Fractions
Key Concepts & Formulas
| # | Concept | Quick Explanation |
|---|---|---|
| 1 | Types of Fractions | Proper (numerator < denominator), Improper (numerator ≥ denominator), Mixed (whole + proper fraction) |
| 2 | Equivalent Fractions | Multiply/divide numerator & denominator by same non-zero number: 2/3 = 4/6 = 6/9 |
| 3 | LCM Method | For addition/subtraction: LCM of denominators = common denominator |
| 4 | Reciprocal Rule | Product of fraction and its reciprocal = 1: (a/b) × (b/a) = 1 |
| 5 | Fraction to Percentage | Multiply by 100: 3/4 = 0.75 × 100 = 75% |
| 6 | Division Rule | Multiply by reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc |
10 Practice MCQs
Q1. What is 3/5 of 250 km railway track? A) 120 km B) 150 km C) 180 km D) 200 km
Answer: B) 150 km
Solution: 3/5 × 250 = (3 × 250)/5 = 750/5 = 150 km
Shortcut: Divide by 5 first: 250 ÷ 5 = 50, then 50 × 3 = 150
Concept: Fractions - Finding fraction of a quantity
Q2. Express 45 minutes as a fraction of 3 hours. A) 1/4 B) 1/3 C) 1/2 D) 2/5
Answer: A) 1/4
Solution: 3 hours = 180 minutes 45/180 = 1/4 (dividing both by 45)
Concept: Fractions - Time conversion to fraction
Q3. Which fraction is smallest? 3/7, 2/5, 1/3, 4/9 A) 3/7 B) 2/5 C) 1/3 D) 4/9
Answer: C) 1/3
Solution: Convert to common denominator (LCM of 7,5,3,9 = 315) 3/7 = 135/315, 2/5 = 126/315, 1/3 = 105/315, 4/9 = 140/315 Smallest: 105/315 = 1/3
Shortcut: Cross multiply: 1×5×7×9 = 315 (smallest numerator)
Concept: Fractions - Comparison
Q4. A train covers 3/8 of 480 km journey. How much distance remains? A) 180 km B) 200 km C) 280 km D) 300 km
Answer: D) 300 km
Solution: Distance covered = 3/8 × 480 = 180 km Remaining = 480 - 180 = 300 km OR: 1 - 3/8 = 5/8 remaining → 5/8 × 480 = 300 km
Concept: Fractions - Remaining quantity
Q5. Simplify: (2/3 + 1/4) ÷ (5/6 - 1/3) A) 11/6 B) 11/10 C) 6/11 D) 10/11
Answer: A) 11/6
Solution: Numerator: 2/3 + 1/4 = 8/12 + 3/12 = 11/12 Denominator: 5/6 - 1/3 = 5/6 - 2/6 = 3/6 = 1/2 Result: (11/12) ÷ (1/2) = 11/12 × 2/1 = 22/12 = 11/6
Concept: Fractions - Combined operations
Q6. If 5/7 of platform length is 35m, what is full length? A) 42m B) 45m C) 49m D) 56m
Answer: C) 49m
Solution: Let full length = x 5/7 × x = 35 x = 35 × 7/5 = 7 × 7 = 49m
Concept: Fractions - Finding whole from part
Q7. A goods train has 3/5 loaded wagons. If 24 wagons are empty, total wagons? A) 40 B) 48 C) 60 D) 72
Answer: C) 60
Solution: Empty wagons = 1 - 3/5 = 2/5 2/5 × Total = 24 Total = 24 × 5/2 = 60
Concept: Fractions - Part-whole relationship
Q8. Find the value of: 1/2 + 1/6 + 1/12 + 1/20 + 1/30 A) 4/5 B) 5/6 C) 7/10 D) 3/4
Answer: B) 5/6
Solution: Pattern: 1/n(n+1) = 1/n - 1/(n+1) = (1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) + (1/5 - 1/6) = 1 - 1/6 = 5/6 (telescoping series)
Concept: Fractions - Series summation
Q9. If x = 3/4 and y = 2/3, find (x² - y²)/(x - y) A) 17/12 B) 1/12 C) 19/12 D) 5/12
Answer: A) 17/12
Solution: (x² - y²)/(x - y) = (x + y)(x - y)/(x - y) = x + y = 3/4 + 2/3 = 9/12 + 8/12 = 17/12
Shortcut: Directly add x + y (difference of squares)
Concept: Fractions - Algebraic manipulation
Q10. A train's speed increases by 1/3. Time saved on 360 km journey is 2 hours. Find original speed. A) 45 km/h B) 60 km/h C) 75 km/h D) 90 km/h
Answer: B) 60 km/h
