11-15-2025, 02:50 PM
Chapter 3 — Fractions Inside Ratios
Most probability mistakes come from ONE thing:
Not understanding how fractions and ratios connect.
Ratios compare PARTS to PARTS.
Fractions compare PART to WHOLE.
To move into probability confidently, we must learn to move between them easily.
This chapter builds that bridge.
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3.1 Ratio vs Fraction — The Key Difference
A ratio compares one part to another part.
Example: 3 : 2 (red to blue)
A fraction compares a part to the whole.
Example: 3/5 red, 2/5 blue
Ratios talk about relationships inside a group.
Fractions talk about the ENTIRE group.
---
3.2 Turning Ratios Into Fractions
Example:
The ratio of red to blue is 3 : 2.
Total parts = 3 + 2 = 5
Red fraction = 3/5
Blue fraction = 2/5
This skill is essential for every probability problem coming later.
---
3.3 Turning Fractions Into Ratios
Example:
3/8 of a bag is green.
How do we write a ratio?
Step 1 — Recognise the fraction means:
3 green
8 total
Step 2 — Subtract to find the other part:
8 − 3 = 5
Step 3 — Ratio:
Green : Other = 3 : 5
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3.4 Fractions Hidden Inside Word Problems
Example
“40% of the class are girls.”
Convert 40% → 40/100 → 2/5
So the ratio girls : boys = 2 : 3
Because:
Girls = 2 parts
Boys = 3 parts
Total = 5 parts
Students often think percentage questions have nothing to do with ratios —
but they ALWAYS do.
---
3.5 Combining Ratios and Fractions (Trickiest Part)
Here is where many students get confused:
Sometimes the ratio is given AND the total is given.
Example:
A bag contains red and blue marbles in ratio 4 : 1.
There are 30 marbles in total.
Find the fraction that are red and the number that are red.
Step 1 — Add ratio parts
4 + 1 = 5
Step 2 — Find each part
30 ÷ 5 = 6 per part
Step 3 — Red
4 × 6 = 24
Fraction red = 24/30 = 4/5
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3.6 Fractions Inside Ratios in Probability Questions
A typical exam-style question:
A box contains red and green counters in ratio 3 : 5.
What is the probability of choosing a green counter?
Step 1 — Total parts = 3 + 5 = 8
Step 2 — Green = 5 parts
Step 3 — Probability = 5/8
Ratios → Fractions → Probability
This is the flow of ALL early probability.
---
3.7 Common Mistakes
Mistake 1: Forgetting to add ratio parts
3 : 7 → total parts = 10, not 3 or 7.
Mistake 2: Mixing up which number belongs where
Ratio of cats to dogs 3 : 2
Some students flip it to 2 : 3.
Mistake 3: Turning a ratio into a fraction incorrectly
3 : 2
Fraction red = 3 / (3+2) = 3/5 — NOT 3/2.
Mistake 4: Thinking ratios must match real amounts
A ratio 4 : 3 could mean
4 & 3
8 & 6
12 & 9
40 & 30
etc.
Ratios scale. Fractions don’t.
---
3.8 Worked Examples
Example 1
The ratio of apples to oranges is 2 : 3.
What fraction are oranges?
Total parts = 5
Oranges = 3
Fraction = 3/5
---
Example 2
A bag has sweets in ratio 5 : 1 (green : purple).
What fraction are green?
Total = 6
Green = 5
Fraction = 5/6
---
Example 3
40% of beads are blue.
Write the ratio blue to not blue.
40% → 40/100 → 2/5
So:
Blue = 2 parts
Not blue = 3 parts
Ratio = 2 : 3
---
Example 4
9/14 of cards are red.
Find the ratio red : non-red.
Red = 9
Total = 14
Non-red = 14 − 9 = 5
Ratio = 9 : 5
---
3.9 Your Turn
1. Ratio red to yellow is 4 : 1.
What fraction are red?
2. 25% of students walk to school.
Write the ratio walkers : non-walkers.
3. A bag has beads in ratio 2 : 7.
What fraction are blue?
4. 3/10 of the animals are cats.
Write a ratio for cats : other animals.
5. A mixture is 60% sugar.
Write sugar : not sugar as a simplified ratio.
---
Chapter Summary
• Ratios compare part to part
• Fractions compare part to whole
• Probability ALWAYS needs part/whole
• You must add ratio parts to get fractions
• Percentages, fractions, and ratios all connect
• This chapter builds the foundation for all probability work ahead
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
Most probability mistakes come from ONE thing:
Not understanding how fractions and ratios connect.
