Fractions
1. What Is a Fraction?
A fraction represents a PART of a WHOLE.
Fraction = Numerator / Denominator
'The DENOMINATOR tells you how many EQUAL parts the whole is divided into. The NUMERATOR tells you how many parts you HAVE.'
Real-Life Examples
- Half a pizza: 1/2
- Three-quarters of a cake: 3/4
- Two-thirds of a glass of water: 2/3
2. Types of Fractions
| Type | Definition | Numerator vs Denominator | Examples |
|---|---|---|---|
| Proper | The numerator is LESS than the denominator | Num < Den | 2/5, 7/9, 3/8 |
| Improper | The numerator is GREATER than or equal to the denominator | Num ≥ Den | 7/4, 11/8, 5/3 |
| Mixed | A whole number + a proper fraction | Whole + proper | 2 1/3, 4 5/8, 1 3/4 |
| Unit | The numerator is 1 | Num = 1 | 1/2, 1/5, 1/10 |
| Like | Same denominator | Same denominator | 2/7, 4/7, 6/7 |
| Unlike | Different denominators | Different denominators | 2/5, 3/7, 1/4 |
'An improper fraction is ALWAYS greater than or equal to 1. A proper fraction is ALWAYS less than 1.'
Converting Mixed to Improper
2 1/3 = (2 × 3 + 1) / 3 = 7/3
'Multiply the whole number by the denominator. Add the numerator. Keep the same denominator.'
Converting Improper to Mixed
11/4 = ?
11 ÷ 4 = 2 remainder 3
11/4 = 2 3/4
3. Equivalent Fractions
Two fractions are EQUIVALENT if they represent the SAME part of a whole.
1/2 = 2/4 = 3/6 = 4/8 = 5/10
'To find an equivalent fraction, MULTIPLY or DIVIDE both numerator and denominator by the SAME number.'
| Original | Multiply by 2 | Multiply by 3 | Multiply by 4 |
|---|---|---|---|
| 2/3 | 4/6 | 6/9 | 8/12 |
| 3/5 | 6/10 | 9/15 | 12/20 |
| 5/8 | 10/16 | 15/24 | 20/32 |
Checking Equivalence
Cross-multiply: a/b = c/d if a × d = b × c
Check if 3/5 = 6/10: 3 × 10 = 30. 5 × 6 = 30. 30 = 30 → YES, they are equivalent.
4. Simplest Form (Lowest Terms)
A fraction is in SIMPLEST FORM when the numerator and denominator have HCF = 1.
Simplify 12/18:
Step 1: Find HCF of 12 and 18. HCF = 6. Step 2: Divide both numerator and denominator by 6. 12 ÷ 6 = 2, 18 ÷ 6 = 3. 12/18 = 2/3 (simplest form)
'To simplify, always divide by the HCF. Dividing in steps works too — but using the HCF gets you to the simplest form in ONE step.'
5. Comparing Fractions
Same Denominator
Compare numerators. Larger numerator = larger fraction.
3/7 < 5/7
Same Numerator
Compare denominators. Smaller denominator = larger fraction.
3/5 > 3/8 ('The whole is divided into FEWER parts, so each part is BIGGER.')
Different Numerators and Denominators
Find the LCM of denominators. Convert to like fractions. Compare numerators.
Compare 2/3 and 3/5:
Step 1: LCM of 3 and 5 = 15. Step 2: 2/3 = 10/15, 3/5 = 9/15. Step 3: 10/15 > 9/15, so 2/3 > 3/5.
| Fractions | LCM | Converted | Comparison |
|---|---|---|---|
| 3/4 and 5/6 | 12 | 9/12 and 10/12 | 3/4 < 5/6 |
| 2/5 and 3/10 | 10 | 4/10 and 3/10 | 2/5 > 3/10 |
| 7/8 and 5/6 | 24 | 21/24 and 20/24 | 7/8 > 5/6 |
6. Addition of Fractions
Like Fractions (Same Denominator)
Add the numerators. Keep the denominator the same.
3/7 + 2/7 = 5/7
Unlike Fractions (Different Denominators)
Find LCM of denominators. Convert. Add numerators.
3/8 + 1/6
Step 1: LCM of 8 and 6 = 24. Step 2: 3/8 = 9/24, 1/6 = 4/24. Step 3: 9/24 + 4/24 = 13/24.