Solution: Let original speed = s km/h New speed = s + s/3 = 4s/3 Time difference: 360/s - 360/(4s/3) = 2 360/s - 270/s = 2 90/s = 2 s = 45 km/h
Concept: Fractions - Speed-time relationship
5 Previous Year Questions
PYQ 1. A train ticket costs ₹240. If 2/3 fare is charged for senior citizens, how much for 3 senior citizens? [RRB NTPC 2021 CBT-1]
Answer: ₹480
Solution: Fare per senior citizen = 2/3 × 240 = ₹160 For 3 senior citizens = 3 × 160 = ₹480
Exam Tip: Calculate fraction first, then multiply by number of people
PYQ 2. Simplify: (5/8 - 3/4 + 1/2) [RRB Group D 2022]
Answer: 3/8
Solution: LCM of 8,4,2 = 8 = 5/8 - 6/8 + 4/8 = (5-6+4)/8 = 3/8
Exam Tip: Always find LCM first for addition/subtraction
PYQ 3. 7/9 of railway employees are technical staff. If 540 are non-technical, find total employees. [RRB ALP 2018]
Answer: 2430
Solution: Non-technical = 1 - 7/9 = 2/9 2/9 × Total = 540 Total = 540 × 9/2 = 2430
Exam Tip: Find complement fraction first
PYQ 4. A 15m platform has 3/5 concrete part, rest steel. Steel length? [RRB JE 2019]
Answer: 6m
Solution: Steel part = 1 - 3/5 = 2/5 2/5 × 15 = 6m
Exam Tip: Remember: Part = Fraction × Whole
PYQ 5. If 3/4 of journey takes 45 minutes, time for full journey? [RPF SI 2019]
Answer: 60 minutes
Solution: 3/4 × Total time = 45 Total time = 45 × 4/3 = 60 minutes
Exam Tip: Use inverse operation to find whole
Speed Tricks & Shortcuts
| Situation | Shortcut | Example |
|---|---|---|
| Finding fraction of number | Divide by denominator, multiply by numerator | 3/7 of 490 = 490÷7×3 = 210 |
| Adding consecutive fractions | Use pattern 1/n(n+1) = 1/n - 1/(n+1) | 1/2+1/6+1/12 = 1-1/4 = 3/4 |
| Comparing fractions | Cross-multiply quickly | 3/5 vs 4/7: 3×7=21, 4×5=20 → 3/5 > 4/7 |
| Mixed to improper | (Whole×denominator)+numerator | 2⅗ = (2×5+3)/5 = 13/5 |
| Percentage conversion | Multiply numerator by 100, divide by denominator | 7/25 = 700÷25 = 28% |
Common Mistakes to Avoid
| Mistake | Why Students Make It | Correct Approach |
|---|---|---|
| Not finding LCM | Trying to add directly | Always find LCM of denominators first |
| Forgetting reciprocal in division | Confusing with multiplication | Remember: ÷ fraction = × reciprocal |
| Simplifying partially | Not dividing by HCF | Always divide by highest common factor |
| Mixing units | Not converting to same units | Convert all to same unit before calculating |
| Wrong operation order | Not following BODMAS | Follow: Brackets → Orders → Division → Multiplication → Addition → Subtraction |
Quick Revision Flashcards
| Front (Question/Term) | Back (Answer) |
|---|---|
| Proper fraction | Numerator < denominator (e.g., 3/5) |
| Improper fraction | Numerator ≥ denominator (e.g., 7/4) |
| Reciprocal of 5/9 | 9/5 |
| 1/3 as percentage | 33.33% |
| LCM of 4,6,8 | 24 |
| Simplest form of 18/24 | 3/4 |
| 5/8 of 1 km | 625 meters |
| Mixed number for 17/5 | 3⅖ |
| Decimal for 7/20 | 0.35 |
| Fraction for 0.125 | 1/8 |
Topic Connections
- Direct Link: Fractions form basis of Ratio & Proportion - both use part-whole concepts
- Combined Questions: Often paired with Percentage (converting fractions to %) and Profit-Loss (fractional profit/loss)
- Foundation For: Advanced topics like Time & Work (fraction of work), Speed-Distance-Time (fractional speeds), and Data Interpretation (pie charts use fractions)
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