R6: Multiplicative Relationships
Express a multiplicative relationship between two quantities as a ratio or a fraction
📋 Key Concepts
Definition: A multiplicative relationship means one quantity is a constant multiple of another. This can be expressed as a ratio, fraction, or multiplier.
Forms of Multiplicative Relationships
- Ratio: a:b where b = k × a
- Fraction: b/a = k (the multiplier)
- Multiplier: To get from a to b, multiply by k
Quantity B = Multiplier × Quantity A
Multiplier = Quantity B ÷ Quantity A
Multiplier = Quantity B ÷ Quantity A
📝 Finding the Multiplier
Method: Divide the second quantity by the first quantity to find the multiplier.
Example 1
Find the multiplier that connects 8 and 12.
Solution:
Multiplier = 12 ÷ 8 = 1.5
So: 12 = 1.5 × 8
Answer: 1.5
Example 2
A recipe uses 150g of flour for 6 cupcakes. How much flour is needed for 10 cupcakes?
Solution:
Multiplier = 10 ÷ 6 = 5/3
Flour = 150 × (5/3) = 250g
Answer: 250g
📝 Expressing as a Ratio
Method: Write both quantities and simplify to find the ratio.
Example 3
Express the relationship between 20 and 50 as a ratio.
Solution:
Ratio = 20:50
Simplify: 20:50 = 2:5 (divide by 10)
Answer: 2:5
This means for every 2 units of A, there are 5 units of B.
📝 Expressing as a Fraction
Method: Write the first quantity as a fraction of the second (or vice versa depending on the question).
Example 4
Express 18 as a fraction of 27. What is the multiplier?
Solution:
Fraction = 18/27 = 2/3 (simplified)
Multiplier to get from 27 to 18 is 2/3
Multiplier to get from 18 to 27 is 3/2 = 1.5
📝 Using Multipliers
Application: Multipliers are useful for scaling quantities up or down proportionally.
Example 5
£1 = $1.27. How many dollars is £45?
Solution:
Multiplier = 1.27
$45 = £45 × 1.27 = $57.15
Example 6
5 books cost £24. How much do 8 books cost?
Solution:
Method 1: Find unit cost first
One book = £24 ÷ 5 = £4.80
8 books = 8 × £4.80 = £38.40
Method 2: Use multiplier
Multiplier = 8/5 = 1.6
Cost = £24 × 1.6 = £38.40
📝 Inverse Multipliers
Inverse Relationships: If A × k = B, then B × (1/k) = A. The inverse multiplier reverses the relationship.
Example 7
A car travels at constant speed. It takes 3 hours to travel 180 miles. How long to travel 240 miles?
Solution:
Multiplier = 240/180 = 4/3
Time = 3 × (4/3) = 4 hours
🧮 Worked Examples — Exam Style
Multiplicative reasoning is ratio thinking with numbers: find the multiplier between two values, then apply it (or its inverse).
3 tins cost £2.10. How much do 7 tins cost?
Multiplier 3 → 7: 7/3 · £2.10 × 7/3 = £2.10 ÷ 3 × 7 = £4.90
M1 for the unitary step, A1 for £4.90. One multiplier (7/3) does the work — ÷3 then ×7 is the same idea.
A map has scale 1 : 50000. How many km does 4 cm represent?
4 × 50000 = 200,000 cm = 2 km
M1 for × 50000, A1 for 2 km (the cm → km conversion is part of the mark).
Exam wording to expect: "How many … for …?", map scales, currency conversions, ingredient scaling.
❓ Practice Questions
Q1: Find the multiplier connecting 12 and 30.
Q2: Express the relationship between 25 and 40 as a ratio in simplest form.
Q3: 4 litres of paint cover 24 m². How many litres are needed for 60 m²?
Q4: £1 = 1.17 euros. How many euros for £85?
Q5: The ratio of teachers to students is 1:15. If there are 285 students, how many teachers are there?
Q6: A map scale is 1 : 25000. Two towns are 6 cm apart on the map. Find the real distance in km.
Q7: 5 identical pens cost £3.75. How many pens can you buy with £9?
