Fractions — Types, Equivalence, Addition and Subtraction
1. What Is a Fraction?
A FRACTION represents a PART of a WHOLE or a PART of a COLLECTION.
'A fraction has two parts — a NUMERATOR and a DENOMINATOR.'
| Term | Meaning | Example |
|---|---|---|
| Numerator | How many parts we have | 3 (in 3/4) |
| Denominator | How many equal parts the whole is divided into | 4 (in 3/4) |
| Fraction bar | The line between numerator and denominator | / or — |
'Read 3/4 as THREE-FOURTHS. The denominator tells you HOW MANY parts. The numerator tells you HOW MANY you have.'
2. Types of Fractions
Proper Fractions:
The numerator is SMALLER than the denominator. Examples: 1/2, 3/4, 2/5, 7/8
'Proper fractions are ALWAYS less than 1.'
Improper Fractions:
The numerator is EQUAL to or GREATER than the denominator. Examples: 5/4, 7/3, 8/8, 12/5
'Improper fractions are EQUAL to or GREATER than 1.'
Mixed Numbers:
A WHOLE number + a PROPER fraction. Examples: 1½, 2¾, 3⅓
'Mixed numbers are another way to write improper fractions. 5/4 = 1¼.'
| Type | Example | Value |
|---|---|---|
| Proper | 3/5 | Less than 1 |
| Improper | 7/5 | More than 1 |
| Mixed | 1⅖ | Same as 7/5 |
| Like fractions | 2/7, 3/7, 5/7 | Same denominator |
| Unlike fractions | 2/3, 3/4, 5/6 | Different denominators |
3. Equivalent Fractions
EQUIVALENT fractions have DIFFERENT numerators and denominators but represent the SAME value.
'1/2 = 2/4 = 3/6 = 4/8 — all represent HALF of something.'
How to Find Equivalent Fractions:
Multiply the numerator and denominator by the SAME number.
| Original | × 2 | × 3 | × 4 |
|---|---|---|---|
| 1/2 | 2/4 | 3/6 | 4/8 |
| 2/3 | 4/6 | 6/9 | 8/12 |
| 3/4 | 6/8 | 9/12 | 12/16 |
How to Reduce (Simplify):
Divide the numerator and denominator by the SAME number.
Example: 8/12 → Divide by 2 → 4/6 → Divide by 2 → 2/3
'8/12 and 2/3 are EQUIVALENT. 2/3 is the SIMPLEST form because 2 and 3 have no common factor except 1.'
Key Fact:
'Whatever you do to the numerator, you must do the SAME to the denominator. Always!'
4. Comparing Fractions
Same Denominator:
Compare the NUMERATORS. Larger numerator = Larger fraction.
Example: 3/7 < 5/7 (3 < 5)
Same Numerator:
Compare the DENOMINATORS. Smaller denominator = Larger fraction.
Example: 3/4 > 3/8 (4 is smaller than 8, so the parts are BIGGER)
'Think: If you share a pizza among 4 people, you get a BIGGER slice than sharing among 8 people.'
Different Numerators AND Denominators:
Convert to EQUIVALENT fractions with the SAME denominator (the LCM of the denominators).
Example: Compare 2/3 and 3/4.
- LCM of 3 and 4 = 12
- 2/3 = 8/12
- 3/4 = 9/12
- 8/12 < 9/12, so 2/3 < 3/4
5. Addition of Like Fractions
Add the NUMERATORS. Keep the DENOMINATOR the SAME.
Examples:
- 2/7 + 3/7 = (2 + 3)/7 = 5/7
- 1/5 + 2/5 = 3/5
- 3/8 + 2/8 + 1/8 = 6/8 = 3/4 (simplify!)
'When adding like fractions, ONLY the numerators change. The denominator stays exactly the same.'
6. Subtraction of Like Fractions
Subtract the NUMERATORS. Keep the DENOMINATOR the SAME.
Examples:
- 5/7 - 2/7 = (5 - 2)/7 = 3/7
- 7/9 - 4/9 = 3/9 = 1/3 (simplify!)
- 6/8 - 2/8 = 4/8 = 1/2
Mixed Number Addition:
2⅓ + 1⅓ = (2 + 1) + (1/3 + 1/3) = 3 + 2/3 = 3⅔
Mixed Number Subtraction:
3¾ - 1¼ = (3 - 1) + (3/4 - 1/4) = 2 + 2/4 = 2½
7. Fraction of a Collection
Formula:
Fraction of a collection = (Number of parts you want / Total parts) × Total items
Example 1:
What is 1/4 of 20?
'1/4 of 20 means 20 ÷ 4 = 5. So 1/4 of 20 = 5.'
Example 2:
What is 2/3 of 24?
Step 1: 24 ÷ 3 = 8 (this is 1/3) Step 2: 8 × 2 = 16 (this is 2/3) Answer: 16
| Fraction | Of | Calculation | Answer |
|---|---|---|---|
| 1/2 | 30 | 30 ÷ 2 | 15 |
| 3/4 | 40 | 40 ÷ 4 = 10, × 3 | 30 |
| 2/5 | 50 | 50 ÷ 5 = 10, × 2 | 20 |
| 5/6 | 36 | 36 ÷ 6 = 6, × 5 | 30 |
8. Common Mistakes
- Adding denominators: 'NEVER add denominators. 2/7 + 3/7 = 5/7, NOT 5/14! The denominator stays the same.'
- Wrong comparison: '2/3 is NOT larger than 3/4. 2/3 = 0.66, 3/4 = 0.75. Draw a picture to see!'
- Forgetting to simplify: 'Always check if the answer can be simplified. 6/8 = 3/4. 4/6 = 2/3.'
- Mixed number confusion: '2½ means 2 + 1/2, not 2 × 1/2!'
9. Key Facts to Remember
- 'The denominator can NEVER be zero. Division by zero is NOT defined.'
- 'If numerator = denominator, the fraction equals 1 (5/5 = 1, 12/12 = 1).'
- 'A fraction with numerator 0 equals 0 (0/5 = 0, 0/12 = 0).'
- 'Equivalent fractions represent the SAME amount — just cut into different numbers of parts.'
- 'When comparing fractions with the SAME numerator, the one with the SMALLER denominator is BIGGER.'
10. Self-Test
Q1: What type of fraction is each? (a) 5/8 (b) 9/7 (c) 2¾
Q2: Find 3 equivalent fractions for 2/5.
Q3: Compare: 5/6 and 7/8. Which is larger?
Q4: Add: 3/10 + 4/10 + 1/10
Q5: Subtract: 8/9 - 2/9
Q6: What is 3/5 of 45?
Q7: Simplify: (a) 12/16 (b) 9/15 (c) 14/21
Q8: Riya ate 1/4 of a pizza and her brother ate 2/4. How much did they eat together? How much is left?
Answers:
A1: (a) Proper — numerator < denominator (b) Improper — numerator > denominator (c) Mixed number A2: 4/10, 6/15, 8/20 (multiply numerator and denominator by 2, 3, 4) A3: 5/6 = 20/24, 7/8 = 21/24. 21/24 > 20/24, so 7/8 > 5/6 A4: 8/10 = 4/5 A5: 6/9 = 2/3 A6: 45 ÷ 5 = 9, 9 × 3 = 27 A7: (a) 3/4 (b) 3/5 (c) 2/3 A8: 1/4 + 2/4 = 3/4. Left: 1 - 3/4 = 1/4 of the pizza.
