Fractions — Class 6 Mathematics

'A fraction is a part of a whole — like sharing a biryani among friends.'

1. Introduction

A fraction represents a part of a whole or a part of a group. When you eat half an apple, share one-fourth of a pizza, or score 3 out of 5 in a test — you are using fractions.

Parts of a Fraction

  3  ← Numerator (how many parts you have)
  ─
  5  ← Denominator (total equal parts)

• The denominator can NEVER be zero
• The numerator CAN be greater than, equal to, or less than the denominator

Why Fractions Matter

• Cooking: 'Use 1/2 cup of rice'
• Shopping: '50% off' = 1/2 off
• Time: 'Quarter past 3' = 3:15
• AP context: Sharing pulihora among family members

2. Types of Fractions

Proper Fraction

Numerator < Denominator. Value is ALWAYS less than 1.

• 1/2, 3/4, 5/8, 9/10
• Pictorially: part of a whole (shaded portion of a shape)

Improper Fraction

Numerator ≥ Denominator. Value is 1 or more.

• 5/3, 7/4, 9/8, 12/5
• 'The numerator is larger — the fraction is worth more than 1 whole'

Mixed Fraction

A whole number + a proper fraction.

• 1½ (one and a half) = 1 + 1/2
• 2¾ (two and three-quarters) = 2 + 3/4
• 3⅔ (three and two-thirds) = 3 + 2/3

Conversion: Improper → Mixed

Divide numerator by denominator. Quotient = whole, Remainder = numerator, Denominator stays.

Example: Convert 11/4 to mixed fraction.

• 11 ÷ 4 = 2 remainder 3
• 11/4 = 2¾

Conversion: Mixed → Improper

Multiply whole by denominator, add numerator, keep same denominator.

Example: Convert 3⅔ to improper fraction.

• 3 × 3 + 2 = 11
• 3⅔ = 11/3

Conversion Examples

3. Fractions on the Number Line

Representing fractions on a number line divides each unit into equal parts.

Example: Represent 3/4 on a number line.

0----|----|----|----1
         3/4

Divide the segment from 0 to 1 into 4 equal parts. Count 3 parts from 0.

Example: Represent 1⅓ (4/3).

0----|----1----|----2
     4/3

Divide each unit into 3 parts. Count 4 parts from 0.

'Between any two fractions, there are infinitely many fractions. The number line is never empty.'

4. Equivalent Fractions

Equivalent fractions are fractions that represent the same value but look different.

How to Find Equivalent Fractions

Multiply (or divide) BOTH numerator AND denominator by the SAME number.

Example: Find 3 equivalent fractions of 2/3.

• 2/3 = (2×2)/(3×2) = 4/6
• 2/3 = (2×3)/(3×3) = 6/9
• 2/3 = (2×4)/(3×4) = 8/12

Checking Equivalence

Cross-multiply. If the products are equal, the fractions are equivalent.

Example: Are 3/5 and 9/15 equivalent?

• Cross multiply: 3 × 15 = 45, 5 × 9 = 45
• 45 = 45 → YES, they are equivalent.

Table of Equivalent Fractions of 1/2

5. Simplest Form (Lowest Terms)

A fraction is in its simplest form when the numerator and denominator have no common factor other than 1.

Method 1: Divide by HCF

1. Find HCF of numerator and denominator
2. Divide both by the HCF

Example: Simplify 12/18.

• HCF of 12 and 18 = 6
• 12 ÷ 6 = 2, 18 ÷ 6 = 3
• 12/18 = 2/3 (simplest form)

Method 2: Repeated Division

Divide numerator and denominator by common factors until no common factor remains.

Example: Simplify 24/36.

• 24/36 → divide by 2: 12/18 → divide by 2: 6/9 → divide by 3: 2/3
• 24/36 = 2/3

Practice

6. Like and Unlike Fractions

Like Fractions

Fractions with the same denominator.

• 2/7, 3/7, 5/7, 6/7 — all like fractions (denominator 7)

Unlike Fractions

Fractions with different denominators.

• 1/2, 2/3, 5/8, 3/4 — all unlike fractions

Converting Unlike to Like

Find LCM of denominators. Convert each fraction to equivalent fraction with LCM as denominator.

Example: Convert 2/3 and 3/5 to like fractions.

• LCM of 3 and 5 = 15
• 2/3 = (2×5)/(3×5) = 10/15
• 3/5 = (3×3)/(5×3) = 9/15
• Now 10/15 and 9/15 are like fractions

7. Comparison of Fractions

Like Fractions

Compare numerators. Larger numerator = larger fraction.

