Ratio
- Simplify ratios to their lowest terms, including after converting to the same units
- Divide quantities in a given ratio using the total parts method
- Combine two separate ratios into a single three-part ratio using LCM
- Apply ratio to real-world contexts including map scales and recipes
- Form and solve equations from ratio problems at Grade 9 level
🔑 Core Concepts
1. Writing and Simplifying Ratios
A ratio compares two or more quantities of the same kind. The notation $a:b$ is read as "$a$ to $b$" and tells you the relative size of each quantity. Ratios are dimensionless — they have no units — so both quantities must be expressed in the same unit before you write the ratio.
To simplify a ratio, divide every term by the Highest Common Factor (HCF) of all the terms. The resulting ratio is in its simplest form when no integer greater than 1 divides all parts.
HCF(24, 36) = 12
$24 \div 12 : 36 \div 12 = \mathbf{2:3}$
b) Simplify 45 cm : 1.2 m.
Convert: 1.2 m = 120 cm, so ratio is 45 : 120
HCF(45, 120) = 15
$45 \div 15 : 120 \div 15 = \mathbf{3:8}$
2. Equivalent Ratios
Just as equivalent fractions represent the same value, equivalent ratios represent the same proportional relationship. You obtain equivalent ratios by multiplying or dividing every term by the same non-zero number.
| Simplest form | ×2 | ×3 | ×5 | ×10 |
|---|---|---|---|---|
| 1 : 3 | 2 : 6 | 3 : 9 | 5 : 15 | 10 : 30 |
| 2 : 5 | 4 : 10 | 6 : 15 | 10 : 25 | 20 : 50 |
| 3 : 4 | 6 : 8 | 9 : 12 | 15 : 20 | 30 : 40 |
| 1 : 2 : 3 | 2 : 4 : 6 | 3 : 6 : 9 | 5 : 10 : 15 | 10 : 20 : 30 |
3. Dividing a Quantity in a Two-Part Ratio
The total parts method is the standard examination approach. The ratio $a:b$ means the total is divided into $a + b$ equal parts. The first quantity receives $a$ parts; the second receives $b$ parts. This gives a clean, reliable procedure for any ratio division problem.
Total parts = 2 + 7 = 9
One part = £180 ÷ 9 = £20
Share&sub1; = 2 × £20 = £40
Share&sub2; = 7 × £20 = £140
Check: £40 + £140 = £180 ✓
4. Three-Part Ratios
Three-part ratios work identically to two-part ratios. The total parts is $a + b + c$, and each person receives their ratio number multiplied by the value of one part. This is the same algorithm — just extended.
Total parts = 3 + 4 + 5 = 12
One part = £360 ÷ 12 = £30
Alice: 3 × £30 = £90
Bob: 4 × £30 = £120
Carol: 5 × £30 = £150
Check: £90 + £120 + £150 = £360 ✓
5. Ratio and Fractions Connection
There is a direct algebraic link between ratios and fractions. Understanding this connection allows you to move fluently between ratio language ("the ratio is $3:7$") and fraction language ("the first part is $\frac{3}{10}$ of the total"). This is crucial for Grade 9 problems that combine ratio with percentage or probability.
Ratio 3:7 → first part is $\frac{3}{3+7} = \frac{3}{10}$, second part is $\frac{7}{10}$.
6. The Unitary Method
The unitary method is a powerful technique for ratio and proportion problems. You first find the value corresponding to one unit, then scale to find any required quantity. It is particularly useful when the total is unknown and one part is given, or in recipe/scaling contexts.
How much flour is needed for 15 biscuits?
Flour for 1 biscuit = 120 ÷ 6 = 20 g
Flour for 15 biscuits = 20 × 15 = 300 g
Value of 1 ratio unit = £45 ÷ 3 = £15
Bea's share = 5 × £15 = £75
Total = (3 + 5) × £15 = £120
7. Combining Two Ratios into a Three-Part Ratio
When two separate ratios share a common term (typically the second term of the first ratio equals the first term of the second ratio), they can be merged into a single three-part ratio. The technique is to equalise the shared term using the LCM, then read off all three values. This is a Grade 7–9 skill that frequently appears in the higher paper.
