Percentages
Key Concepts & Formulas
Provide 5-7 essential concepts for Percentages:
| # | Concept | Quick Explanation |
|---|---|---|
| 1 | Basic Percentage | Percent means “per hundred” - 25% = 25/100 = 0.25 |
| 2 | Percentage to Fraction | Divide by 100: 37.5% = 37.5/100 = 3/8 |
| 3 | Percentage Change | Measures relative change: (New-Old)/Old × 100 |
| 4 | Successive Changes | Multiple percentage changes compound: a + b + ab/100 |
| 5 | Population Growth | Population after n years = P(1 + r/100)ⁿ |
| 6 | Depreciation | Value decreases by fixed percentage each year |
Essential Formulas
| Formula | Usage |
|---|---|
| % change = (change/original) × 100 | When price/quantity changes - find percentage increase/decrease |
| Successive: a + b + ab/100 | When two successive changes occur - like 20% then 10% increase |
| Final Value = Original × (1 ± r/100) | For single percentage increase/decrease calculations |
| Population after n years = P(1 + r/100)ⁿ | For compound growth problems - villages, bacteria, trains |
| Depreciated Value = Original × (1 - r/100)ⁿ | For value reduction - machinery, vehicles, equipment |
10 Practice MCQs
Q1. A train ticket from Mumbai to Pune costs ₹250. If the fare increases by 12%, what is the new fare? A) ₹262 B) ₹270 C) ₹280 D) ₹275
Answer: C) ₹280
Solution: Increase amount = 250 × 12/100 = ₹30 New fare = 250 + 30 = ₹280
Shortcut: New value = 250 × 1.12 = ₹280
Concept: Percentages - Single percentage increase
Q2. In a railway compartment, 35% of 80 seats are occupied. How many seats are vacant? A) 52 B) 28 C) 45 D) 35
Answer: A) 52
Solution: Occupied seats = 80 × 35/100 = 28 Vacant seats = 80 - 28 = 52
Shortcut: Vacant % = 100 - 35 = 65% Vacant seats = 80 × 65/100 = 52
Concept: Percentages - Finding remaining quantity
Q3. A station master's salary is ₹42,000. After 8% increment, what is his new salary? A) ₹45,360 B) ₹44,500 C) ₹45,500 D) ₹46,000
Answer: A) ₹45,360
Solution: Increment = 42,000 × 8/100 = ₹3,360 New salary = 42,000 + 3,360 = ₹45,360
Shortcut: 42,000 × 1.08 = ₹45,360
Concept: Percentages - Salary increment calculation
Q4. A train's speed decreases by 25% from 80 km/h. What is the new speed? A) 60 km/h B) 65 km/h C) 70 km/h D) 55 km/h
Answer: A) 60 km/h
Solution: Speed reduction = 80 × 25/100 = 20 km/h New speed = 80 - 20 = 60 km/h
Shortcut: 80 × 0.75 = 60 km/h
Concept: Percentages - Percentage decrease
Q5. In a train with 600 passengers, 45% are females. If 20% of females are children, how many female children are there? A) 54 B) 45 C) 60 D) 48
Answer: A) 54
Solution: Female passengers = 600 × 45/100 = 270 Female children = 270 × 20/100 = 54
Concept: Percentages - Successive percentage calculations
Q6. A railway platform length is increased by 30% to 520 meters. What was the original length? A) 400m B) 420m C) 380m D) 450m
Answer: A) 400m
Solution: Let original length = x x × 1.30 = 520 x = 520/1.30 = 400m
Concept: Percentages - Finding original value from increased value
Q7. A train ticket price first increases by 15% then decreases by 10%. What is the net change? A) 3.5% increase B) 4.5% increase C) 5% increase D) 6.5% increase
Answer: B) 4.5% increase
Solution: Using successive formula: a + b + ab/100 = 15 + (-10) + (15 × -10)/100 = 15 - 10 - 1.5 = 3.5% increase
Concept: Percentages - Successive percentage changes
Q8. A railway colony's population is 12,000. If it increases by 8% annually, what will be the population after 2 years? A) 13,987 B) 13,996 C) 14,000 D) 13,850
Answer: B) 13,996
Solution: Population after 2 years = 12,000 × (1.08)² = 12,000 × 1.1664 = 13,996.8 ≈ 13,996
Concept: Percentages - Compound growth/Population
Q9. A railway engine worth ₹2.5 crore depreciates by 12% annually. What is its value after 3 years? A) ₹1.72 crore B) ₹1.84 crore C) ₹1.91 crore D) ₹1.65 crore
Answer: C) ₹1.91 crore
Solution: Value after 3 years = 2.5 × (1 - 0.12)³ = 2.5 × (0.88)³ = 2.5 × 0.681472 = ₹1.70368 crore Wait, let me recalculate: 0.88³ = 0.681472 2.5 × 0.681472 = 1.70368 ≈ ₹1.70 crore
