Simplifying Ratios
A ratio compares two quantities. But the same ratio can be written in many ways. For example, 4:6, 2:3, and 8:12 all represent the same comparison.
The simplest form of a ratio is when the two numbers have no common factor other than 1.
In Class 6, you will learn to simplify ratios by dividing both parts by their HCF (Highest Common Factor).
What is Simplifying Ratios - Grade 6 Maths (Ratio and Proportion)?
Definition: A ratio is in its simplest form (or lowest terms) when the HCF of both numbers is 1.
Method to simplify a ratio:
- Find the HCF of both numbers in the ratio.
- Divide both numbers by the HCF.
- The result is the simplest form.
a : b (simplest form) → HCF(a, b) = 1
Simplifying Ratios Formula
Simplifying ratios with different units:
- First convert both quantities to the same unit.
- Then simplify.
- Example: 2 m : 50 cm → 200 cm : 50 cm → 200:50 → divide by 50 → 4:1
Simplifying ratios with decimals:
- Multiply both parts by 10 (or 100) to remove the decimal.
- Then simplify normally.
- Example: 0.5 : 1.5 → multiply by 10 → 5:15 → divide by 5 → 1:3
Simplifying ratios with fractions:
- Multiply both parts by the LCM of the denominators to make them whole numbers.
- Then simplify.
- Example: 1/2 : 3/4 → multiply by 4 → 2:3.
Types and Properties
When is a ratio already in simplest form?
- When both numbers share no common factor except 1.
- Examples: 3:5, 7:11, 1:4, 9:2 — all already simplified.
Common simplification patterns:
- Both even → divide by 2 (keep dividing if still even).
- Both end in 0 or 5 → divide by 5.
- Both divisible by 3 → divide by 3.
- If unsure, find the HCF by prime factorisation.
Solved Examples
Example 1: Basic Simplification
Problem: Simplify 12:18.
Solution:
HCF of 12 and 18 = 6
12 ÷ 6 : 18 ÷ 6 = 2:3
Example 2: Larger Numbers
Problem: Simplify 36:48.
Solution:
HCF of 36 and 48 = 12
36 ÷ 12 : 48 ÷ 12 = 3:4
Example 3: Different Units
Problem: Simplify the ratio 3 kg : 750 g.
Solution:
Convert to same unit: 3 kg = 3000 g
Ratio = 3000 : 750
HCF of 3000 and 750 = 750
3000 ÷ 750 : 750 ÷ 750 = 4:1
Example 4: Ratio with Decimals
Problem: Simplify 0.4 : 1.2.
Solution:
Multiply both by 10: 4 : 12
HCF of 4 and 12 = 4
4 ÷ 4 : 12 ÷ 4 = 1:3
Example 5: Ratio with Fractions
Problem: Simplify 1/3 : 2/5.
Solution:
LCM of denominators 3 and 5 = 15
Multiply both by 15: (1/3 × 15) : (2/5 × 15) = 5 : 6
Answer: 5:6
Example 6: Minutes and Hours
Problem: Simplify 40 minutes : 1 hour.
Solution:
Convert: 1 hour = 60 minutes
Ratio = 40 : 60
HCF = 20
40 ÷ 20 : 60 ÷ 20 = 2:3
Example 7: Three-Part Ratio
Problem: Simplify 6:9:15.
Solution:
HCF of 6, 9, and 15 = 3
6 ÷ 3 : 9 ÷ 3 : 15 ÷ 3 = 2:3:5
Example 8: Already Simplified
Problem: Is 7:11 in simplest form?
Solution:
7 is prime. 11 is prime. They share no common factor other than 1.
Answer: Yes, 7:11 is already in simplest form.
Real-World Applications
Where simplifying ratios is useful:
- Recipes: A recipe calls for flour and sugar in the ratio 500 g : 250 g = 2:1. Easier to scale up or down.
- Maps: Map scale 1:1000 means 1 cm on the map = 1000 cm (10 m) in real life.
- Mixing: Paint mixed in ratio 3:1 — 3 parts colour, 1 part white.
- Speed and Distance: Comparing speeds of two objects.
- Sharing: Dividing money, sweets, or resources in a given ratio.
Key Points to Remember
- A ratio is in simplest form when HCF of both numbers is 1.
- To simplify: divide both parts by their HCF.
- Before simplifying, convert both quantities to the same unit.
- For decimals: multiply by 10 or 100 to make whole numbers first.
- For fractions: multiply by LCM of denominators first.
- A ratio has no units in its final simplified form.
- For three-part ratios, divide all three by their common HCF.
Practice Problems
- Simplify 24:36.
- Simplify 2 km : 500 m.
- Simplify 0.6 : 0.9.
- Simplify 1/4 : 1/6.
- Simplify 45 minutes : 2 hours.
- Simplify the ratio 10:25:35.
Frequently Asked Questions
Q1. What does it mean to simplify a ratio?
It means writing the ratio in its smallest whole numbers by dividing both parts by their HCF. For example, 8:12 simplifies to 2:3.
Q2. Why must both quantities be in the same unit?
A ratio compares two numbers directly. If one is in kg and the other in grams, the comparison is meaningless. Convert to the same unit first, then simplify.
Q3. How do I simplify a ratio with decimals?
Multiply both parts by 10 (for one decimal place) or 100 (for two decimal places) to remove the decimals. Then simplify the whole number ratio using HCF.
Q4. Can a ratio have units?
No. A ratio in simplest form is just two numbers separated by a colon (like 3:5). The units cancel out after conversion.
Q5. Is 2:3 the same as 3:2?
No! The order matters. 2:3 means the first quantity is smaller than the second. 3:2 means the first is larger. Always maintain the correct order.
Q6. What if the HCF is 1?
Then the ratio is already in its simplest form. No further simplification is needed.