Shared term is $B$: value is 4 in first ratio, 2 in second.
LCM(4, 2) = 4
First ratio: $A:B = 3:4$ (already $B = 4$, no change)
Second ratio: $B:C = 2:5$ ×2 → $B:C = 4:10$
∴ $A:B:C = \mathbf{3:4:10}$
Check: HCF(3, 4, 10) = 1, so already in simplest form. ✓
8. Map Scales and Scale Drawings
A map scale is a ratio that compares a measurement on the map or drawing to the actual measurement in real life. A scale of $1:n$ means that 1 unit on the map represents $n$ of the same units in reality. Map scales are always expressed with 1 on the left; scale drawings may use other forms.
Real distance = 8 × 25 000 = 200 000 cm
= 200 000 ÷ 100 = 2 000 m
= 2 000 ÷ 1 000 = 2 km
cm → multiply by scale → real cm → divide by 100 → m → divide by 1000 → km.
Each conversion step can earn a method mark even if the final answer is wrong.
9. Grade 9: Forming and Solving Equations from Ratios
At Grade 9, ratio problems require you to set up algebraic equations. The standard technique is to let one ratio unit equal a variable (commonly $x$ or $k$), express each quantity in terms of that variable, and then use a given condition (difference, sum, or relationship) to form and solve an equation.
Step 2: Express every quantity in terms of $k$ (e.g., Share$_1$ = $3k$, Share$_2$ = $5k$).
Step 3: Use the extra condition to write an equation (e.g., $3k - 5k = 24$ or $3k + 12 = 5k$).
Step 4: Solve for $k$, then find all required quantities.
Step 5: Check by verifying the original ratio and any stated conditions.
Let the numbers be $5k$ and $3k$.
$5k - 3k = 24 \Rightarrow 2k = 24 \Rightarrow k = 12$
Numbers: $5 \times 12 = \mathbf{60}$ and $3 \times 12 = \mathbf{36}$
Check: 60 : 36 = 5 : 3 ✓ and 60 − 36 = 24 ✓
🗺️ Visual Notes
- Convert to same units first
- Find HCF of all terms
- Divide every part by HCF
- No common factor > 1 remains
- Total parts = sum of ratio
- One part = total ÷ total parts
- Multiply by each ratio number
- Check: all shares sum to total
- Identify the shared middle term
- Find LCM of both middle values
- Scale BOTH ratios fully
- Write as single $a:b:c$
- Part A: $\frac{a}{a+b}$ of total
- Part B: $\frac{b}{a+b}$ of total
- Fraction → ratio: part : rest
- Both express proportion
- Map scales (1:n)
- Recipes (unitary method)
- Scale drawings
- Mixing solutions
- Let 1 part = $k$, form equation
- Given one part, find total
- Three-ratio chains
- Connect ratio to geometry
Ratio Methods: Comparison Table
| Situation | Best Method | Key Step | Example |
|---|---|---|---|
| Total is known; divide in ratio $a:b$ | Total parts | One part = total ÷ $(a+b)$ | £100 in 3:2 → £60, £40 |
| One share is known; find others | Unitary | 1 ratio unit = share ÷ ratio number | Part A = £30, ratio 3:5 → Part B = £50 |
| Combine $a:b$ and $b:c$ | LCM scaling | Equalise the shared middle term | $3:4$ and $4:5$ → $3:4:5$ |
| Map / scale drawing | Multiply by scale factor | Real = map × scale | 8 cm, scale 1:25 000 → 2 km |
| Extra condition given (difference, sum) | Algebraic equation | Let 1 part = $k$; form and solve | $5k - 3k = 24 \Rightarrow k = 12$ |
Decision Flowchart: Which Method?
Yes → Total Parts Method
No → continue
Yes → Unitary Method
No → continue
Yes → LCM Combining
No → continue
✏️ Worked Examples
The whole £240 is divided into 8 equal portions.