Example: 5/8 > 3/8 (5 > 3, denominators same)

Unlike Fractions

First convert to like fractions, then compare.

Example: Which is larger: 3/4 or 4/5?

• LCM of 4 and 5 = 20
• 3/4 = 15/20, 4/5 = 16/20
• 16/20 > 15/20, so 4/5 > 3/4

Comparing Mixed Fractions

Compare the whole number part first. If whole parts are equal, compare the fraction parts.

Example: 2⅓ vs 3¼ → 3¼ is larger (whole part 3 > 2) Example: 2½ vs 2⅓ → whole parts equal, compare 1/2 vs 1/3 → 1/2 > 1/3 → 2½ > 2⅓

Quick Comparison Tricks

• Same numerator: smaller denominator = larger fraction (1/2 > 1/3 > 1/4)
• Same denominator: larger numerator = larger fraction
• Comparing with half: if numerator > denominator/2, fraction > 1/2

8. Addition and Subtraction of Fractions

Adding Like Fractions

Add numerators, keep denominator same.

Example: 2/7 + 3/7 = (2+3)/7 = 5/7

Adding Unlike Fractions

1. Find LCM of denominators
2. Convert to like fractions
3. Add numerators

Example: 2/3 + 1/4

• LCM of 3 and 4 = 12
• 2/3 = 8/12, 1/4 = 3/12
• 8/12 + 3/12 = 11/12

Adding Mixed Fractions

Method 1: Add whole parts separately, add fraction parts separately. Method 2: Convert to improper fractions, add, convert back.

Example: 1½ + 2⅓ Method 1: 1+2 = 3, 1/2+1/3 = 3/6+2/6 = 5/6 → 3⅚ Method 2: 3/2 + 7/3 = 9/6 + 14/6 = 23/6 = 3⅚

Subtracting Like Fractions

Subtract numerators, keep denominator same.

Example: 7/9 − 4/9 = 3/9 = 1/3

Subtracting Unlike Fractions

Same as addition but subtract numerators after converting.

Example: 5/6 − 3/8

• LCM of 6 and 8 = 24
• 5/6 = 20/24, 3/8 = 9/24
• 20/24 − 9/24 = 11/24

Worked Example with Mixed Subtraction

Example: 3¼ − 1⅔ Method: 13/4 − 5/3 = 39/12 − 20/12 = 19/12 = 1⁷⁄₁₂

9. Word Problems

Problem 1 (Sharing Food)

Ravi ate 2/5 of a pizza and his sister ate 1/3. How much did they eat together?

Solution: 2/5 + 1/3 = 6/15 + 5/15 = 11/15 Answer: 11/15 of the pizza

Problem 2 (Remaining Work)

Sita completed 3/8 of her homework. How much is left?

Solution: Total = 1 (whole) Completed = 3/8 Remaining = 1 − 3/8 = 8/8 − 3/8 = 5/8 Answer: 5/8

Problem 3 (AP Context — Millets)

A farmer in Anantapur grows ragi on 2/5 of his land, jowar on 1/3, and vegetables on the rest. What fraction is for vegetables?

Solution: Ragi + Jowar = 2/5 + 1/3 = 6/15 + 5/15 = 11/15 Vegetables = 1 − 11/15 = 4/15 Answer: 4/15

10. Common Mistakes — Fix Them Now

11. Exam Focus

Marks Blueprint

Quick Self-Test (5 Questions)

Q1: Which is larger: 3/8 or 2/5?

Q2: Convert 23/6 to a mixed fraction.

Q3: Simplify 24/36 to its simplest form.

Q4: Add: 2¾ + 1⅓

Q5: Subtract: 5/6 − 1/4

12. Chapter Summary

• Fraction: numerator / denominator (denominator ≠ 0)
• Types: proper (<1), improper (≥1), mixed (whole + proper)
• Conversion: Improper→Mixed (divide), Mixed→Improper (whole×den+num)
• Equivalent fractions: multiply/divide numerator AND denominator by same number
• Simplest form: divide by HCF of num and den
• Like fractions: same denominator; Unlike: different
• Comparison: like → compare num; unlike → convert first
• Addition/Subtraction: like → operate on num; unlike → LCM first

'Fractions are everywhere — from recipes to exam scores — and mastering them makes life much easier.'