11-15-2025, 02:29 PM
Chapter 2 — Ratios Made Easy
If there is ONE skill that unlocks probability, it is this:
Understanding ratios.
Most students struggle with probability because they struggle with ratios — not because the probability is hard.
This chapter will take ratios from confusing → clear in a single lesson.
---
2.1 What a Ratio Really Means
A ratio compares PARTS to OTHER PARTS.
Example:
The ratio of boys to girls is 3 : 2
This means:
• for every 3 boys
• there are 2 girls
It does NOT mean “there are 5 children”
(although that might also be true — but only if we're looking at the whole group).
Probability later works with PART : WHOLE,
but ratios are PART : PART.
---
2.2 Ratio DOES NOT mean absolute numbers
3 : 2 does NOT mean:
• 3 boys
• 2 girls
It might be:
• 6 boys, 4 girls
• 9 boys, 6 girls
• 300 boys, 200 girls
The ratio only tells you the relative sizes, not the actual numbers.
---
2.3 Simplifying Ratios
Ratios can ALWAYS be simplified, just like fractions.
Example:
6 : 8
Divide both by 2 → 3 : 4
Another example:
20 : 50
Divide both by 10 → 2 : 5
Simplifying ratios makes probability easier later.
---
2.4 From Ratio to Total Amount
If the ratio of red to blue beads is 3 : 1
and there are 20 beads total…
Step 1 — Add the ratio parts
3 + 1 = 4 parts total
Step 2 — Find the value of each part
20 ÷ 4 = 5 per part
Step 3 — Find amounts
Red = 3 parts = 15
Blue = 1 part = 5
This technique is ESSENTIAL for solving advanced probability problems.
---
2.5 Scaling Ratios
If a recipe uses a ratio 2 : 3 and you want DOUBLE the amount…
Multiply both by 2 → 4 : 6
If you want HALF…
Divide both by 2 → 1 : 1.5
Scaling ratios is one of the most common exam skills.
---
2.6 Ratio in Probability
This is the KEY connection:
Probability is always PART / WHOLE.
But ratios are PART : PART.
To turn a ratio into a probability, you must first find the whole amount.
Example:
A bag contains red and blue balls in the ratio 3 : 2.
Total parts = 3 + 2 = 5
Probability of red = 3/5
Probability of blue = 2/5
This conversion becomes important in Chapter 3.
---
2.7 Common Mistake: Inversion
Many students confuse ratio direction.
Example:
“Girls to boys is 1 : 4”
Some students mistakenly think it means 4 girls and 1 boy.
The order ALWAYS matters:
first label → first number
second label → second number
---
2.8 Worked Examples
Example 1
The ratio of cats to dogs is 5 : 3. There are 32 animals.
How many are cats?
Step 1: Total parts = 5 + 3 = 8
Step 2: Value of each part = 32 ÷ 8 = 4
Step 3: Cats = 5 × 4 = 20
Answer: 20 cats
---
Example 2
The ratio of green to yellow marbles is 3 : 7.
What fraction are yellow?
Total parts = 10
Yellow = 7 parts
Fraction = 7/10
---
Example 3
A mixture is in the ratio 4 : 1. If the larger amount is 48, what is the smaller?
48 corresponds to 4 parts.
1 part = 48 ÷ 4 = 12
Small amount = 12
---
2.9 Your Turn
1. The ratio of water to juice is 3 : 5.
If there are 40 litres of mixture, how much juice?
2. The ratio of wins to losses for a team is 7 : 3.
What fraction of their games do they win?
3. A bag contains red and blue stones in ratio 2 : 9.
How many total stones if there are 22 blue stones?
4. Simplify the ratio 18 : 30.
5. A school has a ratio of teachers to students of 1 : 24.
If there are 72 students, how many teachers?
---
Chapter Summary
• Ratios compare parts to parts
• Ratios can always be simplified
• To work with totals, add the ratio parts
• Probability uses part/whole
• Ratios must be read in the correct order
• Strong ratio skills make probability MUCH easier
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
If there is ONE skill that unlocks probability, it is this:
Understanding ratios.