Ratios compare PARTS to PARTS.
Fractions compare PART to WHOLE.
To move into probability confidently, we must learn to move between them easily.
This chapter builds that bridge.
---
3.1 Ratio vs Fraction — The Key Difference
A ratio compares one part to another part.
Example: 3 : 2 (red to blue)
A fraction compares a part to the whole.
Example: 3/5 red, 2/5 blue
Ratios talk about relationships inside a group.
Fractions talk about the ENTIRE group.
---
3.2 Turning Ratios Into Fractions
Example:
The ratio of red to blue is 3 : 2.
Total parts = 3 + 2 = 5
Red fraction = 3/5
Blue fraction = 2/5
This skill is essential for every probability problem coming later.
---
3.3 Turning Fractions Into Ratios
Example:
3/8 of a bag is green.
How do we write a ratio?
Step 1 — Recognise the fraction means:
3 green
8 total
Step 2 — Subtract to find the other part:
8 − 3 = 5
Step 3 — Ratio:
Green : Other = 3 : 5
---
3.4 Fractions Hidden Inside Word Problems
Example
“40% of the class are girls.”
Convert 40% → 40/100 → 2/5
So the ratio girls : boys = 2 : 3
Because:
Girls = 2 parts
Boys = 3 parts
Total = 5 parts
Students often think percentage questions have nothing to do with ratios —
but they ALWAYS do.
---
3.5 Combining Ratios and Fractions (Trickiest Part)
Here is where many students get confused:
Sometimes the ratio is given AND the total is given.
Example:
A bag contains red and blue marbles in ratio 4 : 1.
There are 30 marbles in total.
Find the fraction that are red and the number that are red.
Step 1 — Add ratio parts
4 + 1 = 5
Step 2 — Find each part
30 ÷ 5 = 6 per part
Step 3 — Red
4 × 6 = 24
Fraction red = 24/30 = 4/5
---
3.6 Fractions Inside Ratios in Probability Questions
A typical exam-style question:
A box contains red and green counters in ratio 3 : 5.
What is the probability of choosing a green counter?
Step 1 — Total parts = 3 + 5 = 8
Step 2 — Green = 5 parts
Step 3 — Probability = 5/8
Ratios → Fractions → Probability
This is the flow of ALL early probability.
---
3.7 Common Mistakes
Mistake 1: Forgetting to add ratio parts
3 : 7 → total parts = 10, not 3 or 7.
Mistake 2: Mixing up which number belongs where
Ratio of cats to dogs 3 : 2
Some students flip it to 2 : 3.
Mistake 3: Turning a ratio into a fraction incorrectly
3 : 2
Fraction red = 3 / (3+2) = 3/5 — NOT 3/2.
Mistake 4: Thinking ratios must match real amounts
A ratio 4 : 3 could mean
4 & 3
8 & 6
12 & 9
40 & 30
etc.
Ratios scale. Fractions don’t.
---
3.8 Worked Examples
Example 1
The ratio of apples to oranges is 2 : 3.
What fraction are oranges?
Total parts = 5
Oranges = 3
Fraction = 3/5
---
Example 2
A bag has sweets in ratio 5 : 1 (green : purple).
What fraction are green?
Total = 6
Green = 5
Fraction = 5/6
---
Example 3
40% of beads are blue.
Write the ratio blue to not blue.
40% → 40/100 → 2/5
So:
Blue = 2 parts
Not blue = 3 parts
Ratio = 2 : 3
---
Example 4
9/14 of cards are red.
Find the ratio red : non-red.
Red = 9
Total = 14
Non-red = 14 − 9 = 5
Ratio = 9 : 5
---
3.9 Your Turn
1. Ratio red to yellow is 4 : 1.
What fraction are red?
2. 25% of students walk to school.
Write the ratio walkers : non-walkers.
3. A bag has beads in ratio 2 : 7.
What fraction are blue?
4. 3/10 of the animals are cats.
Write a ratio for cats : other animals.
5. A mixture is 60% sugar.
Write sugar : not sugar as a simplified ratio.
---
Chapter Summary
• Ratios compare part to part
• Fractions compare part to whole
• Probability ALWAYS needs part/whole
• You must add ratio parts to get fractions
• Percentages, fractions, and ratios all connect
• This chapter builds the foundation for all probability work ahead
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive