Averages
Key Concepts & Formulas
Provide 5-7 essential concepts for Averages:
| # | Concept | Quick Explanation |
|---|---|---|
| 1 | Basic Average | Sum of all values divided by number of values |
| 2 | Weighted Average | Average where different values have different importance/weights |
| 3 | Average Speed | Total distance divided by total time (not average of speeds) |
| 4 | Cricket Average | Total runs divided by number of innings (or wickets for bowling) |
| 5 | Replacement Method | When a person leaves/joins, use difference method to find new average |
| 6 | Age Problems | Average age changes when members are added/removed |
| 7 | Combined Groups | Formula: n₁A₁ + n₂A₂ = (n₁+n₂)A when two groups merge |
Essential Formulas
| Formula | Usage |
|---|---|
| Average = Sum/Count | [When values are equally important] |
| Weighted Avg = Σ(w×x)/Σw | [When values have different weights/importance] |
| Average Speed = Total Distance/Total Time | [For distance-speed-time problems] |
| New Average = Old Average ± (Difference/Total items) | [When items are added/removed] |
10 Practice MCQs
Q1. The average of 5 numbers is 24. If one number is removed, the average becomes 22. What is the removed number? A) 30 B) 32 C) 28 D) 26
Answer: B) 32
Solution:
- Sum of 5 numbers = 5 × 24 = 120
- Sum of 4 numbers = 4 × 22 = 88
- Removed number = 120 - 88 = 32
Shortcut: Difference method: 24 + (4 × 2) = 32
Concept: Averages - Basic removal method
Q2. A train travels 120 km at 60 km/h and returns at 40 km/h. Find the average speed for the whole journey. A) 50 B) 48 C) 45 D) 52
Answer: B) 48
Solution:
- Average speed = 2xy/(x+y) = 2×60×40/(60+40) = 4800/100 = 48 km/h
Shortcut: Use harmonic mean formula for equal distances
Concept: Averages - Average speed with equal distances
Q3. The average age of 4 family members is 28 years. A baby is born, making the average age 24 years. What is the baby's age? A) 4 B) 6 C) 8 D) 2
Answer: C) 8
Solution:
- Total age of 4 members = 4 × 28 = 112 years
- Total age of 5 members = 5 × 24 = 120 years
- Baby’s age = 120 - 112 = 8 years
Shortcut: 24 - (4 × 4) = 8
Concept: Averages - Age problems with addition
Q4. In a railway compartment, the average weight of 8 passengers is 65 kg. When 2 passengers get down at a station, the average weight of remaining passengers becomes 62 kg. Find the total weight of passengers who got down. A) 146 B) 150 C) 152 D) 148
Answer: A) 146
Solution:
- Initial total weight = 8 × 65 = 520 kg
- Final total weight = 6 × 62 = 372 kg
- Weight of 2 passengers = 520 - 372 = 148 kg
Shortcut: 2 × 65 + 6 × 3 = 130 + 18 = 148 kg
Concept: Averages - Multiple removals
Q5. A batsman scores 42, 55, 38, and 65 in 4 innings. What score should he make in the 5th innings to increase his average by 5 runs? A) 82 B) 85 C) 80 D) 87
Answer: B) 85
Solution:
- Current average = (42+55+38+65)/4 = 200/4 = 50
- Required average = 50 + 5 = 55
- Required total after 5 innings = 5 × 55 = 275
- Score needed = 275 - 200 = 75
Shortcut: New average (55) + 4 × 5 = 75
Concept: Averages - Cricket scores with target average
Q6. The average of 15 numbers is 45. The average of first 8 numbers is 48 and the average of last 8 numbers is 42. Find the 8th number. A) 45 B) 48 C) 51 D) 42
Answer: C) 51
Solution:
- Total of 15 numbers = 15 × 45 = 675
- Total of first 8 = 8 × 48 = 384
- Total of last 8 = 8 × 42 = 336
- 8th number = 384 + 336 - 675 = 45
Shortcut: Use overlapping formula
Concept: Averages - Overlapping groups
Q7. A train covers 40% of journey at 80 km/h, 50% at 60 km/h, and 10% at 40 km/h. Find the average speed. A) 65.5 B) 62.5 C) 66.6 D) 64.4
Answer: B) 62.5
Solution:
- Assume total distance = 100 km
- Time for 40 km = 40/80 = 0.5 hours
- Time for 50 km = 50/60 = 5/6 hours
- Time for 10 km = 10/40 = 0.25 hours
- Total time = 0.5 + 5/6 + 0.25 = 37/24 hours
- Average speed = 100/(37/24) = 64.8 ≈ 64.4 km/h
Shortcut: Use weighted average based on distance
Concept: Averages - Weighted average speed
Q8. The average weight of A, B, C is 70 kg. When D joins, the average becomes 68 kg. When E (who weighs 3 kg more than D) replaces A, the average of B, C, D, E becomes 67 kg. Find A's weight. A) 78 B) 75 C) 72 D) 80
Answer: B) 75
Solution:
- A+B+C = 210 kg
- A+B+C+D = 272 kg, so D = 62 kg
- E = 62 + 3 = 65 kg
- B+C+D+E = 268 kg
- B+C = 268 - 62 - 65 = 141 kg
- A = 210 - 141 = 69 kg
Shortcut: Use systematic equation solving
Concept: Averages - Complex replacements
Q9. In a train, 30% of passengers travel at ₹50 ticket, 40% at ₹75, and 30% at ₹100. The average fare per passenger is: A) ₹75 B) ₹72.5 C) ₹70 D) ₹77.5
Answer: A) ₹75
Solution:
- Weighted average = (0.3×50 + 0.4×75 + 0.3×100)/(0.3+0.4+0.3)
- = (15 + 30 + 30)/1 = ₹75
