Notes For All Chapters – Maths Class 6 Ganita Prakash
🍕 Fractions
Chapter 7 — Complete Study Notes with Formulas, Examples, and Exam-Ready Questions
🔢 Numerator & Denominator
🔄 Equivalent Fractions
➕ Addition
➖ Subtraction
📏 Number Line
🏆 Exam Ready
- What is a Fraction?
- Fractional Units and Equal Shares
- Fractions as Parts of a Whole
- Measuring Using Fractional Units
- Fractions on the Number Line
- Mixed Fractions
- Equivalent Fractions
- Simplest Form / Lowest Terms
- Comparing Fractions
- Addition of Fractions (Brahmagupta’s Method)
- Subtraction of Fractions
- A Pinch of History
- Quick Summary
- Important Exam Questions
What is a Fraction?
When whole things are shared equally among some number of people, fractions tell us how much each person’s share is.
If 1 roti is divided equally between 2 children, each child gets 1/2 roti.
If 1 roti is divided equally among 4 children, each child gets 1/4 roti.
More children sharing = smaller share! So 1/2 > 1/4.
🔑 Parts of a Fraction
In any fraction like 5/6:
The number above the line.
Tells us: how many parts we have.
In 5/6, the numerator is 5.
The number below the line.
Tells us: how many equal parts the whole is divided into.
In 5/6, the denominator is 6.
Read 3/4 as “3 times one-fourth” instead of just “three-fourths”. This helps you understand: the fractional unit is 1/4, and we have 3 of them!
7.1 Fractional Units and Equal Shares
When one unit (one roti, one chikki, one strip) is divided into several equal parts, each part is called a fractional unit.
Fractional units are also called unit fractions (fractions with 1 in the numerator).
⚠️ Common Mistake: Comparing Unit Fractions
Many students think 1/9 > 1/5 because 9 > 5. This is WRONG!
Think of it as sharing: If 1 roti is shared among 9 children, each gets less than if shared among 5 children.
More people = smaller share!
So 1/9 < 1/5 ✓
Example: 1/2 > 1/4 > 1/6 > 1/10 > 1/100
In the Rig Veda, the fraction 3/4 is called tripada. In Hindi, 3/4 is called teen paav and in Tamil it is mukkaal. Fractions have been used in India since ancient times!
7.2 Fractions as Parts of a Whole
A fraction represents a part of a whole. The whole can be divided in different ways, but as long as the parts are equal, the fraction is the same.
A whole chikki can be cut into 6 equal pieces in different ways (rectangular cuts or diagonal cuts). Each piece is 1/6 chikki. Even though the shapes look different, all pieces have the same size (same area = same fraction)!
📐 Measuring with Fractional Units
A larger piece can be measured using a smaller fractional unit as a “ruler”:
So the big piece = 3 × 1/4 = 3/4 chikki.
The fraction 3/4 means “3 times the fractional unit 1/4“. We are collecting 3 pieces of size 1/4 each.
7.3 Measuring Using Fractional Units
📄 Paper Folding Activity
Take a paper strip = 1 unit long. Fold it in half → each part = 1/2. Fold again → 4 parts, each = 1/4. Fold once more → 8 parts, each = 1/8.
Fold again (into 4) → each part = 1/4
Fold again (into 8) → each part = 1/8Building up with 1/4:
1 × (1/4) = 1/4
2 × (1/4) = 2/4
3 × (1/4) = 3/4
4 × (1/4) = 4/4 = 1 (whole)
Building up with 1/8:
2 × (1/8) = 2/8, 4 × (1/8) = 4/8
6 × (1/8) = 6/8, 8 × (1/8) = 8/8 = 1 (whole)
🍞 Roti Example
Collecting halves (1/2) of a roti:
| Collection | Fraction | Means |
|---|---|---|
| 1 times half | 1/2 | Half a roti |
| 2 times half | 2/2 = 1 | One whole roti |
| 3 times half | 3/2 | One and a half roti |
| 4 times half | 4/2 = 2 | Two whole rotis |
| 5 times half | 5/2 | Two and a half rotis |
Any fraction = (numerator) × (fractional unit).
5/8 = 5 times 1/8. 7/3 = 7 times 1/3.
7.4 Fractions on the Number Line
Every fraction has a point on the number line, just like whole numbers do. To mark fractions, we divide the space between two whole numbers into equal parts.
📐 How to Mark Fractions
- Identify the gap between two whole numbers (e.g., between 0 and 1).
- Divide this gap into equal parts equal to the denominator.
- Count numerator parts from 0 to find the fraction’s position.
Divide 0–1 into 5 equal parts. Each part = 1/5. Count 3 parts from 0 → that’s 3/5.
🔍 Key Observations
- Fractions with numerator < denominator lie between 0 and 1 (e.g., 3/5, 7/8)
- Fractions with numerator = denominator equal 1 (e.g., 5/5, 8/8)
- Fractions with numerator > denominator are greater than 1 (e.g., 7/5, 11/8)
- Infinitely many fractions lie between 0 and 1!
