SPEC R4, R5 • RATIO, PROPORTION & RATES OF CHANGE
Ratio: Sharing Quantities & Spotting Traps
Notation, simplification, unit conversion and the three-step method for dividing a quantity in any given ratio — the core of every ratio question on every paper.
Why Ratio Matters
Ratio appears on every paper — non-calculator and calculator. It is one of the highest-frequency topics on AQA GCSE Maths and a major source of method marks.
What You'll Master
Writing and reading ratio notation a : b
Simplifying a ratio, and expressing a quantity as a ratio
Converting ratios to the 1 : n unit form
Handling ratios with different units
Dividing a quantity in a given ratio — the three-step method
Telling part : part from part : whole ratios
Exam Signal
Ratio marks are lost most often because students divide by one part instead of the sum of all parts. Fix that habit here and you protect marks across the whole paper.
Ratio Notation: a : b
Definition — Ratio
A ratio is a comparison of two or more quantities in the same unit, written a : b (read as "a to b").
Order matters. 3 apples to 2 bananas is written 3 : 2 — not 2 : 3.
Ratios can compare two, three, or more quantities — e.g. 3 : 5 : 4.
Simplifying & Expressing as a Ratio
Definition — Simplified Ratio
A simplified ratio is one where the parts share no common factor greater than 1. To simplify, divide every part by their highest common factor (HCF).
Worked simplifications
12 : 18 → HCF = 6 → 2 : 320 : 30 : 50 → HCF = 10 → 2 : 3 : 5
Expressing a quantity as a ratio (reverse direction). You may be given raw numbers and asked to write them as a ratio — then simplify.
Example — Class of 30
"In a class of 30 students, 12 are boys. Write the ratio of boys to girls in its simplest form."
Girls = 30 − 12 = 18.
Ratio: 12 : 18 → HCF = 6 → 2 : 3.
Decimals or fractions? Clear them first:
Ratio givenMethodSimplest form
0.4 : 1.6Multiply both by 10 → 4 : 161 : 4
⅔ : ⅚Common denominator 6 → 4 : 54 : 5
Unit Ratios: The 1 : n Form
Definition — Unit Ratio
A unit ratio is a ratio written with the first part equal to 1 (the form 1 : n). To convert, divide both parts by the first part.
Method
3 : 12 ÷ 3 both sides→ 1 : 45 : 22 ÷ 5 both sides→ 1 : 4.4
Why the 1 : n Form?
Unit ratios are used for scaling, map scales, and comparing ratios quickly. Once you see "in the form 1 : n" in a question, divide by the first part.
Different Units — Convert First
A ratio compares quantities in the same unit. If the parts are given in different units, convert first, then simplify.
Common Error
40 cm : 2 m is not 40 : 2.
Convert 2 m to 200 cm first: 40 cm : 200 cm = 40 : 200 = 1 : 5 ✓
Your Unit-Conversion Checklist
Length: 1 m = 100 cm, 1 km = 1000 m
Mass: 1 kg = 1000 g, 1 tonne = 1000 kg
Volume: 1 litre = 1000 ml
Time: 1 hour = 60 min, 1 min = 60 s
Always convert to the smaller unit — it avoids decimals and keeps the arithmetic clean.
Dividing a Quantity in a Given Ratio
This is the single most common ratio question on every paper. Follow a fixed three-step method every time.
1
Add the parts — sum every number in the ratio to find the total number of parts.
2
Find one part — divide the total quantity by the total number of parts.
3
Multiply out — multiply the value of one part by each number in the ratio to get each share.
Quick Example: Share £180 in the ratio 2 : 3 : 4
Total parts = 2 + 3 + 4 = 9. One part = 180 ÷ 9 = £20.
Shares: 2 × 20 = £40, 3 × 20 = £60, 4 × 20 = £80. Check: 40 + 60 + 80 = 180 ✓
Part : Part vs Part : Whole
The AQA spec asks you to share a quantity using either a part : partor a part : whole ratio. The two look similar but the arithmetic differs.
TypeWhat it meansHow to find one part
Part : Part (e.g. boys : girls = 3 : 2)3 boys for every 2 girls. The two numbers are separate groups.Add the parts: 3 + 2 = 5. Then total ÷ 5.
Part : Whole (e.g. boys : total = 3 : 5)3 boys in every 5 students. The second number already is the whole.Do not add. The 5 is already the total of parts. Total ÷ 5 directly.
Example — Part : Whole
"1 in 4 students walk to school. There are 120 students. How many walk?"
This is a part : whole ratio of 1 : 4 (walkers : total).
One part = 120 ÷ 4 = 30 students walk. No summing needed.
Read Carefully
Phrases like "1 in every 5", "3 out of every 7", or "to the total" signal a part : whole ratio. Phrases like "A to B" where A and B are separate groups signal part : part.
MCE: Your Answer Framework
Every multi-step question in this lesson is written using the MCE framework — a three-step habit that protects method marks even when the final answer is wrong.
M
Method — state the plan in one line. "Sum the parts, find one part, multiply out."