Beth's share = 3 × £30 = £90
$B:C = 2:7$ ×5 → $B:C = 10:35$
HCF(6, 10, 35) = 1 → already in simplest form ✓
Total ratio parts = 2 + 3 + 7 = 12
One part = $180° \div 12 = 15°$
Smallest angle = $2 \times 15° = 30°$
Middle angle = $3 \times 15° = 45°$
Largest angle = $7 \times 15° = 105°$
$5k = 75$
$k = 15$
Hmm — this gives 30°, but ratio gives 45°. That means the expression $(k+15)°$ and $k=15$ gives 30°, not 45°. Let me check the question again.
In fact, with $k=15$: middle = $k + 15 = 30$. But ratio gives middle = 45. These are inconsistent, so the expressions in the question are set up to produce a specific $k$.
Revised approach: Let the middle angle expression be $(3k)°$ and largest be $(5k + 30)°$, with ratio 2:3:7.
Middle angle = $3 \times 15 = 45° \Rightarrow 3k = 45 \Rightarrow k = 15$
Largest angle = $5(15) + 30 = 75 + 30 = 105°$ ✓
Middle angle = $3k = 45°$ ✓
Largest angle = $5k + 30 = 105°$ ✓
Smallest angle = $2 \times 15° = 30°$
Sum: $30° + 45° + 105° = 180°$ ✓
Ratio check: $30:45:105 = 2:3:7$ ✓
❓ Exam Questions
Simplify the ratio 36 : 48.
HCF(36, 48) = 12
$36 \div 12 : 48 \div 12 = 3:4$
Mark scheme:
B1: 3:4
Divide £350 in the ratio 3 : 4. State clearly how much each person receives.
Total parts = 3 + 4 = 7
One part = £350 ÷ 7 = £50
Share&sub1; = 3 × £50 = £150
Share&sub2; = 4 × £50 = £200
Check: £150 + £200 = £350 ✓
Mark scheme:
M1: correct method (finding total parts OR dividing by 7)
A1: both correct answers £150 and £200
A map has a scale of 1 : 40 000. Two cities are 12.5 cm apart on the map. Find the real distance between the cities in kilometres.
Real distance = 12.5 × 40 000 = 500 000 cm
= 500 000 ÷ 100 = 5 000 m
= 5 000 ÷ 1 000 = 5 km
Mark scheme:
M1: real distance = 12.5 × 40 000 (= 500 000 cm)
M1: correct unit conversion shown (cm → m → km)
A1: 5 km
A bag contains red and blue counters in the ratio 5 : 3. There are 24 more red counters than blue counters. How many counters are there in total?
Let one ratio unit = $k$.
Red = $5k$, Blue = $3k$
Difference: $5k - 3k = 24$
$2k = 24 \Rightarrow k = 12$
Total = $(5 + 3) \times 12 = 8 \times 12 = \mathbf{96}$
Mark scheme:
M1: forming expression for difference ($5k - 3k$ or equivalent)
M1: correct equation $2k = 24$ and solving for $k$
A1: $k = 12$
A1: total = 96
Given that $A:B = 4:7$ and $B:C = 3:5$, find the ratio $A:B:C$. Give your answer in its simplest form.
Shared term is $B$: value is 7 in first ratio, 3 in second.
LCM(7, 3) = 21
$A:B = 4:7$ ×3 → $12:21$
$B:C = 3:5$ ×7 → $21:35$
Combined: $A:B:C = 12:21:35$
HCF(12, 21, 35) = 1 → simplest form ✓
Mark scheme:
M1: recognising that $B$ needs to be equalised
M1: finding LCM(7, 3) = 21 and scaling both ratios correctly
A1: 12:21 and 21:35 (or one of these)
A1: $A:B:C = 12:21:35$ (with both scales correct)
Three business partners Alice, Ben, and Carol share annual profits in the ratio $2:5:8$.
(a) In one year, Alice's share is £3 000 less than Carol's share. Find the total profit that year. (3 marks)
(b) Carol reinvests $\dfrac{3}{5}$ of her share back into the business. Find the amount Carol keeps for herself. (3 marks)
Let one ratio unit = $p$.
