6.1 Ratios
What is a Ratio?
A ratio compares the sizes of two or more quantities, by showing how much of one thing there is compared to another.
A ratio compares the sizes of two or more quantities, by showing how much of one thing there is compared to another.
⚡ Ratio Notation:
Ratios can be written in different ways:
• Using a colon: $3:2$ (read as "3 to 2")
• Using words: "3 to 2"
• As a fraction: $\frac{3}{2}$
The order matters! $3:2$ is different from $2:3$
• Using a colon: $3:2$ (read as "3 to 2")
• Using words: "3 to 2"
• As a fraction: $\frac{3}{2}$
The order matters! $3:2$ is different from $2:3$
Example: Understanding Ratio $3:2$
If we have blue and orange counters in the ratio $3:2$:
This means:
• For every 3 blue counters, there are 2 orange counters
• Total of 5 counters (3 + 2 = 5)
• Blue counters make up $\frac{3}{5}$ of the total
• Orange counters make up $\frac{2}{5}$ of the total
Blue
Blue
Blue
Orange
Orange
This means:
• For every 3 blue counters, there are 2 orange counters
• Total of 5 counters (3 + 2 = 5)
• Blue counters make up $\frac{3}{5}$ of the total
• Orange counters make up $\frac{2}{5}$ of the total
🎯 Ratio Practice
In a bag of sweets, the ratio of chocolates to toffees is $4:3$.
If there are 12 chocolates, how many toffees are there?
Just like fractions, ratios can be simplified by dividing all parts by their highest common factor (HCF).
⚡ How to Simplify:
Step 1: Find the HCF of all the numbers
Step 2: Divide all parts of the ratio by the HCF
Step 3: Check your answer can't be simplified further
Step 2: Divide all parts of the ratio by the HCF
Step 3: Check your answer can't be simplified further
Example 1: Simplify $12:8$
Step 1: Find HCF of 12 and 8
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 8: 1, 2, 4, 8
HCF = 4
Step 2: Divide both parts by 4
$12 \div 4 = 3$
$8 \div 4 = 2$
Answer: $12:8 = 3:2$
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 8: 1, 2, 4, 8
HCF = 4
Step 2: Divide both parts by 4
$12 \div 4 = 3$
$8 \div 4 = 2$
Answer: $12:8 = 3:2$
Example 2: Simplify $15:25:30$
Step 1: Find HCF of 15, 25, and 30
The HCF is 5
Step 2: Divide all parts by 5
$15 \div 5 = 3$
$25 \div 5 = 5$
$30 \div 5 = 6$
Answer: $15:25:30 = 3:5:6$
The HCF is 5
Step 2: Divide all parts by 5
$15 \div 5 = 3$
$25 \div 5 = 5$
$30 \div 5 = 6$
Answer: $15:25:30 = 3:5:6$
Example 3: Simplify $4:6$ then write in the form $1:n$
Step 1: Simplify by HCF = 2
$4:6 = 2:3$
Step 2: To get the form $1:n$, divide both parts by 2
$2 \div 2 = 1$
$3 \div 2 = 1.5$
Answer: $1:1.5$
$4:6 = 2:3$
Step 2: To get the form $1:n$, divide both parts by 2
$2 \div 2 = 1$
$3 \div 2 = 1.5$
Answer: $1:1.5$
💡 Ratios with Units:
Important: Convert to the same units first.
Simplify the ratio $2 \text{ km} : 500 \text{ m}$
Step 1: Convert to same units
$2 \text{ km} = 2000 \text{ m}$
So we have $2000:500$
Step 2: Simplify (HCF = 500)
$2000 \div 500 = 4$
$500 \div 500 = 1$
Answer: $4:1$
Simplify the ratio $2 \text{ km} : 500 \text{ m}$
Step 1: Convert to same units
$2 \text{ km} = 2000 \text{ m}$
So we have $2000:500$
Step 2: Simplify (HCF = 500)
$2000 \div 500 = 4$
$500 \div 500 = 1$
Answer: $4:1$
🎯 Simplification Practice
Dividing Amounts Using Ratios
When we share an amount in a given ratio, we divide it into parts based on that ratio.
When we share an amount in a given ratio, we divide it into parts based on that ratio.
