What is a proportion?

A proportion states that two ratios are equal. Written as a:b :: c:d (read as 'a is to b as c is to d'). Example: 2:3 :: 8:12 because 2/3 = 8/12. The product of means equals the product of extremes: b × c = a × d.

What is the unitary method?

The unitary method finds the value of one unit first, then multiplies to find the value of many units. Example: if 5 pens cost ₹35, then 1 pen costs ₹35 ÷ 5 = ₹7. So 8 pens cost 8 × ₹7 = ₹56.

How do you check if four numbers are in proportion?

Check if the product of extremes = product of means. For a:b::c:d, check if a × d = b × c. Example: 2:4::3:6 → 2×6 = 12 and 4×3 = 12. Equal, so yes — they are in proportion.

CBSE Class 6 Maths Chapter 12 Notes — Ratio and Proportion

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What is a Ratio?

A ratio compares two quantities of the same kind and same unit.

Written as a : b (read as "a is to b")

Example: In a class, there are 18 girls and 12 boys.

Ratio of girls to boys = 18 : 12 = 3 : 2 (simplified)
Ratio of boys to girls = 12 : 18 = 2 : 3

Important: Ratio has no units. Quantities must be in the same unit.

Example: Ratio of 50 paise to ₹2 = 50 paise to 200 paise = 50 : 200 = 1 : 4

Properties of Ratio

1. Order matters: a:b ≠ b:a (in general) 2. Ratio can be simplified (divide both by HCF):

3. Equivalent ratios (multiply/divide both parts by same number):

15:25 = 3:5 (HCF = 5)
2:3 = 4:6 = 6:9 = 8:12

Simplifying Ratios

Divide both terms by their HCF.

Example: Simplify 24 : 36

HCF(24, 36) = 12
24 ÷ 12 = 2; 36 ÷ 12 = 3
Simplest form: 2 : 3

Comparison of Ratios

Convert to equivalent fractions (common denominator) and compare.

Example: Which is greater — 3:4 or 5:7?

3/4 = 21/28; 5/7 = 20/28
21/28 > 20/28, so 3:4 > 5:7

Proportion

Proportion: When two ratios are equal.

Written as a : b :: c : d (a, b, c, d are in proportion)

Extremes: first and last terms (a and d)
Means: middle terms (b and c)

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

Example: Are 2, 4, 3, 6 in proportion?

2:4 :: 3:6 → 2 × 6 = 12 and 4 × 3 = 12 → Yes ✅

Example: Find x if 3:x :: 6:14

3 × 14 = 6 × x
42 = 6x → x = 7

Unitary Method

Find the value of one unit first, then scale up or down.

Steps:

1. Find the value for 1 unit. 2. Multiply to find the value for the required number.

Example 1: 4 kg of rice costs ₹120. Find cost of 7 kg.

Cost of 1 kg = 120 ÷ 4 = ₹30
Cost of 7 kg = 30 × 7 = ₹210

Example 2: A car travels 90 km in 3 hours. How far in 5 hours?

Distance in 1 hour = 90 ÷ 3 = 30 km
Distance in 5 hours = 30 × 5 = 150 km

Dividing a Quantity in a Given Ratio

Example: Divide ₹600 in the ratio 2:3.

Total parts = 2 + 3 = 5
One part = 600 ÷ 5 = 120
First share = 2 × 120 = ₹240
Second share = 3 × 120 = ₹360

Check: 240 + 360 = 600 ✅

Key Vocabulary

Term	Meaning
Ratio	Comparison of two quantities of same kind
Terms of ratio	The two numbers in a ratio (a and b in a:b)
Equivalent ratio	Ratio obtained by multiplying/dividing both terms by same number
Proportion	Two equal ratios
Extremes	First and last terms of a proportion
Means	Middle two terms of a proportion
Unitary method	Finding value of 1 unit first, then multiplying

Exam Tips

Units must match before writing a ratio — convert first.
a:b ≠ b:a — order of comparison matters.
Proportion check: cross-multiply → product of extremes = product of means.
In "divide ₹500 in ratio 3:7": total parts = 10, one part = ₹50. Don't forget to add the parts first.
Unitary method — always clearly state "cost/distance/weight of 1 unit = …"
Simplest form of ratio: HCF method — same as simplifying fractions.

Frequently Asked Questions

A ratio compares two quantities of the same kind. Written as a:b or a/b. Example: if there are 3 red balls and 5 blue balls, ratio of red to blue = 3:5. A ratio has no units — the quantities must be in the same unit before comparing.