Alice = $2p$, Carol = $8p$
$8p - 2p = 3000 \Rightarrow 6p = 3000 \Rightarrow p = 500$
Total profit = $(2 + 5 + 8) \times 500 = 15 \times 500 = \mathbf{\pounds7\,500}$
Check: Alice = £1 000, Carol = £4 000, difference = £3 000 ✓
Part (b): Carol keeps £1 600
Carol's full share = $8p = 8 \times 500 = \pounds4\,000$
Amount reinvested = $\tfrac{3}{5} \times \pounds4\,000 = \pounds2\,400$
Amount kept = $\pounds4\,000 - \pounds2\,400 = \mathbf{\pounds1\,600}$
Alternatively: keeps $\tfrac{2}{5} \times \pounds4\,000 = \pounds1\,600$ ✓
Mark scheme:
(a) M1: forming equation from the difference ($8p - 2p = 3000$)
A1: $p = 500$
A1: total = £7 500
(b) M1: finding Carol's share = £4 000
M1: finding $\frac{3}{5}$ or $\frac{2}{5}$ of Carol's share
A1: £1 600
⭐ Grade 9 Model Answers
Below is a fully annotated, examination-standard response for Question 6 (the hardest question, worth 6 marks). Study the annotation to understand exactly what earns marks and why.
Q6: Three business partners Alice, Ben, and Carol share profits in the ratio $2:5:8$. Alice's share is £3 000 less than Carol's. Find: (a) the total profit; (b) the amount Carol keeps if she reinvests $\frac{3}{5}$.
Part (a):
Let one ratio unit $= p$ [Define variable clearly — this shows structured algebraic thinking]
$\Rightarrow$ Alice $= 2p$, Ben $= 5p$, Carol $= 8p$ [Express all quantities in terms of $p$ — M1 for correct setup]
$8p - 2p = 3000 \Rightarrow 6p = 3000 \Rightarrow p = 500$ [Form equation from the stated condition — M1 for correct equation; A1 for $p=500$]
Total profit $= 15p = 15 \times 500 = \mathbf{\pounds7\,500}$ [A1 for correct total — must multiply ALL parts, not just two]
Check: Alice $= \pounds1\,000$, Carol $= \pounds4\,000$, difference $= \pounds3\,000$ ✓ [Explicit check earns no extra mark but demonstrates mathematical rigour]
Part (b):
Carol's share $= 8 \times 500 = \pounds4\,000$ [M1 — must correctly calculate Carol's share from $p=500$]
Carol reinvests $\frac{3}{5} \times \pounds4\,000 = \pounds2\,400$ [M1 — correctly applying the fraction to Carol's share]
Carol keeps $\pounds4\,000 - \pounds2\,400 = \mathbf{\pounds1\,600}$ [A1 — correct final answer]
Alternative: Carol keeps $\frac{2}{5} \times \pounds4\,000 = \pounds1\,600$ (since $1 - \frac{3}{5} = \frac{2}{5}$) — this is equally acceptable and slightly faster.
What makes this a Grade 9 answer?