⚡ The Method:
To share an amount in ratio $a:b$:
Step 1: Add the parts: $a + b = $ total parts
Step 2: Divide the amount by total parts: $\frac{\text{amount}}{\text{total parts}} = $ value of 1 part in the ratio
Step 3: Multiply to find each ratio value
• First share: $a \times $ value of 1 part
• Second share: $b \times $ value of 1 part
Step 1: Add the parts: $a + b = $ total parts
Step 2: Divide the amount by total parts: $\frac{\text{amount}}{\text{total parts}} = $ value of 1 part in the ratio
Step 3: Multiply to find each ratio value
• First share: $a \times $ value of 1 part
• Second share: $b \times $ value of 1 part
Example 1: Share £60 in the ratio $3:2$
Step 1: Total parts = $3 + 2 = 5$
Step 2: Value of 1 part = $60 \div 5 = £12$
Step 3: Calculate each share
• First share: $3 \times £12 = £36$
• Second share: $2 \times £12 = £24$
Check: $£36 + £24 = £60$ ✓
Step 2: Value of 1 part = $60 \div 5 = £12$
Step 3: Calculate each share
• First share: $3 \times £12 = £36$
• Second share: $2 \times £12 = £24$
Check: $£36 + £24 = £60$ ✓
First Share
£36
(3 parts)
Second Share
£24
(2 parts)
Example 2: Share 280 sweets between Amy, Ben and Charlie in the ratio $2:3:5$
Step 1: Total parts = $2 + 3 + 5 = 10$
Step 2: Value of 1 part = $280 \div 10 = 28$ sweets
Step 3: Calculate each share
• Amy gets: $2 \times 28 = 56$ sweets
• Ben gets: $3 \times 28 = 84$ sweets
• Charlie gets: $5 \times 28 = 140$ sweets
Check: $56 + 84 + 140 = 280$ ✓
Step 2: Value of 1 part = $280 \div 10 = 28$ sweets
Step 3: Calculate each share
• Amy gets: $2 \times 28 = 56$ sweets
• Ben gets: $3 \times 28 = 84$ sweets
• Charlie gets: $5 \times 28 = 140$ sweets
Check: $56 + 84 + 140 = 280$ ✓
Amy
56
(2 parts)
Ben
84
(3 parts)
Charlie
140
(5 parts)
Example 3: Finding the Total Amount
Tom and Jerry share some money in the ratio $4:5$. Tom gets £32. How much money is there altogether?
Step 1: Tom has 4 parts = £32
So 1 part = $£32 \div 4 = £8$
Step 2: Total parts = $4 + 5 = 9$ parts
Total amount = $9 \times £8 = £72$
Check: Jerry gets $5 \times £8 = £40$
$£32 + £40 = £72$ ✓
Step 1: Tom has 4 parts = £32
So 1 part = $£32 \div 4 = £8$
Step 2: Total parts = $4 + 5 = 9$ parts
Total amount = $9 \times £8 = £72$
Check: Jerry gets $5 \times £8 = £40$
$£32 + £40 = £72$ ✓
💡 Ratio Problems - Key Tips:
• Always check your answers add up to the total
• Label your working clearly (what does each part represent?)
• Watch out for questions asking "how much more" or "what's the difference"
• Label your working clearly (what does each part represent?)
• Watch out for questions asking "how much more" or "what's the difference"
🎯 Sharing Practice
First share:
Second share:
Using Algebra with Ratios
When we don't know the actual values, we can use algebra to represent ratio problems.
When we don't know the actual values, we can use algebra to represent ratio problems.
⚡ Representing Ratios with Algebra:
If the ratio is $a:b$, we can write:
• First quantity = $ax$ (where $x$ is the value of 1 part)
• Second quantity = $bx$
• Total = $ax + bx = (a + b)x$
• First quantity = $ax$ (where $x$ is the value of 1 part)
• Second quantity = $bx$
• Total = $ax + bx = (a + b)x$
Example 1: Two numbers are in the ratio $3:5$. Their sum is 96. Find the numbers.
Step 1: Let the numbers be $3x$ and $5x$
Step 2: Write an equation
$3x + 5x = 96$
$8x = 96$
Step 3: Solve for $x$
$x = 96 \div 8 = 12$
Step 4: Find the numbers
First number: $3x = 3 \times 12 = 36$
Second number: $5x = 5 \times 12 = 60$
Check: $36 + 60 = 96$ ✓ and $36:60 = 3:5$ ✓
Step 2: Write an equation
$3x + 5x = 96$
$8x = 96$
Step 3: Solve for $x$
$x = 96 \div 8 = 12$
Step 4: Find the numbers
First number: $3x = 3 \times 12 = 36$
Second number: $5x = 5 \times 12 = 60$
Check: $36 + 60 = 96$ ✓ and $36:60 = 3:5$ ✓
Example 2: The angles in a triangle are in the ratio $2:3:4$. Find each angle.