📍 Fractions greater than 1: numerator > denominator
7.5 Mixed Fractions
A mixed fraction (mixed number) has two parts: a whole number part and a fractional part (which is less than 1).
Example: 2 2/3 = 2 + 2/3 = “two and two-thirds”
🔄 Improper Fraction → Mixed Fraction
An improper fraction has numerator > denominator. To convert:
- Divide numerator by denominator.
- The quotient is the whole number part.
- The remainder is the new numerator.
- Keep the denominator the same.
8 ÷ 3 = 2 remainder 2
So 8/3 = 2 and 2/3 (written as 2²⁄₃)Convert 11/5 to mixed fraction:
11 ÷ 5 = 2 remainder 1
So 11/5 = 2 and 1/5 (written as 2¹⁄₅)
Convert 9/4 to mixed fraction:
9 ÷ 4 = 2 remainder 1
So 9/4 = 2 and 1/4 (written as 2¼)
🔄 Mixed Fraction → Improper Fraction
To convert a mixed fraction back to an improper fraction:
= (3 × 4 + 1) / 4 = (12 + 1) / 4 = 13/4
= (7 × 3 + 2) / 3 = (21 + 2) / 3 = 23/3
📊 Quick Reference Table
| Mixed Fraction | Improper Fraction | How? |
|---|---|---|
| 1 1/2 | 3/2 | (1×2+1)/2 |
| 2 1/3 | 7/3 | (2×3+1)/3 |
| 3 1/4 | 13/4 | (3×4+1)/4 |
| 4 1/2 | 9/2 | (4×2+1)/2 |
| 2 3/5 | 13/5 | (2×5+3)/5 |
7.6 Equivalent Fractions
Equivalent fractions are fractions that look different but represent the same value (the same point on the number line, the same share).
🍞 Equal Shares — the “why”
• 1 roti shared equally by 2 → each gets 1/2
• 2 rotis shared equally by 4 → each gets 2/4
• 3 rotis shared equally by 6 → each gets 3/6
Each child gets the same amount! So 1/2 = 2/4 = 3/6.
🔢 How to Find Equivalent Fractions
Multiply both numerator and denominator by the same number.
2/3 = (2×2)/(3×2) = 4/6
2/3 = (2×3)/(3×3) = 6/9
Divide both numerator and denominator by the same number.
6/9 = (6÷3)/(9÷3) = 2/3
8/12 = (8÷4)/(12÷4) = 2/3
If you multiply OR divide BOTH numerator and denominator by the SAME non-zero number, you get an equivalent fraction!
🧱 Fraction Wall — Key Equivalences
| Fraction | Some Equivalent Fractions |
|---|---|
| 1/2 | 2/4, 3/6, 4/8, 5/10, 6/12 |
| 1/3 | 2/6, 3/9, 4/12, 5/15 |
| 2/3 | 4/6, 6/9, 8/12, 10/15 |
| 3/4 | 6/8, 9/12, 12/16, 15/20 |
| 1/4 | 2/8, 3/12, 4/16, 5/20 |
Simplest Form / Lowest Terms
A fraction is in its simplest form (lowest terms) when the numerator and denominator have no common factor except 1.
📘 Step-by-Step Method
- Find the HCF (Highest Common Factor) of numerator and denominator.
- Divide both numerator and denominator by the HCF.
- Check: the new fraction should have no common factor except 1.
HCF of 16 and 20 = 4
16/20 = (16÷4)/(20÷4) = 4/5
4 and 5 have no common factor → 4/5 is the simplest form.
Step 1: Both even → divide by 2: 36/60 = 18/30
Step 2: Both even again → divide by 2: 18/30 = 9/15
Step 3: Both divisible by 3 → divide by 3: 9/15 = 3/5
3 and 5 have no common factor → 3/5 is the simplest form.
7.7 Comparing Fractions
🔷 Case 1: Same Denominator
When denominators are the same, simply compare the numerators. Larger numerator = larger fraction.
Same denominator (7). Numerator 5 > 3, so 5/7 > 3/7.
🔷 Case 2: Same Numerator
When numerators are the same, larger denominator = smaller fraction (more people sharing = less per person).
Same numerator (4). Denominator 7 < 8, so 4/7 > 4/8.
🔷 Case 3: Different Denominators — Use Equivalent Fractions
Convert both fractions to have the same denominator, then compare numerators.
- Find a common multiple of both denominators (LCM or product of denominators).
- Convert both fractions to equivalent fractions with this common denominator.
- Compare the numerators of the equivalent fractions.
Common denominator = 5 × 9 = 45
4/5 = 36/45 and 7/9 = 35/45
Since 36 > 35, 4/5 > 7/9 ✓
Common denominator = 20
3/4 = 15/20 and 7/10 = 14/20
Since 15 > 14, 3/4 > 7/10 ✓
You can only directly compare numerators when the denominators are the same (same fractional unit). Never compare numerators of fractions with different denominators directly!