| Feature | Why it matters |
|---|---|
| Variable defined at the outset ("Let $p = $ one ratio unit") | Shows clear algebraic structure; avoids ambiguity |
| All three quantities expressed in terms of $p$ | Demonstrates systematic approach; sets up equation correctly |
| Equation formed and solved in one line | Efficient, clean algebra; earns method marks even with arithmetic error |
| Total uses sum of ALL ratio parts (15$p$) | Common error is to add only the known parts; full method earns A1 |
| Explicit check after part (a) | Demonstrates mathematical confidence; vital in 6-mark questions |
| Alternative approach noted for part (b) | Shows flexibility of thinking, a Grade 9 trait |
📋 Revision Sheet
| Term | Meaning |
|---|---|
| Ratio | Comparison of two or more quantities; no units |
| Simplest form | HCF of all terms = 1 |
| Equivalent ratio | Same relationship, different numbers (multiply/divide all parts) |
| Total parts | Sum of all ratio terms ($a + b$ or $a + b + c$) |
| Unitary method | Find value of 1 unit first, then scale |
| Map scale $1:n$ | 1 unit on map = $n$ units in real life |
Divide $Q$ in ratio $a:b$:
$$\text{One part} = \frac{Q}{a+b}$$
$$\text{Share}_1 = \frac{a}{a+b} \cdot Q$$
Fraction from ratio $a:b$:
$$\text{Part A} = \frac{a}{a+b} \text{ of total}$$
Combine $a:b$ and $b:c$:
Scale both so $b = \text{LCM}$, then write $a:b:c$
Map scale:
Real distance = Map distance × scale factor
- "HCF to simplify; LCM to combine" — two main techniques, two different averages
- "Part over Total" — ratio $a:b$ → fraction $\frac{a}{a+b}$
- "Scale ALL, not ONE" — when equalising shared terms, multiply every ratio value
- "One unit first" — unitary method: divide to find 1, then multiply
- "Check by adding" — shares must sum to original total
- "cm to km: ÷ 100 000" — map scale unit conversion shortcut
- Convert all quantities to the same unit before writing a ratio
- Always show the total parts step — it earns a method mark
- Write an explicit check (shares sum to total) in 3+ mark questions
- For "given one share" problems, find 1 ratio unit before multiplying
- For combining ratios, always state the LCM in your working
- In map problems, write each unit conversion on a separate line
- In Grade 9 equation questions, define your variable ("let $k = $ one part") before using it
🔄 Flashcards
Click each card to reveal the answer. Use these for last-minute revision before an exam.
✗ Common Mistakes
Why marks are lost: The ratio is meaningless because the units differ. The denominator is already in different units.
How to avoid it: Always convert to the smaller unit first. 30 min : 2 h → 30 min : 120 min = 30 : 120 = 1 : 4.
Why marks are lost: $\frac{3}{5}$ is the ratio of the two parts, not the fraction of the total. The fraction of the total is $\frac{3}{3+5} = \frac{3}{8}$.
How to avoid it: Always divide by the sum of ratio parts when converting to a fraction of the total.
Why marks are lost: The ratio $10:5$ is equivalent to $2:1$, not $2:5$. The entire ratio is distorted.
How to avoid it: Multiply all terms in the ratio by the same factor. $2:5 \times 5 = 10:25$.
Why marks are lost: The one-part value is only an intermediate step. Both shares are multiples of one part.
How to avoid it: After finding one part, always multiply by each ratio number to find each share.
Why marks are lost: Arithmetic errors in ratio problems are common. An explicit check catches mistakes before they cost marks.
How to avoid it: Always add your shares at the end. If they don't equal the original total, find and fix the error.
Why marks are lost: The final answer is wrong and the question specifically asks for km. The M1 for the multiplication may still be awarded, but the A1 is lost.
How to avoid it: Write out the full conversion chain: × scale → real cm → ÷100 → m → ÷1000 → km.
✅ Final Checklist
Tick each skill as you master it. Your progress is saved automatically.
- I can simplify a two-part ratio by finding and dividing by the HCF
- I can simplify a ratio after converting both quantities to the same unit
- I can write and identify equivalent ratios by scaling
- I can divide a quantity in a two-part ratio using the total parts method
- I can divide a quantity in a three-part ratio
- I can find a missing share when one share and the ratio are given (unitary method)
- I can convert a ratio $a:b$ into fractions of the total ($\frac{a}{a+b}$ and $\frac{b}{a+b}$)
- I can apply the unitary method to recipe and scaling problems
- I can use a map scale to convert between map distance and real distance
- I can combine two ratios by equalising the shared term using the LCM
- I can form and solve an algebraic equation from a ratio problem
- I can find angles in a triangle or other shape given their ratio
- I always check my answer by verifying shares sum to the original total
- I can tackle multi-step Grade 9 ratio problems combining ratio with algebra or fractions