Step 1: Let the angles be $2x°$, $3x°$ and $4x°$
Step 2: Use angle sum property (angles in triangle = $180°$)
$2x + 3x + 4x = 180$
$9x = 180$
Step 3: Solve for $x$
$x = 180 \div 9 = 20$
Step 4: Find each angle
• First angle: $2x = 2 \times 20 = 40°$
• Second angle: $3x = 3 \times 20 = 60°$
• Third angle: $4x = 4 \times 20 = 80°$
Check: $40° + 60° + 80° = 180°$ ✓
Step 2: Use angle sum property (angles in triangle = $180°$)
$2x + 3x + 4x = 180$
$9x = 180$
Step 3: Solve for $x$
$x = 180 \div 9 = 20$
Step 4: Find each angle
• First angle: $2x = 2 \times 20 = 40°$
• Second angle: $3x = 3 \times 20 = 60°$
• Third angle: $4x = 4 \times 20 = 80°$
Check: $40° + 60° + 80° = 180°$ ✓
Example 3: Finding Differences
Two numbers are in the ratio $7:4$. The difference between them is 21. Find the larger number.
Step 1: Let the numbers be $7x$ and $4x$
Step 2: Write equation for difference
$7x - 4x = 21$
$3x = 21$
Step 3: Solve for $x$
$x = 21 \div 3 = 7$
Step 4: Find larger number
Larger number: $7x = 7 \times 7 = 49$
(Smaller number: $4x = 4 \times 7 = 28$)
Check: $49 - 28 = 21$ ✓
Step 1: Let the numbers be $7x$ and $4x$
Step 2: Write equation for difference
$7x - 4x = 21$
$3x = 21$
Step 3: Solve for $x$
$x = 21 \div 3 = 7$
Step 4: Find larger number
Larger number: $7x = 7 \times 7 = 49$
(Smaller number: $4x = 4 \times 7 = 28$)
Check: $49 - 28 = 21$ ✓
🎯 Algebraic Ratios
Real-World Ratio Applications
Example 1: Recipe Scaling
A cake recipe for 6 people uses 200g flour and 150g sugar. How much flour and sugar is needed for 10 people?
Method 1: Using Ratios
People ratio = $6:10$ or simplified $3:5$
Scale factor = $10 \div 6 = \frac{5}{3}$
Flour needed: $200 \times \frac{5}{3} = \frac{1000}{3} \approx 333g$
Sugar needed: $150 \times \frac{5}{3} = 250g$
Method 2: Find per person
Per person: Flour = $200 \div 6 = 33.3g$, Sugar = $150 \div 6 = 25g$
For 10 people: Flour = $33.3 \times 10 = 333g$, Sugar = $25 \times 10 = 250g$
Method 1: Using Ratios
People ratio = $6:10$ or simplified $3:5$
Scale factor = $10 \div 6 = \frac{5}{3}$
Flour needed: $200 \times \frac{5}{3} = \frac{1000}{3} \approx 333g$
Sugar needed: $150 \times \frac{5}{3} = 250g$
Method 2: Find per person
Per person: Flour = $200 \div 6 = 33.3g$, Sugar = $150 \div 6 = 25g$
For 10 people: Flour = $33.3 \times 10 = 333g$, Sugar = $25 \times 10 = 250g$
Example 2: Map Scales
A map has a scale of $1:50000$. Two towns are 8cm apart on the map. What is the actual distance?
Step 1: Understand the scale
1 cm on map = 50000 cm in real life
Step 2: Calculate
8 cm on map = $8 \times 50000 = 400000$ cm
Step 3: Convert to sensible units
400000 cm = 4000 m = 4 km
Answer: The towns are 4 km apart
Step 1: Understand the scale
1 cm on map = 50000 cm in real life
Step 2: Calculate
8 cm on map = $8 \times 50000 = 400000$ cm
Step 3: Convert to sensible units
400000 cm = 4000 m = 4 km
Answer: The towns are 4 km apart
Example 3: Mixing Solutions
Orange juice concentrate and water are mixed in the ratio $1:4$ to make orange juice. How much water is needed with 250ml of concentrate?
Step 1: Identify the ratio parts
Concentrate : Water = $1:4$
Step 2: Scale up
If 1 part = 250ml concentrate
Then 4 parts = $4 \times 250 = 1000$ ml water
Answer: 1000ml (or 1 litre) of water is needed
Total drink: $250 + 1000 = 1250$ ml (or 1.25 litres)
Step 1: Identify the ratio parts
Concentrate : Water = $1:4$
Step 2: Scale up
If 1 part = 250ml concentrate
Then 4 parts = $4 \times 250 = 1000$ ml water
Answer: 1000ml (or 1 litre) of water is needed
Total drink: $250 + 1000 = 1250$ ml (or 1.25 litres)
Real Life Uses:
• Cooking: Scaling recipes up or down
• Maps: Understanding distances and scales
• Art: Mixing paints to get the right color
• Chemistry: Mixing solutions and compounds
• Business: Profit sharing among partners
• Construction: Mixing concrete (cement:sand:gravel)
• Cooking: Scaling recipes up or down
• Maps: Understanding distances and scales
• Art: Mixing paints to get the right color
• Chemistry: Mixing solutions and compounds
• Business: Profit sharing among partners
• Construction: Mixing concrete (cement:sand:gravel)