Introduction to Ratio
Your mother is making tea. She uses 2 spoons of sugar for every 1 spoon of tea leaves. The comparison of sugar to tea leaves is 2 to 1. This comparison is called a ratio.
A ratio compares two quantities by showing how many times one quantity contains the other. We see ratios everywhere — in recipes, maps, mixtures, and even in sharing things fairly among friends.
When someone says "mix lemon juice and water in the ratio 1 : 4," it means for every 1 part of lemon juice, you add 4 parts of water. If you take 1 cup of lemon juice, you add 4 cups of water. If you take 2 cups of lemon juice, you add 8 cups of water. The ratio stays the same.
Ratios are also used to compare sizes. If a model car is made in the ratio 1 : 20 of the real car, it means every 1 cm on the model represents 20 cm on the real car. Maps use ratios to show how distances on paper relate to actual distances on the ground.
In this chapter, you will learn what a ratio is, how to write it, how to simplify it, how to find equivalent ratios, and how to divide a quantity in a given ratio. This topic is part of the Ratio and Proportion chapter in Grade 6 Maths (NCERT/CBSE).
What is Introduction to Ratio - Grade 6 Maths (Ratio and Proportion)?
Definition: A ratio is a way of comparing two quantities of the same kind by division.
If there are 3 boys and 5 girls in a group:
- The ratio of boys to girls = 3 to 5
- Written as 3 : 5
- Read as "3 is to 5"
Key points about ratios:
- A ratio has no units — it is a pure number.
- The order matters. The ratio of boys to girls (3 : 5) is different from the ratio of girls to boys (5 : 3).
- Both quantities must be in the same unit before forming the ratio.
- A ratio can also be written as a fraction: 3 : 5 = 3/5.
Parts of a ratio:
- In the ratio a : b, a is called the antecedent (the first term).
- b is called the consequent (the second term).
Introduction to Ratio Formula
Writing a ratio:
Ratio of a to b = a : b = a/b
Simplifying a ratio:
Divide both terms by their HCF (Highest Common Factor).
a : b = (a ÷ HCF) : (b ÷ HCF)
Example:
- 12 : 18 → HCF of 12 and 18 = 6
- 12 ÷ 6 : 18 ÷ 6 = 2 : 3
Equivalent ratios:
Multiply or divide both terms by the same number:
- 2 : 3 = 4 : 6 = 6 : 9 = 8 : 12 (all equivalent)
Derivation and Proof
The concept of ratio comes from the idea of fair comparison.
Suppose you and your friend collect stickers. You have 10 and your friend has 15. To compare, you could subtract: 15 − 10 = 5 (your friend has 5 more). But this does not tell you the relative size.
A ratio does: 10 : 15 = 2 : 3. This tells you that for every 2 stickers you have, your friend has 3. The ratio gives a better picture of the comparison.
Why must both quantities be in the same unit?
Suppose you want to compare 2 metres and 50 centimetres. You cannot write 2 : 50 — that would be misleading. First convert: 2 m = 200 cm. Now the ratio is 200 : 50 = 4 : 1. This makes sense.
Why does order matter?
- If there are 4 apples and 7 oranges:
- Ratio of apples to oranges = 4 : 7
- Ratio of oranges to apples = 7 : 4
- These are different ratios!
Connection to fractions:
A ratio a : b is closely related to the fraction a/b. Simplifying a ratio works exactly like simplifying a fraction — divide both parts by their HCF.
Types and Properties
Type 1: Writing a ratio from given information
- Given two quantities, write the ratio in the correct order.
- Example: 8 boys, 12 girls → Ratio of boys to girls = 8 : 12 = 2 : 3
Type 2: Simplifying a ratio
- Divide both terms by their HCF.
- Example: 24 : 36 → HCF = 12 → 2 : 3
Type 3: Equivalent ratios
- Find ratios equal to a given ratio by multiplying/dividing.
- Example: 3 : 5 = 6 : 10 = 9 : 15
Type 4: Converting units before finding ratio
- Convert both quantities to the same unit first.
- Example: 1 hour and 45 minutes → 60 min : 45 min = 4 : 3
Type 5: Dividing a quantity in a given ratio
- Divide a total into parts as per the ratio.
- Example: Divide 40 in the ratio 3 : 5 → Parts = 15 and 25
Type 6: Word problems
- Real-life problems involving sharing, mixing, or comparing.
Solved Examples
Example 1: Example 1: Writing a Ratio
Problem: In a class of 30 students, 18 are girls and 12 are boys. Find the ratio of: (a) boys to girls (b) girls to total students.
Solution:
- (a) Ratio of boys to girls = 12 : 18
- HCF of 12 and 18 = 6
- 12 ÷ 6 : 18 ÷ 6 = 2 : 3
- (b) Ratio of girls to total = 18 : 30
- HCF of 18 and 30 = 6
- 18 ÷ 6 : 30 ÷ 6 = 3 : 5
Example 2: Example 2: Simplifying Ratios
Problem: Simplify: (a) 15 : 25 (b) 48 : 64 (c) 100 : 250
Solution:
- (a) HCF of 15, 25 = 5 → 15 ÷ 5 : 25 ÷ 5 = 3 : 5
- (b) HCF of 48, 64 = 16 → 48 ÷ 16 : 64 ÷ 16 = 3 : 4
- (c) HCF of 100, 250 = 50 → 100 ÷ 50 : 250 ÷ 50 = 2 : 5
Example 3: Example 3: Unit Conversion Before Ratio
Problem: Find the ratio of 500 g to 2 kg.
Solution:
Step 1: Convert to the same unit.
- 2 kg = 2000 g
Step 2: Write the ratio.
- 500 : 2000
- HCF = 500
- 500 ÷ 500 : 2000 ÷ 500 = 1 : 4
Answer: The ratio is 1 : 4.
Example 4: Example 4: Ratio with Time
Problem: Find the ratio of 40 minutes to 1 hour.
Solution:
Step 1: Convert to the same unit.
- 1 hour = 60 minutes
Step 2: Write the ratio.
- 40 : 60
- HCF = 20
- 40 ÷ 20 : 60 ÷ 20 = 2 : 3
Answer: The ratio is 2 : 3.
Example 5: Example 5: Dividing in a Given Ratio
Problem: Divide Rs. 560 between Anita and Bina in the ratio 3 : 5.
Solution:
Step 1: Find the total parts.
- Total parts = 3 + 5 = 8
Step 2: Find the value of each part.
- Value of 1 part = 560 ÷ 8 = Rs. 70
Step 3: Find each person's share.
- Anita's share = 3 × 70 = Rs. 210
- Bina's share = 5 × 70 = Rs. 350
Check: 210 + 350 = 560 ✓
Example 6: Example 6: Recipe Problem
Problem: A lemonade recipe uses sugar and lemon juice in the ratio 2 : 5. If you use 6 spoons of sugar, how many spoons of lemon juice do you need?
Solution:
- Ratio = Sugar : Lemon juice = 2 : 5
- Sugar used = 6 spoons
- Multiplying factor = 6 ÷ 2 = 3
- Lemon juice = 5 × 3 = 15 spoons
Answer: You need 15 spoons of lemon juice.
Example 7: Example 7: Comparing Ratios
Problem: Are the ratios 4 : 6 and 6 : 9 equivalent?
Solution:
Simplify both:
- 4 : 6 → HCF = 2 → 2 : 3
- 6 : 9 → HCF = 3 → 2 : 3
Both simplify to 2 : 3.
Answer: Yes, 4 : 6 and 6 : 9 are equivalent ratios.
Example 8: Example 8: Length Ratio
Problem: Pencil A is 15 cm long. Pencil B is 25 cm long. Find the ratio of their lengths.
Solution:
- Ratio = 15 : 25
- HCF = 5
- 15 ÷ 5 : 25 ÷ 5 = 3 : 5
Answer: The ratio of Pencil A to Pencil B is 3 : 5.
Example 9: Example 9: Three-Way Ratio
Problem: In a bag, there are 6 red, 10 blue, and 4 green balls. Find the ratio of red to blue to green balls.
Solution:
- Ratio = 6 : 10 : 4
- HCF of 6, 10, 4 = 2
- 6 ÷ 2 : 10 ÷ 2 : 4 ÷ 2 = 3 : 5 : 2
Answer: The ratio is 3 : 5 : 2.
Example 10: Example 10: Finding a Quantity from a Ratio
Problem: The ratio of Raj's savings to his spending is 2 : 7. If he spends Rs. 3,500, how much does he save?
Solution:
- Savings : Spending = 2 : 7
- Spending = Rs. 3,500 (this corresponds to 7 parts)
- Value of 1 part = 3500 ÷ 7 = Rs. 500
- Savings = 2 × 500 = Rs. 1,000
Answer: Raj saves Rs. 1,000.
Real-World Applications
Ratios are used everywhere in daily life:
- Cooking: Recipes use ratios — rice to water is 1 : 2 means for every cup of rice, use 2 cups of water.
- Maps: A map scale of 1 : 1000 means 1 cm on the map = 1000 cm (10 m) in real life.
- Sharing: If three friends share a pizza in the ratio 2 : 3 : 5, the ratio tells you how many slices each person gets.
- Mixing paints: To get the right colour shade, paints are mixed in specific ratios — like yellow and blue in 1 : 3 to get a green shade.
- Sports: Win-loss ratio tells how a team is performing. A ratio of 7 : 3 means 7 wins for every 3 losses.
- Business: Profit sharing among partners uses ratios based on their investment amounts.
Key Points to Remember
- A ratio compares two quantities of the same kind by division.
- Ratio is written as a : b and read as "a is to b."
- The order in a ratio matters: a : b is NOT the same as b : a.
- A ratio has no units — both quantities must be in the same unit before forming the ratio.
- To simplify a ratio, divide both terms by their HCF.
- A ratio in simplest form has terms with no common factor other than 1.
- Equivalent ratios are obtained by multiplying or dividing both terms by the same number.
- To divide a quantity in ratio a : b, total parts = a + b, then each share = (part/total) × quantity.
- A ratio can be written as a fraction: a : b = a/b.
- The first term of a ratio is called the antecedent; the second is the consequent.
Practice Problems
- There are 24 mangoes and 36 apples in a basket. Find the ratio of mangoes to apples in simplest form.
- Find the ratio of 75 cm to 1.5 m.
- Simplify the ratio 45 : 60.
- Divide 72 chocolates between two children in the ratio 4 : 5.
- A recipe needs flour and sugar in the ratio 5 : 2. If you use 300 g of flour, how much sugar do you need?
- Are 8 : 12 and 14 : 21 equivalent ratios?
- The ratio of Manu's age to Tanu's age is 3 : 4. If Tanu is 16 years old, find Manu's age.
- In a school, the ratio of teachers to students is 1 : 30. If there are 18 teachers, how many students are there?
Frequently Asked Questions
Q1. What is a ratio?
A ratio is a comparison of two quantities of the same kind by division. It tells you how many times one quantity is contained in another. For example, if there are 4 cats and 6 dogs, the ratio of cats to dogs is 4 : 6 = 2 : 3.
Q2. Does the order in a ratio matter?
Yes, the order matters. The ratio of boys to girls (3 : 5) is different from the ratio of girls to boys (5 : 3). Always write the ratio in the order that the question asks.
Q3. Can a ratio have units?
No. A ratio is a pure number with no units. Both quantities must be converted to the same unit before forming the ratio. For example, to find the ratio of 2 m to 50 cm, first convert: 2 m = 200 cm. Ratio = 200 : 50 = 4 : 1.
Q4. How do you simplify a ratio?
Divide both terms of the ratio by their HCF. For example, 18 : 24 → HCF = 6 → 18 ÷ 6 : 24 ÷ 6 = 3 : 4.
Q5. What is the difference between a ratio and a fraction?
A ratio a : b and a fraction a/b are closely related. The ratio 3 : 4 can be written as the fraction 3/4. The difference is that a ratio compares two separate quantities, while a fraction usually represents a part of a whole.
Q6. What are equivalent ratios?
Equivalent ratios have the same simplified form. For example, 2 : 3, 4 : 6, and 6 : 9 are all equivalent because they all simplify to 2 : 3. You get equivalent ratios by multiplying or dividing both terms by the same number.
Q7. Can a ratio have more than two terms?
Yes. A ratio can compare three or more quantities. For example, the ratio of red, blue, and green balls = 3 : 5 : 2. This means for every 3 red balls, there are 5 blue and 2 green.
Q8. How do you divide a number in a given ratio?
To divide a quantity in the ratio a : b: find total parts = a + b. Value of one part = total quantity ÷ total parts. Then first share = a × one part, second share = b × one part.
