before substituting numbers |
End-of-Topic Practice Questions
Question 1 (Foundation): The cost of hiring a car is directly proportional to the number of days. It costs £120 to hire for 3 days. How much does it cost to hire for 8 days?
Question 2 (Foundation): The time to paint a fence is inversely proportional to the number of painters. 4 painters can paint it in 6 hours. How long will it take 3 painters?
Question 3 (Foundation): If y is directly proportional to x, and y = 36 when x = 9, find the value of x when y = 20.
Question 4 (Higher tier only): The surface area of a sphere is proportional to the square of its radius. A sphere with radius 4 cm has surface area 201 cm². Find the surface area of a sphere with radius 6 cm.
Question 5 (Higher tier only): The pressure in a gas is inversely proportional to its volume. When the volume is 50 cm³, the pressure is 8 atmospheres. Find the pressure when the volume is compressed to 20 cm³.
Question 6 (Foundation): y is inversely proportional to x. Complete this table:
| x | 2 | ? | 8 | 20 |
|---|---|---|---|---|
| y | 30 | 12 | ? | ? |
Question 7 (Higher tier only): The time for a pendulum to complete one swing is proportional to the square root of its length. A 1 metre pendulum takes 2 seconds per swing. How long does a 4 metre pendulum take?
Question 8 (Foundation): A recipe for 6 people uses 450g of flour. The amount of flour is directly proportional to the number of people. How much flour is needed for 10 people?
Question 9 (Higher tier only): The intensity of radiation is inversely proportional to the square of the distance from the source. At 3 metres, the intensity is 400 units. At what distance will the intensity be 100 units?
Question 10 (Foundation): Two quantities p and q are related. When p = 5, q = 24. When p = 8, q = 15. Are p and q in direct proportion, indirect proportion, or neither? Show your working.
End-of-Topic Solutions
Solution 1:
Cost ∝ days, so Cost = k × days
Find k: £120 = k × 3, so k = £40 per day
For 8 days: Cost = £40 × 8 = £320
Solution 2:
Time ∝ 1/painters, so Time = k/painters
Find k: 6 = k/4, so k = 24
For 3 painters: Time = 24/3 = 8 hours
Solution 3:
y = kx, so 36 = k × 9, giving k = 4
When y = 20: 20 = 4x, so x = 5
Solution 4:
Surface area ∝ r², so A = kr²
Find k: 201 = k × 4², so k = 201/16
For r = 6: A = (201/16) × 6² = (201/16) × 36 = 452.25 cm²
Solution 5:
Pressure ∝ 1/volume, so P = k/V
Find k: 8 = k/50, so k = 400
For V = 20: P = 400/20 = 20 atmospheres
Solution 6:
y ∝ 1/x, so xy = k = 2 × 30 = 60
When y = 12: x = 60/12 = 5
When x = 8: y = 60/8 = 7.5
When x = 20: y = 60/20 = 3
Complete table:
| x | 2 | 5 | 8 | 20 |
|---|---|---|---|---|
| y | 30 | 12 | 7.5 | 3 |
Solution 7:
Time ∝ √length, so T = k√L
Find k: 2 = k√1, so k = 2
For L = 4: T = 2√4 = 2 × 2 = 4 seconds
Solution 8:
Flour ∝ people, so F = k × P
Find k: 450 = k × 6, so k = 75g per person
For 10 people: F = 75 × 10 = 750g
Solution 9:
Intensity ∝ 1/distance², so I = k/d²
Find k: 400 = k/3², so k = 3600
When I = 100: 100 = 3600/d², so d² = 36, giving d = 6 metres
Solution 10:
Check if pq = constant (indirect) or p/q = constant (direct)
When p = 5, q = 24: pq = 120, p/q = 5/24
When p = 8, q = 15: pq = 120, p/q = 8/15
Since pq is constant but p/q is not, p and q are in indirect proportion.
Summary
- Direct proportion means quantities change in the same direction at a constant rate (y = kx)
- Indirect proportion means quantities change in opposite directions with constant product (y = k/x)
- Always find the constant k first, then use it to solve for unknowns
- Check your answers by substituting back into the original relationship
- Higher tier students must also handle proportion involving powers and roots
- These relationships appear everywhere in real life, from shopping to science
By Atomic