'You CANNOT add fractions with different denominators directly. You MUST convert them to LIKE fractions first.'
Mixed Numbers
2 1/3 + 1 2/5
Method 1: Convert to improper: 7/3 + 7/5 = 35/15 + 21/15 = 56/15 = 3 11/15. Method 2: Add wholes, then fractions: 2 + 1 = 3, 1/3 + 2/5 = 5/15 + 6/15 = 11/15. Total = 3 11/15.
7. Subtraction of Fractions
Like Fractions
7/10 − 3/10 = 4/10 = 2/5 (simplified)
Unlike Fractions
5/6 − 3/8
Step 1: LCM of 6 and 8 = 24. Step 2: 5/6 = 20/24, 3/8 = 9/24. Step 3: 20/24 − 9/24 = 11/24.
From a Whole Number
3 − 1/4 = 12/4 − 1/4 = 11/4 = 2 3/4
'Write the whole number as a fraction with the same denominator as the fraction you are subtracting.'
8. Multiplication of Fractions
Fraction × Whole Number
3/7 × 4 = (3 × 4) / 7 = 12/7 = 1 5/7
'Multiply the numerator by the whole number. Keep the denominator the same.'
Fraction × Fraction
2/3 × 4/5 = (2 × 4) / (3 × 5) = 8/15
'Multiply the numerators. Multiply the denominators. Simplify if needed. You can CANCEL common factors BEFORE multiplying — it makes the numbers smaller.'
Key Facts to Remember
- Every whole number can be written as a fraction with denominator 1.
- 'The value of a fraction does NOT change when you multiply or divide both numerator and denominator by the same non-zero number.'
- A fraction with numerator 0 equals 0 (0/5 = 0).
- A fraction with denominator 1 equals the numerator (7/1 = 7).
- A fraction where numerator = denominator equals 1 (8/8 = 1).
Common Mistakes
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| Adding denominators | 1/3 + 1/4 = 2/7 is WRONG | LCM = 12. 4/12 + 3/12 = 7/12 |
| Forgetting to simplify | 4/8 is correct but not simplest | Divide by HCF: 4/8 = 1/2 |
| Comparing by cross-multiplying the wrong way | Wrong order gives wrong comparison | Multiply first numerator × second denominator |
| Treating mixed number as separate when subtracting | 3 1/4 − 1 3/4 — some subtract wholes and fractions separately without borrowing | Borrow from the whole, or convert to improper |
Exam Focus (ICSE Class 5)
| Topic | Marks (Typical) | Question Type |
|---|---|---|
| Types of fractions and conversion | 2-3 marks | Convert between mixed and improper |
| Equivalent fractions and simplest form | 2-3 marks | Find equivalent / reduce to simplest |
| Comparing fractions | 3 marks | Arrange in ascending/descending order |
| Addition and subtraction | 4-5 marks | Compute and simplify |
| Multiplication | 3 marks | Word problems involving fractions |
Self-Test: 5 Questions
Q1. Convert 17/6 to a mixed number and 4 2/5 to an improper fraction.
Q2. Find three equivalent fractions for 2/7.
Q3. Arrange in ascending order: 2/3, 5/6, 3/4, 7/12.
Q4. Simplify: 5 1/3 + 2 3/4 − 1 1/2.
Q5. Riya ate 2/5 of a cake and her brother ate 1/3 of it. How much cake did they eat together? How much is left?
Answers
A1. 17/6 = 2 5/6. 4 2/5 = 22/5.
A2. 2/7 = 4/14 = 6/21 = 8/28 (multiply numerator and denominator by 2, 3, and 4).
A3. LCM of 3, 6, 4, 12 = 12. 2/3 = 8/12, 5/6 = 10/12, 3/4 = 9/12, 7/12 = 7/12. Order: 7/12 < 2/3 < 3/4 < 5/6.
A4. 5 1/3 = 16/3, 2 3/4 = 11/4, 1 1/2 = 3/2. LCM of 3, 4, 2 = 12. 16/3 = 64/12, 11/4 = 33/12, 3/2 = 18/12. 64/12 + 33/12 − 18/12 = 79/12 = 6 7/12.
A5. 2/5 + 1/3 = 6/15 + 5/15 = 11/15. Left = 1 − 11/15 = 4/15.