C
Calculate — show every step with numbers. Do not skip lines.
E
Evaluate — check the answer makes sense and explicitly answer the question asked (including units).
Why This Wins Marks
AQA mark schemes award method marks for the sum of parts (M1), the value of one part (M1), and the final scaled share (A1). If you only write the answer, you lose those method marks when you slip up.
Worked Example — Sharing £480
EX
3 Marks — Paper 1 or 2 style
Share £480 between Alan, Ben and Clara in the ratio 3 : 5 : 4. How much more does Clara get than Alan?
Alan 3 parts
Ben 5 parts
Clara 4 parts
Total = 12 parts = £480
M
Sum the parts, divide £480 by total, multiply out, then subtract Alan's share from Clara's.
C
Total parts = 3 + 5 + 4 = 12. One part = 480 ÷ 12 = £40.
C
Alan: 3 × 40 = £120. Ben: 5 × 40 = £200. Clara: 4 × 40 = £160.
C
Difference: 160 − 120 = £40.
E
Clara gets £40 more than Alan. Check: 120 + 200 + 160 = 480 ✓
Answer the Specific Question
A ratio question rarely stops at "find each share". The final mark almost always depends on a second step: a difference, a fraction, a conversion.
Question asksExtra step
"How much more does Clara get than Alan?"Subtract Alan's share from Clara's
"What fraction of the total does Ben get?"Ben's parts ÷ total parts
"Alan gives half to Dan. How much does Dan receive?"Find Alan's share, then × ½
"The largest angle on the line is ?"Pick the largest share — not 'one part'
Examiner Alert — stopping at 'one part'
Students sometimes find "one part" and give that as the final answer. In an AQA June 2023 Paper 1 question, angles in ratio 1 : 5 : 3 on a line (180°) produced one part = 20. The required answer was the largest angle = 5 × 20 = 100°, not 20. Read the last line of the question again before you circle your answer.
Examiner Alert — Unit Conversion
Top Source of Lost Marks
Students write "30 cm : 1 m = 30 : 1" — forgetting that 1 m = 100 cm. The correct ratio is 30 cm : 100 cm = 3 : 10.
Weak vs Strong Answer
Weak responseStrong responseWrites 30 : 1, simplifies to 30 : 1. No conversion.Converts 1 m → 100 cm. Writes 30 : 100 = 3 : 10.
Habit to Build
Before you write any ratio, ask: are both parts in the same unit? If not, convert, then write the ratio, then simplify.
Examiner Alert — Sum of Parts
Most Common Ratio Error
Share £180 in the ratio 3 : 5.
Weak student divides 180 by 3 (or by 5) — wrong. Strong student adds the parts first: 3 + 5 = 8, then 180 ÷ 8 = £22.50 per part.
In a real AQA Paper 1 question…
Three angles on a line were in ratio 1 : 5 : 3. Examiners reported students divided 180 by 8, 5, or 3 — only dividing by 1 + 5 + 3 = 9 earns the method mark.
The Fix — Three Words
"Sum all parts." Write the sum on its own line before you divide anything. It earns a method mark on its own.
Practice Question
A roll of red ribbon is 2.4 metres long. It is cut into two pieces in the ratio 3 : 5. How many centimetres long is the shorter piece?[3 marks]
M
Convert 2.4 m to cm, sum the parts, find one part, then multiply by the smaller ratio number.
C
2.4 m = 2.4 × 100 = 240 cm.
C
Total parts = 3 + 5 = 8. One part = 240 ÷ 8 = 30 cm.
C
Shorter piece = 3 × 30 = 90 cm.
E
Answer: 90 cm. Check: shorter + longer = 90 + 150 = 240 cm = 2.4 m ✓. Units in cm as required.
Ratio Pitfalls: Quick Recap
TrapFix
Dividing by one ratio part (not the sum)Always sum all parts first
Different units left unconvertedConvert to the smaller unit before writing the ratio
Stopping at "one part" as the answerRe-read the question — multiply, subtract, or add as asked
Using fraction of first part (not of total)In 3 : 5, first part is 3/8 of total — not 3/5
Writing ratios in the wrong orderMatch names or labels to numbers in the order given
Summing a part:whole ratio unnecessarilyIf the ratio is already part : total , do not add — divide by the second number directly
30-Second Self-Check
Before finalising any ratio answer: (1) Are units the same? (2) Did I sum all parts? (3) Did I answer what was asked?
Lesson Summary
✓ Notation
A ratio compares quantities in the same unit, written a : b. Order matters. Ratios can have 2, 3 or more parts.
✓ Simplification
Divide every part by their HCF. Multiply to clear decimals; use a common denominator to clear fractions. Unit form 1 : n is found by dividing both parts by the first.
✓ Different Units
Convert to the same unit before writing the ratio — always convert to the smaller unit.
✓ Dividing a Quantity
Step 1: Sum all parts. Step 2: Divide total by sum to get one part. Step 3: Multiply one part by each ratio number.
✓ MCE Framework
Method → Calculate → Evaluate. Show working on every line to capture every method mark.
Next: Why Ratio Matters