Class 6 Fractions
Understand fractions as equal shares, fractional units, parts of a whole, and number line points. Master mixed numbers, the interactive fraction wall, simplest forms, common denominator comparisons, Brahmagupta’s universal addition & subtraction method (628 CE), and ancient Indian mathematical history.
What are Fractions in Class 6 Mathematics?
In NCERT Ganita Prakash Grade 6 Chapter 7, a fraction represents the exact equal share when whole units are divided equally. Written as a/b, a is the numerator (count of fractional units) and b is the denominator (number of equal parts making 1 whole).
Fractions with numerator 1 (1/2, 1/3, 1/4...). Note: 1/5 > 1/9 because sharing among fewer people gives a larger share.
Combines a whole part & fractional part (<1). e.g., 8/3 = 2 + 2/3 = 2 2/3 ('two and two thirds').
Denoting the same quantity: 1/2 = 2/4 = 3/6 = 4/8 = 5/10. Simplest form has HCF(numerator, denominator) = 1.
To add/subtract fractions, convert to a common denominator (LCM), combine numerators, and reduce to lowest terms.
Interactive Chapter Roadmap & Table of Contents
Fractional Units and Equal Shares
The Roti Sharing Discovery (Shabnam & Mukta)
When 1 roti is shared equally between 2 children, each child receives ½ roti ('one half' or 'one upon two'). When shared equally among 4 children, each receives ¼ roti.
Why is ½ > ¼?
In the second group, more children share the same single roti, so each child receives a smaller individual portion!
½ > ¼ and ⅕ > ⅑ and ¹⁄₁₀₀ > ¹⁄₂₀₀
What is a Fractional Unit?
When 1 whole unit is divided into several equal parts, each single part is called a fractional unit (or unit fraction):
½, ⅓, ¼, ⅕, ⅙, ..., ¹⁄₁₀, ..., ¹⁄₅₀, ..., ¹⁄₁₀₀
Knowledge from the Past! Vedic & Traditional Indian Fraction Names
In the Rig Veda, the fraction ¾ is referred to as tri-pada. This ancient root directly connects to words used across Indian languages today: teen paav in Hindi, mukkaal in Tamil.
Parts of a Whole, Paper Folding & Number Line
Area Invariance
Cutting a whole chikki into 6 equal parts in different ways yields pieces of different shapes, but each piece has the exact same area (⅙ chikki).
Collecting Fractional Units
Folding a 1-unit strip: 1 fold = 2 parts of ½; 2 folds = 4 parts of ¼ (2 times ¼ = ²⁄₄, 3 times ¼ = ¾, 4 times ¼ = 1).
Number Line Density
Fractions are actual numbers with exact geometric points on the number line. There are an infinite (uncountable) number of fractions between 0 and 1!
Mixed Numbers and Fractions Greater Than One
Proper Fractions vs Improper Fractions
- Proper Fraction (< 1 unit): Numerator is strictly smaller than the denominator (e.g., ½, ⅗, ⁷⁄₁₀).
- Improper Fraction (> 1 unit): Numerator is larger than the denominator (e.g., ³⁄₂, ⁸⁄₃, ⁴⁷⁄₉).
- Whole Number Multiples: ⁸⁄₄ = 2 is a pure whole number (not a mixed number).
Converting Between Forms (NCERT Verified Examples)
Equivalent Fractions and the Interactive Fraction Wall
Interactive Fraction Wall (1 to 1/10)
Interactive Fraction Simplifier & Lowest Terms Explorer
Enter any numerator and denominator to reduce to simplest form by dividing by HCF!
Comparing, Adding & Subtracting Fractions
Brahmagupta Arithmetic Simulator
Step-by-step common denominator reduction and exact calculation!
Step 1: Common Denominator (LCM): LCM(3, 4) = 12.
Equivalent Fractions: 2/3 = 8/12, and 3/4 = 9/12.
Step 2: Combine Numerators: (8 + 9) / 12 = 17/12.
Step 3: Lowest Terms: 17/12 (1 5/12).
A Pinch of History & Egyptian Unit Fraction Puzzles
Origins of Global Fraction Notation
In Sanskrit, fractions were called bhinna ('broken'), bhaga, or ansha ('part').
• Bakshali Manuscript (c. 300 CE): Wrote fractions vertically with numerator over denominator without a bar.
• Brahmagupta (628 CE): Codified complete rules of fraction operations in Brahmasphuṭasiddhānta (Verse 12.2).
• Al-Hassar (12th century): Moroccan mathematician who added the horizontal fraction bar.
The Egyptian Unit Fraction Sum Puzzle
Can distinct unit fractions add up to exactly 1?
½ + ⅓ + ⅙ = 11) ½ + ⅓ + ¹⁄₇ + ¹⁄₄₂ = 1
2) ½ + ⅓ + ⅛ + ¹⁄₂₄ = 1
3) ½ + ⅓ + ⅑ + ¹⁄₁₈ = 1
4) ½ + ⅓ + ¹⁄₁₀ + ¹⁄₁₅ = 1
5) ½ + ¼ + ⅕ + ¹⁄₂₀ = 1
6) ½ + ¼ + ⅙ + ¹⁄₁₂ = 1
12 Comprehensive Worked Examples
Example 1: Unit Fraction Sharing
What we know: 1 kg rice packed equally into 4 packets
Method / Formula: Share = Total Quantity ÷ Number of Packets
Step 1: 1 kg ÷ 4 = 1/4 kg per packet
Step 2: Each packet weighs exactly a quarter kilogram
Answer: Weight of each packet = 1/4 kg
Example 2: Multiple Item Equal Sharing
What we know: 4 friends share 3 glasses of juice equally
Method / Formula: Share = 3 ÷ 4
Step 1: Divide each of the 3 glasses into 4 quarters
Step 2: Each friend receives 3 quarters = 3/4 glass
Answer: Each friend drinks 3/4 glass of sugarcane juice
Example 3: Adding Unit Weights
What we know: Big fish = 1/2 kg, Small fish = 1/4 kg
Method / Formula: Total = 1/2 + 1/4
Step 1: Convert 1/2 to 2/4
Step 2: 2/4 + 1/4 = 3/4 kg
Answer: Together they weigh 3/4 kg
Example 4: Converting to Mixed Fraction
What we know: Improper fraction 47/9
Method / Formula: Numerator ÷ Denominator = Quotient + (Remainder/Denominator)
Step 1: 47 ÷ 9 = 5 with remainder 2
Step 2: Whole part = 5, Fractional part = 2/9
Answer: 47/9 = 5 2/9
Example 5: Converting Mixed Number to Improper
What we know: Mixed number 7 2/3
Method / Formula: (Whole × Denominator + Numerator) / Denominator
Step 1: 7 × 3 + 2 = 21 + 2 = 23
Step 2: Place over denominator 3: 23/3
Answer: 7 2/3 = 23/3
Example 6: Finding Lowest Terms using HCF
What we know: Fraction 64/144
Method / Formula: Divide both by HCF(64, 144) = 16
Step 1: 64 ÷ 16 = 4
Step 2: 144 ÷ 16 = 9
Answer: 64/144 in lowest terms = 4/9
Example 7: Stepwise Reduction to Lowest Terms
What we know: Fraction 36/60
Method / Formula: Divide by common factors 2, 2, 3
Step 1: 36/60 ÷ 2 = 18/30; 18/30 ÷ 2 = 9/15
Step 2: 9/15 ÷ 3 = 3/5
Answer: 36/60 in lowest terms = 3/5
Example 8: Comparing Fractions with Different Denominators
What we know: Compare 4/5 and 7/9
Method / Formula: Convert to common denominator LCM(5, 9) = 45
Step 1: 4/5 = (4×9)/45 = 36/45
Step 2: 7/9 = (7×5)/45 = 35/45. Since 36/45 > 35/45
Answer: 4/5 > 7/9
Example 9: Brahmagupta Addition (Different Denominators)
What we know: Add 2/3 + 1/5
Method / Formula: LCM(3, 5) = 15; convert to 15ths
Step 1: 2/3 = 10/15, and 1/5 = 3/15
Step 2: 10/15 + 3/15 = 13/15
Answer: 2/3 + 1/5 = 13/15
Example 10: Brahmagupta Subtraction (Different Denominators)
What we know: Subtract 2/3 from 3/4 (3/4 − 2/3)
Method / Formula: LCM(4, 3) = 12; convert to 12ths
Step 1: 3/4 = 9/12, and 2/3 = 8/12
Step 2: 9/12 − 8/12 = 1/12
Answer: 3/4 − 2/3 = 1/12
Example 11: Real-Life Paint Mixture Word Problem
What we know: 2/3 L yellow paint + 3/4 L blue paint
Method / Formula: Total Volume = 2/3 + 3/4
Step 1: 2/3 = 8/12 L, 3/4 = 9/12 L
Step 2: 8/12 + 9/12 = 17/12 L = 1 5/12 L
Answer: Rahim made 1 5/12 litres of green paint
Example 12: School Walking Distance Word Problem
What we know: Total distance = 7/10 km, auto ride = 1/2 km
Method / Formula: Walking distance = 7/10 − 1/2
Step 1: 1/2 = 5/10 km
Step 2: 7/10 − 5/10 = 2/10 km = 1/5 km
Answer: Jaya walks 1/5 km daily
20 Core Practice Questions
Three guavas together weigh 1 kg. If they are of equal size, what does each guava weigh?
Which of the following unit fractions represents the largest share?
Four friends share 3 glasses of sugarcane juice equally. How much juice does each friend drink?
In the fraction 5/6, what is the number 5 called?
