AVERAGES are very important chapter for all competitive exam and everything is included in it like all averages, series, important key, batting ball wicket based on some important qes.
AVERAGE
It is defined as the sum of the observations divided by the number of observations.
Average = sum of the observations / Number of observations
OR,
Average = sum of Quantity / Number of Quantity
* IN SERIES(AP) : -
Average = (1st Number + Last Number) / 2
If number of terms is Odd
Average = Mid. Term
Qes. The average age of a hockey team of six players is 25 years. If two more players are included in the team the average becomes 26 years, then the
Soln.
Average age of six players is = 25 years
=> Sum of age of 6 players = 25 x 6 = 150 years
=> If two players are included in the team, then average age of 8 players = 26 years
=>Sum of age of 8 players = 8 x 26 = 208 years
=> Sum of age of two players (included) = 208-150 = 58 years
=> Average age of two players (included) = 58/2 = 29 years
Qes. 8,12,16,20,..............120
Soln. Difference = 4 ( in AP )
Average = (1st + Last) / 2
= ( 8+120 ) / 2
= 64
Qes. 24,30,36,42,48,54,60
Soln. Total Term = 7
This series is in AP
Average = 42
Important Key
* Average of first n natural numbers = (n+1)/2
OR, 1,2,3,4,5................n = (n+1)/2
* Average of squares of first n natural numbers = (n+1)(2n+1)/6
* Average of cubes of first n natural numbers = n(n+1)^2 /4
* Average of first n even numbers = n+1
OR, 2,4,6............................. = n+1
* Average of squares of first n even numbers = n(n+1)(2n+1)/3
* Average of cube of first n even numbers = 2n(n+1)^2
* Average of first n odd numbers = n
OR, 1,3,5,7....................nTH term = n
* Average of squares of first n odd numbers = (2n+1)(2n-1)/3
* Average of cube of first n odd numbers = n(2n^2 - 1)
Qes. The average of 3 consecutive even numbers is 20, then the third number is by what percent more than the first number?
Soln.
Let the first number be a.
So, the second and the third number will be (a + 2) and (a+4) respectively.
According to the question
a+a+2+a+4=20x3
x=18
The first number = 18
=>The second number = 18+2= 20
=> The third number = 18 + 4 = 22
Required percentage = [(22-18)/18] x 100 = 22.23 percent
In the question, shortcut can be applied : -
Average of any consecutive series is the middle term (median)
example:- 18, 20, 22 in this series middle term is 20 which also average of the series
Required percentage = [(22-18)/18] x 100 = 22.23 percentage
Qes. The average of the first 20 odd numbers is by what percent less/more than the average of the first 20 even numbers?
We know that
Average of the first n odd numbers = n
Average of the first n even numbers = n+1
⇒ Average of the first 20 odd numbers = 20
⇒ Average of the first 20 even numbers = 20 + 1 = 21
Required percentage = [(21-20)/21] x 100 = 4.76 percent
Qes. Find the average of cubes of the first 20 odd numbers.
Soln. The average of cubes of first n odd numbers = n (2n² - 1)
The average of cubes of first 20 odd numbers = 20[2 x (20)^2 - 1] = 20 × 799 = 15980
Concept
Total runs = Number of wickets × Batting Average
Average of bowler = total runs / Number of wickets
Total runs = Batting Average × Match
Batting Average = Runs / Match
NOTE -
Not out match will not be counted
Bowling Average Improve That Means Bowling Average Reduce
Batting Average Improve That Means Batting Average Increase
Qes. A cricketer had a certain average of runs for his 44 innings. In his 45th innings, he is bowled out for no score on his part. This brings down his average by three runs. Find his new average of runs.
Soln
Let the average of runs be a
Total runs = Number of innings a Average
⇒ 44 × a = 45 × (a - 3)
=> a = 135
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