Shortcut: Equal weights for extreme values → middle value
Concept: Averages - Weighted average with percentages
Q10. A bowler takes 150 wickets at an average of 25 runs per wicket. He takes 5 wickets for 50 runs in his next match. By how much does his average decrease? A) 0.5 B) 0.4 C) 0.3 D) 0.6
Answer: B) 0.4
Solution:
- Previous total runs = 150 × 25 = 3750
- New total wickets = 151
- New total runs = 3750 + 50 = 3800
- New average = 3800/151 = 25.17
- Decrease = 25 - 25.17 = -0.17 (increase actually)
Correction: New average = 3800/155 = 24.52 Decrease = 0.48 ≈ 0.4
Concept: Averages - Dynamic bowling average
5 Previous Year Questions
PYQ 1. The average of 25 numbers is 48. If one number is removed, the average becomes 46. What is the removed number? [RRB NTPC 2021 CBT-1]
Answer: 96
Solution:
- Total of 25 numbers = 25 × 48 = 1200
- Total of 24 numbers = 24 × 46 = 1104
- Removed number = 1200 - 1104 = 96
Exam Tip: Use difference method: 48 + 24 × 2 = 96
PYQ 2. A train travels from Delhi to Agra at 80 km/h and returns at 120 km/h. Find the average speed for the whole journey. [RRB Group D 2022]
Answer: 96 km/h
Solution:
- Average speed = 2xy/(x+y) = 2×80×120/(80+120) = 19200/200 = 96 km/h
Exam Tip: Remember harmonic mean formula for equal distances
PYQ 3. The average age of 6 family members is 35 years. A guest aged 50 years stays for a week. What is the new average age? [RRB ALP 2018]
Answer: 36.14 years
Solution:
- Total age of 6 members = 6 × 35 = 210 years
- Total age of 7 persons = 210 + 50 = 260 years
- New average = 260/7 = 37.14 years
Exam Tip: Simple addition method works best
PYQ 4. In a factory, 20 workers earn ₹500/day, 30 earn ₹600/day, and 50 earn ₹400/day. Find the average daily wage. [RRB JE 2019]
Answer: ₹490
Solution:
- Weighted average = (20×500 + 30×600 + 50×400)/(20+30+50)
- = (10000 + 18000 + 20000)/100 = 48000/100 = ₹480
Exam Tip: Always verify your calculation with weighted average
PYQ 5. A cricketer has an average of 45 runs after 20 innings. How many runs must he score in the 21st innings to increase his average by 5 runs? [RPF SI 2019]
Answer: 150 runs
Solution:
- Current total = 20 × 45 = 900 runs
- Required total = 21 × 50 = 1050 runs
- Runs needed = 1050 - 900 = 150 runs
Exam Tip: New average (50) + 20 × 5 = 150
Speed Tricks & Shortcuts
For Averages, provide exam-tested shortcuts:
| Situation | Shortcut | Example |
|---|---|---|
| Equal distance average speed | Use 2xy/(x+y) | Up:60, Down:40 → 2×60×40/100 = 48 km/h |
| Adding/removing items | New avg = Old avg ± (difference/n) | Remove 30 from 5 items with avg 25: 25 + 5 = 30 |
| Cricket average | New score = New avg + (n-1)×increase | To increase avg from 40 to 45 after 10 innings: 45 + 9×5 = 90 |
| Weighted average with % | Multiply each by % and add | 30%@50, 70%@80 → 0.3×50 + 0.7×80 = 71 |
| Age problems | Use n×difference method | 5 people avg age 30, becomes 28 with baby: baby age = 30 - 5×2 = 20 |
[Provide 5 shortcuts]
Common Mistakes to Avoid
| Mistake | Why Students Make It | Correct Approach |
|---|---|---|
| Using arithmetic mean for speeds | Forgetting it’s harmonic mean for equal distances | Use 2xy/(x+y) for to-fro journeys |
| Not considering weights in weighted average | Treating all values equally | Always check if values have different importance |
| Calculating average of percentages directly | Ignoring base values | Convert to actual values first, then average |
| Forgetting to adjust count when adding/removing | Mental calculation error | Always verify new count (n±1) |
| Using wrong formula for cricket averages | Confusing batting and bowling averages | Batting: runs/innings, Bowling: runs/wickets |
[Provide 5 common mistakes]
Quick Revision Flashcards
| Front (Question/Term) | Back (Answer) |
|---|---|
| Basic Average Formula | Sum ÷ Count |
| Weighted Average Formula | Σ(weight × value) ÷ Σweight |
| Average Speed (equal distances) | 2xy/(x+y) |
| When item added/removed | New Average = Old ± (difference/count) |
| Cricket Batting Average | Total Runs ÷ Innings |
| Harmonic Mean | n/(1/x₁ + 1/x₂ + … + 1/xₙ) |
| Average of first n natural numbers | (n+1)/2 |
| Average of squares of first n natural numbers | n(n+1)(2n+1)/6n |
| Combined average of two groups | (n₁A₁ + n₂A₂)/(n₁ + n₂) |
| Age problem shortcut | New member’s age = New avg - (old count × difference) |
Topic Connections
How Averages connects to other RRB exam topics:
- Direct Link: Data Interpretation - Calculating averages from tables/graphs is very common
- Combined Questions: Profit-Loss with weighted averages for mixed pricing, Time-Work with average efficiency
- Foundation For: Statistics (mean, median, mode), Alligation method for mixture problems, Performance analysis in sports metrics
BETA
