UGC NET Paper 1 Unit 5 Topic 3: Mathematical Aptitude - Fractions

➗ Fractions

Master the fundamental concepts of fractions with types, operations, conversions, and shortcut techniques. Includes UGC NET pattern questions on simplification, comparison, and word problems.

🔑 Basic Definition: A fraction represents parts of a whole written as a/b where:
• a = numerator (parts considered)
• b = denominator (total parts)
• Example: 3/4 means 3 parts out of 4 equal parts

1. Types of Fractions

Type	Definition	Example	Visualization (▢ = 1 part)
Proper	Top < Bottom	2/5	▢▢▣▣▣
Improper	Top ≥ Bottom	7/4	▢▢▢▢ ▢▢▣
Mixed	Whole + Fraction	1¾	▢ + ▢▢▣
Equivalent	Same value	1/2 = 2/4	▢▣ = ▢▢▣▣

Key: ▣ = shaded part, ▢ = unshaded part (each ▢ or ▣ = 1 fractional unit)

Conversion Formulas:
• Improper → Mixed: 7/4 = 1¾ (7÷4=1 with remainder 3)
• Mixed → Improper: 1¾ = (1×4 + 3)/4 = 7/4

2. Fraction Operations

Core Operations:

Operation	Rule	Example
Addition	a/b + c/d = (ad + bc)/bd	1/2 + 1/3 = (3+2)/6 = 5/6
Subtraction	a/b - c/d = (ad - bc)/bd	1/2 - 1/4 = (4-2)/8 = 2/8 = 1/4
Multiplication	a/b × c/d = ac/bd	2/3 × 3/5 = 6/15 = 2/5
Division	a/b ÷ c/d = a/b × d/c	3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8

Shortcut Techniques:

• Cancellation: Simplify before multiplying (6/7 × 14/15 = 2/1 × 2/5 = 4/5)
• LCD Addition: Find Least Common Denominator first (1/6 + 1/4 = 2/12 + 3/12 = 5/12)
• Reciprocal Division: "Keep-Change-Flip" method for division

3. Decimal and Percentage Conversions

Fraction	Decimal	Percentage	Memory Tip
1/2	0.5	50%	Half dollar
1/4	0.25	25%	Quarter coin
1/3	0.333...	33.33%	Repeating three
3/4	0.75	75%	Three quarters
1/5	0.2	20%	Fifth of 100

Conversion Formulas:
• Fraction → Decimal: Divide numerator by denominator
• Decimal → Fraction: Write as numerator/10ⁿ and simplify
• Fraction → Percentage: Multiply by 100%
• Percentage → Fraction: Divide by 100 and simplify

4. Comparison of Fractions

Comparison Methods:

1. Cross-Multiplication:
   Compare 3/5 vs 4/7 → 3×7=21 vs 5×4=20 → 21>20 ∴ 3/5 > 4/7
2. Common Denominator:
   Convert to LCD: 2/3=8/12 vs 3/4=9/12 → 9/12 > 8/12
3. Decimal Conversion:
   1/2=0.5 vs 3/7≈0.428 → 0.5 > 0.428

Comparison	Technique	Result
5/8 vs 7/12	Cross-multiply (5×12=60 vs 8×7=56)	60>56 ∴ 5/8 > 7/12
2/3, 5/6, 7/12	LCD=12 → 8/12, 10/12, 7/12	5/6 > 2/3 > 7/12

5. Word Problems

Problem-Solving Approach:

1. Identify what's given and what's asked
2. Determine if addition/subtraction/multiplication/division is needed
3. Convert mixed numbers to improper fractions if necessary
4. Perform operations step-by-step
5. Simplify the final answer

🔥 Previous Year UGC NET Questions:

1. If ⅜ of a number is 24, what is ½ of that number?
   Solution: Let number be x → 3x/8=24 → x=64 → ½×64=32
2. A student reads 2/5 of a book on Day 1 and 1/3 on Day 2. What fraction remains?
   Solution: 2/5 + 1/3 = 11/15 → Remaining = 4/15
3. Which is greater: 7/12 or 9/16?
   Solution: Cross-multiply 7×16=112 vs 12×9=108 → 7/12 > 9/16
4. Simplify: (1 + ½)/(1 - ½)
   Solution: Numerator=3/2, Denominator=1/2 → (3/2)÷(1/2)=3

🚀 Shortcut Techniques

Fast Calculation Methods:

• Percentage Equivalents:
  1/8 = 12.5% | 1/6 ≈ 16.67% | 5/6 ≈ 83.33%
• Reciprocal Multiples:
  1/7 ≈ 0.142857 | 1/9 ≈ 0.1111 | 1/11 ≈ 0.0909
• Addition Shortcut:
  a/b + a/c = a(b+c)/bc → 2/3 + 2/5 = 2(8)/15=16/15

Common Mistakes to Avoid:

• Adding denominators directly (1/2 + 1/3 ≠ 2/5)
• Forgetting to simplify fractions to lowest terms
• Mixing operations without proper parentheses
• Dividing by forgetting to take reciprocal

📝 Practice Essentials

Recommended Practice:

1. Convert between mixed/improper fractions daily
2. Memorize fraction-decimal-percentage equivalents up to 1/12
3. Solve 5 comparison problems using different methods
4. Practice word problems involving part-whole relationships