✅ Answers
- 2.5 (30 ÷ 12)
- 5:8
- 10 litres
- 99.45 euros
- 19 teachers
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. Multiplier 30 ÷ 12 = 2.5
Q2. 25 : 40 = ÷5 → 5 : 8
Q3. 24 ÷ 4 = 6 m² per litre → 60 ÷ 6 = 10 litres
Q4. 85 × 1.17 = £99.45 (99.45 euros)
Q5. 285 ÷ 15 = 19 teachers
Q6. 6 × 25000 = 150,000 cm = 1.5 km
Q7. £3.75 ÷ 5 = £0.75 each → £9 ÷ 0.75 = 12 pens
📌 Key Facts to Memorise
- Multiplier = new ÷ old — apply the SAME multiplier to every connected quantity
- Inverse multiplier: to reverse a ×2.5, use ÷2.5 (or ×2/5)
- Map scale 1 : n → real distance = map distance × n
- Currency: × the rate one way, ÷ the rate back
- Check direction: more items → more money (direct), faster speed → less time (inverse)
🗺️ In the Exam
Multiplicative reasoning underpins best buys, exchange rates, recipes and maps: 3–5 marks per paper. It is the arithmetic engine of the ratio/proportion strand.
Links to: proportion (R7/R10), ratio (R4/R5), percentages (R9).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- To find a multiplier, divide: new value ÷ original value
- A multiplier greater than 1 increases, less than 1 decreases
- The inverse multiplier is the reciprocal (1/k)
- Multiplicative relationships are the basis of proportional reasoning
- Check: original × multiplier should give the new value
🧠 Problem-Solving Strategies
Problem-Solving
Multiplicative relationships connect quantities through a constant multiplier. Use the "constant of proportionality" (k) where y = kx. Identify whether the relationship is multiplicative by checking if one quantity scales with the other. Use a multiplier table to track the scaling between known and unknown values.
Multi-Step Problem
4 identical books cost £14.60. How much do 7 books cost? What is the cost per book?
Solution: Find the multiplier: cost = price per book × quantity. Price per book = 14.60 ÷ 4 = £3.65. For 7 books: 7 × 3.65 = £25.55.
⚠️ Common Errors
Watch Out!
1. Wrong: Finding the cost of 7 books by adding 14.60 + 14.60 = 29.20 (cost of 8 books) then subtracting one book's price Correct: Find the unit cost first (14.60 ÷ 4 = 3.65) then multiply by 7 = £25.55
2. Wrong: Thinking that if 3 items cost £12, then 9 items must cost £36 by adding £12 three times without checking Correct: Use the unitary method: 1 item = £4, 9 items = 9 × 4 = £36. This works, but always find one unit first.
3. Wrong: Confusing additive and multiplicative relationships (e.g. adding 3 each time instead of ×3) Correct: If 2 → 6, the relationship could be +4 (additive) or ×3 (multiplicative). Check which pattern fits all data pairs.
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A recipe for 6 people uses 450 g of flour and 3 eggs. Tom wants to make the recipe for 10 people, but eggs only come in packs of 6. How much flour does he need, and how many packs of eggs must he buy? What weight of flour will be wasted if he uses whole packs of eggs to make the maximum number of servings?
Step 1: Flour per person = 450 ÷ 6 = 75 g. For 10 people: 10 × 75 = 750 g flour.
Step 2: Eggs per person = 3 ÷ 6 = 0.5. For 10 people: 10 × 0.5 = 5 eggs.
Step 3: He must buy 1 pack of 6 eggs (nearest pack size above 5).
Step 4: With 6 eggs, he can serve 6 ÷ 0.5 = 12 people. Flour needed = 12 × 75 = 900 g. Wasted flour = 900 − 750 = 150 g (if making only 10 portions from 12-serving batch).
Mark scheme: M1 for unitary method, A1 for 750g, M1 for eggs calculation, A1 for 5 eggs/1 pack, M1 for 12 servings, A1 for waste
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A car uses 8 litres of petrol per 100 km. Petrol costs £1.48 per litre.
(a) How much does a 350 km journey cost in petrol?
(b) A hybrid car uses 4.2 litres per 100 km. How much money is saved on the same 350 km journey?
(c) The hybrid car costs £3000 more to buy. After how many kilometres of driving does the fuel saving cover the extra cost?
Answers: (a) 350 ÷ 100 × 8 = 28 litres. Cost = 28 × 1.48 = £41.44 (b) Hybrid uses 350 ÷ 100 × 4.2 = 14.7 litres. Cost = 14.7 × 1.48 = £21.76. Saving = £41.44 − £21.76 = £19.68 (c) Saving per km = 19.68 ÷ 350 = £0.0562/km. 3000 ÷ 0.0562 ≈ 53,381 km
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