A big fish weighs 1/2 kg and a small fish weighs 1/4 kg. What is their total combined weight?
Convert the improper fraction 47/9 into a mixed fraction.
Convert the mixed fraction 3 1/4 into an improper fraction.
How many pieces of length 1/6 will make a total length of 1/2?
What is the simplest form (lowest terms) of the fraction 36/60?
What is the simplest form of 17/51?
Which fraction is strictly greater: 4/5 or 7/9?
Which of the following represents the fractions 7/10, 11/15, and 2/5 in ascending order?
Find the sum of 1/4 + 1/3 using Brahmagupta's method.
Find the sum: 2/3 + 5/6.
Calculate the difference: 3/4 − 2/3.
Subtract 13/4 from 10/3.
Rahim mixes 2/3 L of yellow paint with 3/4 L of blue paint. What is the total volume of green paint?
Jaya walks 7/10 km to school. She takes an auto for 1/2 km and walks the rest. How far does she walk?
Which Indian treatise first formally codified general rules for fraction arithmetic in 628 CE?
What is the ONLY combination of 3 distinct unit fractions that adds up to 1?
10 Challenge & Figure It Out Questions
The Uniqueness Proof for Three Unit Fractions (Section 7.9)
Prove why 1/2 + 1/3 + 1/6 = 1 is the ONLY possible solution where three distinct unit fractions sum to 1.
Egyptian Fractions: Four Unit Fractions Summing to 1 (Section 7.9)
Find all 6 distinct combinations of four unit fractions that sum to 1.
Chikki Sharing Invariant: Equal Area across Irregular Shapes (Section 7.2)
A rectangular chikki is cut into 6 equal parts in two different ways: one by 5 parallel horizontal cuts, and another by a grid of 2 rows and 3 columns. Explain mathematically why pieces of completely different geometric shapes have the exact same fractional value 1/6.
Geeta and Shamim Tablecloth Border Challenge (Section 7.8)
Geeta bought 2/5 metre of lace and Shamim bought 3/4 metre of the same lace to put a complete border on a tablecloth with perimeter 1 metre. Show the calculation to verify whether their combined lace is sufficient and by how much.
Jeevika vs Namit Park Lap Race (Section 7.8)
Jeevika takes 10/3 minutes to complete a round of the park, while Namit takes 13/4 minutes. Who is faster, and by what fraction of a minute?
Three-Fraction LCM Addition Challenge (Section 7.8)
Evaluate the three-term fraction sum: 2/3 + 4/5 + 3/7 using Brahmagupta's method. Show all steps and simplify.
Stepwise vs HCF Fraction Reduction (Section 7.6)
Show how to reduce 525/112 to lowest terms using: (a) Stepwise division by prime factors, and (b) Finding the Highest Common Factor (HCF).
Multi-Term Rational Subtraction (Section 7.8)
Subtract 18/5 from 23/3. Show the complete common denominator derivation.
Infinite Fractions Density between 0 and 1 (Section 7.4)
Prove that between any two distinct fractions a/b and c/d on the number line, there is always another fraction.
Sanskrit Fraction Terminology and Linguistic Roots (Section 7.9)
Explain the mathematical and historical significance of the terms 'bhinna', 'tri-pada', 'teen paav', and 'mukkaal'.
Common Mistakes & How to Avoid Them
Mistake: Saying ⅑ > ⅕ because 9 > 5.
Correction: Dividing among more people gives smaller shares, so ⅕ > ⅑.
Mistake: ⅖ + ⅕ = ³⁄₁₀.
Correction: The fractional unit remains ⅕; only add numerators: ⅖ + ⅕ = ⅗.
Mistake: ¼ + ⅓ = ²⁄₇.
Correction: Must first find a common denominator: ³⁄₁₂ + ⁴⁄₁₂ = ⁷⁄₁₂.
Fractions – Key Takeaways
Summary of rules, definitions, and theorems from Ganita Prakash Chapter 7.
1. Fractional Units
A fractional unit is 1/d. Every fraction a/b is a collection of 'a times 1/b'.
2. Mixed Numbers
Fractions > 1 can be decomposed into whole units and a proper fractional remainder.
3. Equivalent Fractions
Multiply or divide numerator and denominator by the same non-zero number to maintain identical value.
4. Brahmagupta's Addition
Reduce to common denominator, add/subtract numerators, and reduce to lowest terms.
5. Indian Heritage
Fractions originated in Vedic texts (*tri-pada*) and were formalized by Brahmagupta (628 CE).
6. Egyptian Unit Sums
½ + ⅓ + ⅙ = 1 is the unique 3-unit fraction solution for 1.
16 FAQs on Class 6 Fractions
Computational Thinking with Rational Numbers & Precision Arithmetic
In computer science, fractional values drive floating-point binary representation, rational number data structures, proportional screen scaling, RGB colour mixing ratios, and probability calculations in artificial intelligence.