Fractions and Decimals
1. Types of Fractions
Classification
| Type | Definition | Example |
|---|---|---|
| Proper fraction | Numerator < Denominator | 3/7, 2/5 |
| Improper fraction | Numerator ≥ Denominator | 7/3, 5/2 |
| Mixed fraction | Whole number + proper fraction | 2 1/3, 4 3/5 |
| Like fractions | Same denominator | 2/7, 5/7 |
| Unlike fractions | Different denominators | 3/4, 5/6 |
| Unit fraction | Numerator = 1 | 1/2, 1/8 |
Converting Mixed to Improper
2 1/3 = (2 × 3 + 1)/3 = 7/3
Converting Improper to Mixed
7/3 → 7 ÷ 3 = 2 remainder 1 → 2 1/3
2. Multiplication of Fractions
Rule
Product of fractions = (Product of numerators) / (Product of denominators)
Multiplication of a Fraction by a Whole Number
a × (b/c) = (a × b)/c
Example: 5 × (3/4) = 15/4 = 3 3/4.
Multiplication of a Fraction by a Fraction
(a/b) × (c/d) = (a × c)/(b × d)
Worked Example (ICSE 2024, 2 marks)
Simplify: 2 1/3 × 1 2/5.
Solution: Convert to improper: 2 1/3 = 7/3, 1 2/5 = 7/5. (7/3) × (7/5) = 49/15 = 3 4/15.
Cancellation (Cross-Cancelling)
Before multiplying, cancel common factors between ANY numerator and ANY denominator. Example: (4/9) × (3/8) = (4÷4)/(9÷3) × (3÷3)/(8÷4) = 1/3 × 1/2 = 1/6.
3. Division of Fractions
Rule
To divide by a fraction, MULTIPLY by its RECIPROCAL. a/b ÷ c/d = a/b × d/c (where c ≠ 0 and d ≠ 0).
Worked Example (ICSE 2023, 2 marks)
Simplify: 3 1/2 ÷ 1 3/4.
Solution: 3 1/2 = 7/2, 1 3/4 = 7/4. (7/2) ÷ (7/4) = (7/2) × (4/7) = 28/14 = 2.
Word Problem (ICSE Focus, 3 marks)
'A ribbon of length 15 3/4 m is cut into pieces of 1 3/4 m each. How many pieces?' Total length = 63/4 m. Each piece = 7/4 m. Number = (63/4) ÷ (7/4) = (63/4) × (4/7) = 63/7 = 9 pieces.
4. Decimal Numbers
Place Value Chart
| Thousands | Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|---|
| 1000 | 100 | 10 | 1 | . | 1/10 | 1/100 | 1/1000 |
Example: 345.678 = 3 × 100 + 4 × 10 + 5 × 1 + 6/10 + 7/100 + 8/1000.
Types of Decimals
- Terminating: Division ends. 3/8 = 0.375.
- Non-terminating recurring: Digits repeat. 1/3 = 0.333... = 0.\bar{3}.
- Non-terminating non-recurring: NOT rational. These are IRRATIONAL numbers.
5. Decimal Operations
Addition and Subtraction
Line up DECIMAL POINTS. Add/Subtract as whole numbers. Place decimal in answer.
Example: 12.35 + 4.7 = 12.35 + 4.70 = 17.05.
Multiplication
- Ignore decimal points and multiply as whole numbers.
- Count TOTAL decimal places in both factors.
- Place decimal in product (from RIGHT).
Example: 2.5 × 0.04 = ? 25 × 4 = 100. Total decimal places: 1 + 2 = 3. 100 → 0.100 = 0.1.
Division by a Decimal
- Move decimal in divisor to make it a whole number.
- Move decimal in dividend the SAME number of places.
- Divide as usual.
Example: 4.2 ÷ 0.07 = 420 ÷ 7 = 60.
Worked Example (ICSE 2023, 3 marks)
Simplify: 12.5 × 0.8 + 3.6 ÷ 0.09. 12.5 × 0.8 = 10.0. 3.6 ÷ 0.09 = 360 ÷ 9 = 40. 10 + 40 = 50.
6. Fraction-Decimal Conversions
Fraction → Decimal
Divide numerator by denominator. Examples: 3/8 = 0.375, 5/6 = 0.8333... = 0.8\bar{3}.
Decimal → Fraction (Terminating)
Write as fraction with power of 10 denominator. Simplify. Example: 0.375 = 375/1000 = 3/8.
Decimal → Fraction (Recurring)
Let x = 0.\bar{3}. 10x = 3.\bar{3}. 10x - x = 3. 9x = 3. x = 1/3.
7. ICSE Exam Focus
Common Mistakes
- Adding/subtracting decimals WITHOUT aligning decimal points.
- Forgetting to count TOTAL decimal places in multiplication.
- Cross-cancelling in addition/subtraction (only allowed in multiplication).
- Dividing numerator by numerator and denominator by denominator (wrong — take reciprocal).
| Topic | Marks | Frequency |
|---|---|---|
| Fraction multiplication and division | 3 marks | Very High |
| Decimal operations | 2-3 marks | Very High |
| Fraction-decimal conversion | 2 marks | High |
| Word problems | 3-4 marks | High |
Self-Test (5 Questions)
Q1. Simplify: 2 1/4 × 1 1/3. (2 marks)
- A) 3
- B) 2 1/3
- C) 3 1/2
- D) 4
Q2. Divide: 5 1/6 ÷ 2 1/3. (2 marks)
Q3. Find: 3.6 × 0.25 + 4.8 ÷ 0.16. (3 marks)
- A) 30.9
- B) 30.1
- C) 29.9
- D) 31
Q4. Convert 0.625 to a fraction in simplest form. (2 marks)
Q5. 'A rectangular field is 12.5 m long and 8.4 m wide. Find its perimeter.' (2 marks)
Answers
A1. A) 3. (9/4 × 4/3 = 36/12 = 3.) A2. 2 3/14. (31/6 ÷ 7/3 = 31/6 × 3/7 = 93/42 = 31/14 = 2 3/14.) A3. A) 30.9. (3.6 × 0.25 = 0.9. 4.8 ÷ 0.16 = 30. 0.9 + 30 = 30.9.) A4. 5/8. (0.625 = 625/1000 = 5/8.) A5. 41.8 m. (Perimeter = 2(l + w) = 2(12.5 + 8.4) = 2 × 20.9 = 41.8 m.)
