Average Maths formulaes and shortcut methods
Important Average Formulaes:
1. Average or Arithmetic mean of n quantities =Sum of n quantities/n Here, n = Number of quantities.
2. Average of n consecutive natural numbers = (n + 1)/2{Total of n consecutive natural number = n (n + 1)/2For e.g. mathematically, 1 + 2 + 3 +…………. + 20 = 20 (20 + 1)/2 = (20 21)/2 = 210
Using formula; Here, n = 20, Therefore, Total = 20 (20 + 1)/2 = 210}.
3. Average of squares of n consecutive natural numbers = (n + 1) (2n + 1)/6
4. Average cubes of n consecutive natural numbers = n (n + 1)²/4
5. Average of n consecutive even numbers = (n + 1){ For e.g. mathematically, 2 + 4 + 6 + 8 + 10 + 12 = 42/6 = 7
Using formula; Here, n = 6, Therefore, average = (n + 1) = (6 + 1) = 7 }
6. Average of squares of n consecutive even numbers = 2 (n + 1) (2n + 1)/3
7. Average of n consecutive odd numbers = n{ For e.g. mathematically, 1 + 3 + 5 + 7 + 9 + 11 = 36/6 = 6Using formula; Here, n = 6, Therefore, average is n = 6 }
8. Average speed = distance traveled/Time taken
9. Average of two different speeds ‘x’ and ‘y’ to travel same distance = 2xy/(x + y){ Suppose a man covers a certain distance at x kmph and an equal distance at y kmph, then the average speed during the whole journey is 2xy/(x + y) kmph. }
10. Average of three different speeds x, y, z to travel equal distance = 3xyz/(xy + yz + zx)
11. If a person is replace by another person by which;
(I) Average is increased, then Age of new comer = Age of person left + no. of persons × increase in average age.
(II) Average is decreased, then Age of new comer = Age of person left – no. of persons × decrease in average age
12. If a person joins a group without replacing any person by which;
(I) Average increased, then Age of new comer = Previous average age + no. of persons (including new comer) × increase in average age.
Or, Age of new comer = Increased average age + no. of persons originally in the group × increase in average age.
(II) Average decreased, then Age of new comer = Previous average age – no. of persons (including new comer) × decrease in average age.
13. If a person leaves the group but nobody joins the group by which;
(I) Average increased, then Age of man left = Previous average age + no. of present persons × increase in average age.
(II) Average decreased, then Age of man left = Previous average age + no. of present persons × decrease in average age.
14. If the average marks obtained by ‘x’ candidates in an examination is ‘n’. If the average mark of passed candidates is ‘p’ and that of failed candidates is ‘q’. Then
(I) No. of passed candidates: = [Total candidates (Total average. – Failed average.)]/(Passed average – Failed Average)
(II) No. of failed candidates: = [Total candidates (Passed average. – Total average.)]/(Passed average – Failed Average)
15. The geometric mean of numbers, x₁, x₂, x₃… xn is given by,
Example: 01-The average of 5 numbers is 7. When 3 new numbers are added, the average of the 8 numbers is 8.5. The average of the 3 new numbers is:
Solution:Sum of 3 new numbers = (8 ×8.5) – (5 × 7) = 33∴ Their average = 33/3 = 11
Example: 02-A cricketer has a certain average for 9 innings. In the tenth inning, he scores 100 runs, thereby increasing his average by 8 runs. His new average is:
Solution:Let, average of 9 innings be x
Then, (9x + 100)/10 = x + 8or, 10x + 80 = 9x + 100or, x = 20∴ The new average is = (x + 8) = 28 runs.
Example: 03-In a cricket team, the average of 11 players is 28 years. Out of these, the average ages of 3 groups of 3 players each are 25 years, 28 years and 30 years respectively. If in these groups, the captain and the youngest player are not included and the captain is 11 years older than the youngest player. What is the age of the captain?
Solution:Let the age of youngest player be x
Then, (3 × 25) + (3 × 28) + (3 × 30) + x + x + 11 = 11 × 28 or, 75 + 84 + 90 + 2x + 11 = 308or, 2x = 48or, x = 24
∴ Age of the captain = (x + 11) = (24 + 11) = 35 years.