7.8 Addition of Fractions — Brahmagupta’s Method
➕ Case 1: Same Denominator (Easy!)
When denominators are the same, just add the numerators and keep the denominator.
4/7 + 6/7 = 10/7 = 1 + 3/7 = 1 3/7
➕ Case 2: Different Denominators — Brahmagupta’s Method
This method was first described by the Indian mathematician Brahmagupta in 628 CE! It is still what we use today.
Step 1: Find equivalent fractions with the same denominator (use LCM or product of denominators)
Step 2: Add the numerators, keep the same denominator
Step 3: Simplify to lowest terms if needed
LCM of 4 and 3 = 12
1/4 = 3/12 and 1/3 = 4/12
3/12 + 4/12 = 7/12 → Answer: 7/12
LCM of 3 and 5 = 15
2/3 = 10/15 and 1/5 = 3/15
10/15 + 3/15 = 13/15 → Answer: 13/15
LCM of 3 and 4 = 12
2/3 = 8/12 and 3/4 = 9/12
8/12 + 9/12 = 17/12 = 1 5/12 → Answer: 1 5/12 litres
Subtraction of Fractions — Brahmagupta’s Method
➖ Case 1: Same Denominator
Subtract the numerators, keep the denominator the same.
5/8 − 3/8 = 2/8 = 1/4 (simplified)
➖ Case 2: Different Denominators — Brahmagupta’s Method
Step 1: Find equivalent fractions with the same denominator
Step 2: Subtract the numerators, keep the same denominator
Step 3: Simplify to lowest terms if needed
LCM of 4 and 3 = 12
3/4 = 9/12 and 2/3 = 8/12
9/12 − 8/12 = 1/12 → Answer: 1/12
Walk = 7/10 − 1/2
LCM of 10 and 2 = 10
7/10 − 5/10 = 2/10 = 1/5 → She walks 1/5 km
Compare: 10/3 vs 13/4
10/3 = 40/12 and 13/4 = 39/12
40/12 > 39/12 → Namit takes less time.
Difference = 40/12 − 39/12 = 1/12 min → Namit is faster by 1/12 minute
7.9 A Pinch of History — Fractions in India
The way we write fractions today (numerator over denominator) originated in India. The Bakshali manuscript (c. 300 CE) used almost the same notation we use today!
📜 Timeline of Fractions in India
| Year (CE) | Mathematician | Contribution |
|---|---|---|
| ~300 CE | Bakshali Manuscript | Early fraction notation |
| 499 CE | Aryabhata | Used fraction notation |
| 628 CE | Brahmagupta | Formalized rules for +, −, ×, ÷ of fractions |
| ~750 CE | Sridharacharya | Extended fraction methods |
| ~850 CE | Mahaviracharya | Further developments |
| 12th century | Al-Hassar (Morocco) | Added the fraction bar (÷ line) |
| 17th century | Europe | Adopted Indian fraction methods via Arabs |
In Sanskrit, a fraction was called bhinna (meaning “broken”) or bhaga / ansha (meaning “part” or “piece”). These beautiful ancient words show how fractions were thought of as broken pieces of a whole.
Ancient Egyptians only used unit fractions (1/n). They wrote all fractions as sums of different unit fractions.
Example: 19/24 = 1/2 + 1/6 + 1/8
This “Egyptian fraction” representation leads to beautiful mathematical puzzles!
Quick Summary — All Key Concepts
p/q where p = numerator, q = denominator (q ≠ 0).
Represents parts of a whole.
1/n — one whole divided into n equal parts.
Bigger n = smaller fraction.
Whole part + fractional part.
Example: 2⅓ = 2 + 1/3 = 7/3
Same value, different form.
Multiply/divide both by same number.
HCF of numerator & denominator = 1.
Divide both by their HCF.
p/q < 1 if p < q.
p/q > 1 if p > q.
p/q = 1 if p = q.
Same denom: add numerators.
Diff denom: make same denom first.
Same denom: subtract numerators.
Diff denom: make same denom first.
Same denom → compare numerators.
Diff denom → convert first, then compare.
Indian mathematician who first formally described rules for all fraction operations.
Important Exam Questions with Solutions
🔷 Fraction Basics
🔷 Equivalent Fractions & Simplest Form
🔷 Comparing Fractions
4/9 = 28/63 and 3/7 = 27/63
28 > 27, so 4/9 > 3/7.
7/10 = 21/30, 11/15 = 22/30, 2/5 = 12/30
Ascending: 2/5 < 7/10 < 11/15
🔷 Addition and Subtraction
3/4 = 9/12 and 1/3 = 4/12
9/12 + 4/12 = 13/12 = 1 1/12
5/6 = 15/18 and 4/9 = 8/18
15/18 − 8/18 = 7/18 → Answer: 7/18
2/5 = 8/20 and 3/4 = 15/20
Total = 8/20 + 15/20 = 23/20 = 1 3/20 m.
Since 1 3/20 > 1 m, yes, the lace is sufficient.
Very best