🚀 Averages: Quick Revision & Shortcuts
Master these formulas to solve questions in 10 seconds!
🔹 1. Basic Concept & Core Formula
Average = (Sum of Observations) / (Number of Observations)
Sum = Average × Number
👉 Shortcut: Use multiplication directly when Avg and N are given.
🔹 2. Replacement & Change Concept
New Value = Old Value + (New Avg - Old Avg) × n
👉 Shortcut: Total Impact = Change in Average × Total Members.
🔹 3. Combined Average (Group Merging)
Combined Avg = (n1a1 + n2a2) / (n1 + n2)
🔹 4. Consecutive Numbers Shortcut
Odd Numbers: Average = Middle Number
Even Numbers: Avg = (First + Last) / 2
🔹 5. Average Speed (Same Distance)
Average Speed = 2xy / (x + y)
🔥 Example (Shortcut Based)
Q: Avg of 5 numbers is 20. One replaced by 30, new avg becomes 22. Find replaced number.
- Old Sum: 20 × 5 = 100
- New Sum: 22 × 5 = 110
- Increase: 110 - 100 = 10
- Result: 30 - 10 = 20
Averages: Master Class
Practice Set by Wisdom Helps
1. The average marks of 8 students was 51 and by 9 other students was 68. The average marks of all 17 students was?
Solution: Total = (8 × 51) + (9 × 68) = 408 + 612 = 1020. Average = 1020 / 17 = 60.
2. The average of 30 results is 20 and the average of other 20 results is 30. What is the average of all results?
Solution: Total = (30 × 20) + (20 × 30) = 600 + 600 = 1200. Average = 1200 / 50 = 24.
3. Average weight of 6 students is 50 kg; 2 students is 51 kg; and 2 students is 55 kg. Find total average?
Solution: Total Sum = (6×50)+(2×51)+(2×55) = 300+102+110 = 512. Average = 512 / 10 = 51.2 kg.
4. Average weight of 5 persons in a boat is 38 kg. Average of boat + persons is 52kg. Weight of boat?
Solution: Total (Boat+5) = 6 × 52 = 312. Total (5 men) = 5 × 38 = 190. Boat = 312 - 190 = 122 kg.
5. 12 kg rice at 30/kg mixed with 8 kg at 40/kg. Average price of mixed rice?
Solution: Total cost = (12×30) + (8×40) = 360 + 320 = 680. Total kg = 20. Avg = 680/20 = 34.
6. Average of 20 numbers is 15; average of first five is 12. Average of rest?
Solution: Total (20) = 300. Total (5) = 60. Sum of rest (15) = 240. Avg = 240/15 = 16.
7. Avg marks of 50 students is 72. Avg for boys is 70 and girls is 75. Number of boys?
Solution: Using Alligation: (75-72):(72-70) = 3:2. Boys = (3/5) × 50 = 30.
8. Avg of 9 numbers is 30. First 5 is 25, last 3 is 35. What is the 6th number?
Solution: Total = 270. Sum of 8 = (125 + 105) = 230. 6th No = 270 - 230 = 40.
9. Avg of 15 numbers is 7. First 8 is 6.5, last 8 is 9.5. Middle number?
Solution: Sum (8+8) = (52 + 76) = 128. Total (15) = 105. Middle = 128 - 105 = 23.
10. Avg of 5 numbers is 27. If one is excluded, avg becomes 25. Excluded number?
Solution: Sum (5) = 135. Sum (4) = 100. Excluded = 135 - 100 = 35.
11. Avg marks of 28 students is 50. 8 leave, avg increases by 5. Avg of those who left?
Solution: Total = 1400. New Total (20×55) = 1100. Sum left = 300. Avg = 300 / 8 = 37.5.
12. Avg of 50 numbers is 38. If 45 and 55 are discarded, avg of remaining?
Solution: Total = 1900. New Sum = 1900 - 100 = 1800. New Avg = 1800 / 48 = 37.5.
13. Avg of 14 girls + teacher is 15. Teacher excluded, avg drops by 1. Teacher's age?
Solution: Total = 225. Girls only = 14 × 14 = 196. Teacher = 225 - 196 = 29.
14. Avg weight of 25 students is 50 kg. Teacher included, avg increases by 1 kg. Teacher's weight?
Solution: New Total = 26 × 51 = 1326. Old = 1250. Teacher = 1326 - 1250 = 76 kg.
15. Avg age of 4 brothers is 12. Mother included, avg increases by 5. Mother's age?
Solution: Total (5) = 5 × 17 = 85. Sum (4) = 48. Mother = 85 - 48 = 37 years.
16. Mean of 20 observations was 75; 97 was misread as 79. Correct mean?
Solution: Difference = 18. Mean increase = 18/20 = 0.9. Correct mean = 75.9.
17. Cricketer avg for 8 innings increases by 9 runs after scoring 100 in 9th. New average?
Solution: Let x be old avg. (8x + 100)/9 = x+9. 8x + 100 = 9x + 81. x=19. New avg = 28.
18. Batsman scores 90 in 12th innings; avg decreases by 5. New average?
Solution: Let x be old avg. 11x + 90 = 12(x-5). 11x + 90 = 12x - 60. x=150. New avg = 145.
19. Avg age father & mother is 35. With only son, avg is 27. Son's age?
Solution: Sum (3) = 81. Sum (2) = 70. Son = 81 - 70 = 11.
20. Avg of 8 students is 48. Add four (44 kg) and four (58 kg). Find avg of 16 students?
Solution: Sum = (8×48) + (4×44) + (4×58) = 384 + 176 + 232 = 792. Avg = 792/16 = 49.5.