Most students struggle with probability because they struggle with ratios — not because the probability is hard.
This chapter will take ratios from confusing → clear in a single lesson.
---
2.1 What a Ratio Really Means
A ratio compares PARTS to OTHER PARTS.
Example:
The ratio of boys to girls is 3 : 2
This means:
• for every 3 boys
• there are 2 girls
It does NOT mean “there are 5 children”
(although that might also be true — but only if we're looking at the whole group).
Probability later works with PART : WHOLE,
but ratios are PART : PART.
---
2.2 Ratio DOES NOT mean absolute numbers
3 : 2 does NOT mean:
• 3 boys
• 2 girls
It might be:
• 6 boys, 4 girls
• 9 boys, 6 girls
• 300 boys, 200 girls
The ratio only tells you the relative sizes, not the actual numbers.
---
2.3 Simplifying Ratios
Ratios can ALWAYS be simplified, just like fractions.
Example:
6 : 8
Divide both by 2 → 3 : 4
Another example:
20 : 50
Divide both by 10 → 2 : 5
Simplifying ratios makes probability easier later.
---
2.4 From Ratio to Total Amount
If the ratio of red to blue beads is 3 : 1
and there are 20 beads total…
Step 1 — Add the ratio parts
3 + 1 = 4 parts total
Step 2 — Find the value of each part
20 ÷ 4 = 5 per part
Step 3 — Find amounts
Red = 3 parts = 15
Blue = 1 part = 5
This technique is ESSENTIAL for solving advanced probability problems.
---
2.5 Scaling Ratios
If a recipe uses a ratio 2 : 3 and you want DOUBLE the amount…
Multiply both by 2 → 4 : 6
If you want HALF…
Divide both by 2 → 1 : 1.5
Scaling ratios is one of the most common exam skills.
---
2.6 Ratio in Probability
This is the KEY connection:
Probability is always PART / WHOLE.
But ratios are PART : PART.
To turn a ratio into a probability, you must first find the whole amount.
Example:
A bag contains red and blue balls in the ratio 3 : 2.
Total parts = 3 + 2 = 5
Probability of red = 3/5
Probability of blue = 2/5
This conversion becomes important in Chapter 3.
---
2.7 Common Mistake: Inversion
Many students confuse ratio direction.
Example:
“Girls to boys is 1 : 4”
Some students mistakenly think it means 4 girls and 1 boy.
The order ALWAYS matters:
first label → first number
second label → second number
---
2.8 Worked Examples
Example 1
The ratio of cats to dogs is 5 : 3. There are 32 animals.
How many are cats?
Step 1: Total parts = 5 + 3 = 8
Step 2: Value of each part = 32 ÷ 8 = 4
Step 3: Cats = 5 × 4 = 20
Answer: 20 cats
---
Example 2
The ratio of green to yellow marbles is 3 : 7.
What fraction are yellow?
Total parts = 10
Yellow = 7 parts
Fraction = 7/10
---
Example 3
A mixture is in the ratio 4 : 1. If the larger amount is 48, what is the smaller?
48 corresponds to 4 parts.
1 part = 48 ÷ 4 = 12
Small amount = 12
---
2.9 Your Turn
1. The ratio of water to juice is 3 : 5.
If there are 40 litres of mixture, how much juice?
2. The ratio of wins to losses for a team is 7 : 3.
What fraction of their games do they win?
3. A bag contains red and blue stones in ratio 2 : 9.
How many total stones if there are 22 blue stones?
4. Simplify the ratio 18 : 30.
5. A school has a ratio of teachers to students of 1 : 24.
If there are 72 students, how many teachers?
---
Chapter Summary
• Ratios compare parts to parts
• Ratios can always be simplified
• To work with totals, add the ratio parts
• Probability uses part/whole
• Ratios must be read in the correct order
• Strong ratio skills make probability MUCH easier
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive