Ratio and Proportion — Class 6 Mathematics

'Life is about comparisons — ratio and proportion help us compare things fairly and accurately.'

1. Introduction

A ratio compares two quantities. A proportion says two ratios are equal. The unitary method finds the value of one unit.

Why This Chapter Matters

• Cooking: 2 cups of rice to 3 cups of water
• Maps: 1 cm = 10 km
• Shopping: comparing prices per kg
• AP context: mixing fertilisers, comparing crop yields

2. Ratio

A ratio is a comparison of two quantities of the same kind using division.

Notation: Ratio of a to b is written as a : b or a/b

Example: There are 8 boys and 12 girls in a class.

• Ratio of boys to girls = 8 : 12 = 2 : 3 (simplified)
• Ratio of girls to boys = 12 : 8 = 3 : 2
• Ratio of boys to total students = 8 : 20 = 2 : 5

Rules for Writing Ratios

1. Both quantities must be in the SAME unit (convert if needed)
2. Ratio has NO unit (it's a pure number)
3. Simplify like a fraction (divide both by HCF)
4. Order matters! 'Ratio of a to b' means a:b, NOT b:a

Examples

Common Mistake

WRONG: Ratio of 40 cm to 1 m = 40:1 CORRECT: Convert to same unit. 1 m = 100 cm. 40:100 = 2:5.

'Always check units before writing a ratio. You cannot compare centimetres to metres directly.'

3. Equivalent Ratios

Just like equivalent fractions, ratios can be multiplied or divided by the same number.

Finding Equivalent Ratios

Multiply or divide both terms by the SAME number.

Example: Find 3 equivalent ratios of 2:3

• 2:3 = 4:6 (×2)
• 2:3 = 6:9 (×3)
• 2:3 = 10:15 (×5)

Comparing Ratios

Convert to equivalent ratios with same second term, or convert to fractions.

Example: Which is larger: 2:3 or 3:4? 2:3 = 8:12 (×4) 3:4 = 9:12 (×3) Since 9 > 8, 3:4 > 2:3

Simplifying Ratios

Divide both terms by their HCF.

Example: Simplify 24:36 HCF of 24 and 36 = 12 24÷12 = 2, 36÷12 = 3 24:36 = 2:3

4. Proportion

A proportion is an equation stating that two ratios are equal.

Notation: a : b = c : d or a/b = c/d

Read as: 'a is to b as c is to d'

Terms of a Proportion

a : b = c : d
│              │
└──────┬────────┘
    Extremes
       │
       └─── Inner terms (means)

• Extremes: a and d (first and fourth terms)
• Means: b and c (second and third terms)

Key Property

Product of extremes = Product of means a × d = b × c

Checking if Two Ratios Form a Proportion

Cross-multiply. If the products are equal, they form a proportion.

Example: Do 2:3 and 8:12 form a proportion? Extremes: 2 × 12 = 24 Means: 3 × 8 = 24 24 = 24 → YES, they form a proportion.

Example: Do 4:5 and 10:12 form a proportion? Extremes: 4 × 12 = 48 Means: 5 × 10 = 50 48 ≠ 50 → NO, they do not form a proportion.

Finding the Missing Term

Example: Find x: 3:5 = 12:x Product of extremes = 3 × x Product of means = 5 × 12 = 60 3 × x = 60 x = 20

Example: Find x: x:7 = 8:14 x × 14 = 7 × 8 14x = 56 x = 4

5. Unitary Method

The unitary method finds the value of one unit, then uses it to find the value of any number of units.

Steps

1. Find the value of 1 unit (divide)
2. Multiply to find the value of required units

Example 1 (Shopping)

12 pens cost Rs. 60. Find the cost of 20 pens.

Solution: Step 1 (Find 1 unit): 1 pen = 60 ÷ 12 = Rs. 5 Step 2 (Find 20 units): 20 pens = 20 × 5 = Rs. 100 Answer: Rs. 100

Example 2 (Work)

5 workers can build a wall in 30 days. How long will 6 workers take?

Solution: Step 1 (1 worker): 1 worker takes 5 × 30 = 150 days Step 2 (6 workers): 6 workers take 150 ÷ 6 = 25 days Answer: 25 days

'More workers → fewer days. This is inverse proportion. More items → more cost. This is direct proportion.'

Direct vs Inverse Proportion

6. Word Problems

Problem 1 (Ratio in a Class)

In a class of 45 students, the ratio of boys to girls is 2:3. Find the number of boys and girls.

Solution: Sum of ratio parts = 2 + 3 = 5 One part = 45 ÷ 5 = 9 students Boys = 2 × 9 = 18 Girls = 3 × 9 = 27 Answer: 18 boys, 27 girls

Problem 2 (AP Context — Fertiliser Mix)

A farmer in Guntur mixes fertiliser in the ratio 3:2 for nitrogen and phosphorus. If she uses 18 kg of nitrogen, how much phosphorus does she need?

Solution: Ratio N:P = 3:2 3 parts = 18 kg, so 1 part = 6 kg 2 parts of P = 2 × 6 = 12 kg Answer: 12 kg of phosphorus

Problem 3 (Proportion)

A map scale shows 1 cm = 25 km. If two towns are 6 cm apart on the map, what is the actual distance?

Solution: 1:25 = 6:x 1 × x = 25 × 6 x = 150 km Answer: 150 km

Problem 4 (Unitary — Cooking)

A recipe requires 3 cups of water for 2 cups of rice. How much water is needed for 5 cups of rice?

Solution: 2 cups rice = 3 cups water 1 cup rice = 3/2 = 1.5 cups water 5 cups rice = 5 × 1.5 = 7.5 cups water Answer: 7.5 cups

7. Common Mistakes — Fix Them Now

8. Exam Focus

Marks Blueprint

Quick Self-Test (5 Questions)

Q1: Simplify the ratio 36:48.

Q2: Do 5:8 and 15:24 form a proportion?

Q3: Find x: 6:x = 3:9

Q4: 8 kg of rice cost Rs. 360. Find the cost of 5 kg.

Q5: Divide Rs. 800 between Ravi and Sita in the ratio 3:5.

9. Chapter Summary

• Ratio: comparison of two same-unit quantities (a:b)
• Simplify: divide by HCF of both terms
• Equivalent ratios: multiply/divide both terms by same number
• Proportion: a:b = c:d → product of extremes = product of means (a×d = b×c)
• Unitary method: find 1 unit, then multiply
• Direct proportion: more A → more B
• Inverse proportion: more A → less B

'Ratios, proportions, and the unitary method are tools for fair comparison and smart calculation — you'll use them in shopping, cooking, travel, and business.'