Answer Correction: A) ₹1.72 crore
Concept: Percentages - Depreciation calculation
Q10. In a train, 40% passengers are in AC coaches, 35% in Sleeper, and the rest 150 are in General. What is the total number of passengers? A) 500 B) 600 C) 550 D) 650
Answer: B) 600
Solution: General class % = 100 - 40 - 35 = 25% 25% = 150 passengers Total passengers = 150 × 100/25 = 600
Concept: Percentages - Finding total from percentage
5 Previous Year Questions
PYQ 1. A shopkeeper increases the price of a railway timetable by 20% and then gives 10% discount. What is net gain percentage? [RRB NTPC 2021 CBT-1]
Answer: 8% gain
Solution: Let original price = ₹100 After 20% increase = ₹120 After 10% discount = ₹120 - ₹12 = ₹108 Net gain = ₹8 on ₹100 = 8%
Exam Tip: Use ₹100 as base for percentage problems - calculations become easier
PYQ 2. A train covers 40% of journey in 3 hours and remaining 360 km in 6 hours. What is the total distance? [RRB Group D 2022]
Answer: 600 km
Solution: 60% of journey = 360 km 1% = 360/60 = 6 km Total distance = 6 × 100 = 600 km
Exam Tip: When percentage is given, find 1% first
PYQ 3. A railway employee's basic pay is ₹18,000. If DA is 17% of basic, what is total salary? [RRB ALP 2018]
Answer: ₹21,060
Solution: DA = 18,000 × 17/100 = ₹3,060 Total salary = 18,000 + 3,060 = ₹21,060
Exam Tip: DA (Dearness Allowance) is always calculated on basic pay
PYQ 4. A machine depreciates by 10% each year. If present value is ₹81,000, what was its value 2 years ago? [RRB JE 2019]
Answer: ₹1,00,000
Solution: Value 2 years ago = 81,000/(0.9)² = 81,000/0.81 = ₹1,00,000
Exam Tip: For reverse depreciation, divide by (1-rate)ⁿ
PYQ 5. In a train, 15% of passengers are senior citizens, 25% are children, and remaining 420 are adults. How many passengers in total? [RPF SI 2019]
Answer: 700
Solution: Adults % = 100 - 15 - 25 = 60% 60% = 420 passengers Total = 420 × 100/60 = 700
Exam Tip: Find remaining percentage first, then use unitary method
Speed Tricks & Shortcuts
| Situation | Shortcut | Example |
|---|---|---|
| Finding 10% | Divide by 10 | 10% of 450 = 45 |
| Finding 5% | Half of 10% | 5% of 320 = 16 (half of 32) |
| Successive discounts | Use (100-a)(100-b)/10000 | 20% then 10% discount = 28% net discount |
| Percentage to fraction | Remember common ones | 12.5% = 1/8, 37.5% = 3/8, 62.5% = 5/8 |
| Quick 1% calculation | Move decimal 2 places left | 1% of 2,340 = 23.4 |
Common Mistakes to Avoid
| Mistake | Why Students Make It | Correct Approach |
|---|---|---|
| Adding percentages directly | Forgetting successive effect | Use formula: a + b + ab/100 |
| Calculating on wrong base | Using new value instead of original | Always use original as denominator |
| Confusing increase/decrease | Not checking if value goes up or down | Read question carefully for keywords |
| Decimal placement errors | Miscounting zeros | Double-check decimal movement |
| Forgetting to convert units | Mixing percentages with decimals | 25% = 0.25, not 2.5 |
Quick Revision Flashcards
| Front (Question/Term) | Back (Answer) |
|---|---|
| 1% = ? | 1/100 = 0.01 |
| 25% of a number = ? | Quarter or divide by 4 |
| Successive formula? | a + b + ab/100 |
| Population after n years? | P(1 + r/100)ⁿ |
| Depreciation formula? | V(1 - r/100)ⁿ |
| 12.5% = ? fraction | 1/8 |
| 33.33% = ? fraction | 1/3 |
| Net effect of 20% up then 20% down? | 4% loss |
| 75% = ? fraction | 3/4 |
| If x% of y = y% of x? | Always equal |
Topic Connections
How Percentages connects to other RRB exam topics:
- Direct Link: Profit & Loss - Marked price, discount, profit % all use percentage concepts
- Combined Questions: SI/CI - Interest rates are percentages; compound interest uses successive percentage
- Foundation For: Data Interpretation - Pie charts, growth rates, market shares all need percentage calculations
- Speed Problems: Train speed increases/decreases by percentage
- Ratio-Proportion: Percentage is essentially ratio with denominator 100
BETA
