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Average concept

Average concept

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Average

 
An average or an arithmetic mean of given data is the sum of the given observations divided by number of observations.
 
Important Formulae Related to Average of numbers
 
1. Average of first n natural number=(n+1)/2

 

2. Average of first n even number= (n+1)

 
3. Average of first n odd number= n
 
4. Average of consecutive number= (Firtst number+Last number)/2
 
5. Average of 1 to n odd numbers=  (Last odd number+1)/2
 
6. Average of 1 to n even numbers=  (Last even number+2)/2
 
7. Average of squares of first n natural numbers=[(n+1)(2n+1)]/6
 
8. Average of the cubes of first n natural number=[n(n+1)^2]/4
 
9. Average of n multiples of any number=[Number*(n+1)]/2
 
 
Concept 1 
 
If the average of n_1 observations is a_1; the average of n_2 observations is a_2 and so on, then
Average of all the observations=(n_1* a_1+n_2 *a_2+......)/(n_1+n_2+........)
 
Concept 2 
 
If the average of m observations is a and the average of n observations taken out of  is b, then
Average of rest of the observations=(ma-nb)/(m-n)
 
Example 1 : A man bought 20 cows in RS. 200000. If the average cost of 12 cows is Rs. 12500, then what will be the average cot of remaining cows?
 
Here m = 20 , n = 12 , a = 10000 , b = 12500 
 
average cost of remaining cows ( 20-8) cows = (20*10000 - 12*12500)/(20-8) =Rs  6250
 
Concept 3 
 
If the average of n students in a class is a, where average of passed students is x and average of failed students is y, then
Number of students passed=[Total Students (Total average-Average of failed students)]/(Average of passed students-Average of failed students)
=[n(a-y)]/(x-y)
 
Example 2: In a class, there are 75 students are their average marks in the annual examination is 35. If the average marks of passed students is 55 and average marks of failed students is 30, then find out the number of students who failed.
 
Here , n = 75 , a = 35 , x = 55 , y = 30 
Number of students who passed = 75(35- 30)/(55- 30) = 15
Number of students who failed  = 75- 15 = 60
 
Concept 4
 
If the average of total components in a group is a, where average of n components (1st part) is b and average of remaining components (2nd part) is c, then Number of remaining components (2nd part)=[n(a-b)]/(c-a)
 
Example 3 : The average salary of the entire staff in an offfice is Rs. 200 per day. The average salary of officers is Rs. 550 and that of non-officers is Rs. 120. If the number of officers is 16, then find the numbers of non-officers in the office.
 
Here n= 16 , a = 200 , b = 550 , c = 120 
 
 
Number of non - officer = 16(200- 550)/(120- 200) = 70 
 
Average Speed
Average speed is defined as total distance travelled divided by total time taken. 
Average speed=Total distance travelled/ Total time taken
 
Case 1
 
If a person covers a certain distance at a speed of A km/h and again covers the same distance at a speed of B km/h, then the average speed during the whole journey 
will be 
2AB/A+B
 
Case II
 
If a person covers three equal distances at the speed of A km/h, B km/h and C km/h respectively, then the average speed during the whole Journey will be 
3ABC/(AB+BC+CA)
 
Case III
 
If distance P is covered with speed x, distance Q is covered with speed y and distance R is covered with speed z, then for the whole journey,
Average speed=(P+Q+R+.....)/(P/x+Q/y+R/z+...)
 
Example 4 : A person covers 20 km distance with a speed of 5 km/h, then he covers the next 15 km with a speed of 3 km/h and the last 10 km is covered by him with a speed of 2 km/h. Find out his average speed for the whole journey. 
 
 Average speed  =  ( 20 +15 +10)/(20/5+15/3+10/2) = 3(3/14)
 
Case IV
 
If a person covers P part of his total distance with speed of x, Q part of total distance with speed of y and R part of total distance with speed of z,then
Average speed=1/(P/x+Q/y+R/z+......)
 

Tricks and Rules of Average in Details

 

Averages = (sum of all terms)/ number of terms


Average is the estimation of the middle number of any series of numbers. 

For example average of 1,2,3,4, 5 is 3.
Average can be calculated by sum of all numbers divided by the total number of numbers
Average of  1,2,3,4,5= (1+2+3+4+5)/5 = 15/5 = 3
Which is also the middle number of the series , from here we can also say that in an A.P. i.e arithmetic progression the middle term is the average of the series .



Rule 1: In the Arithmetic Progression there are two cases when the number of terms is odd and second one is when number of terms is even.

So when the number of terms is odd the average will be the middle term.
And when the number of terms is even then the average will be the average of two middle terms.

Examples 1: what will be the average of 13, 14, 15, 16, 17?  
Solution: Average is the middle term when the number of terms is odd, but before that let’s checks whether it is in A.P or not, since the common difference is same so the series is in A.P.

So the middle term is 15 which is our average of the series.

Let’s check it in another way.
In the first statement of the article we have written that the average of a set of terms is equal to:
Sum of all terms / Number of terms
So the sum of all terms in this case is 75 and the number of terms is 5 so the average is 15.

Now come to the second form when the number of terms are even

Example 2: What will be the average of 13, 14, 15, 16, 17, 18?
Solution: We have discussed that when the number of terms are even then the average will be the average of two middle terms.

Now the two middle terms are 15 and 16, but before that the average we must check that the series should be A.P. Since the common difference is same for each of the term we can say that the series is in A.P.
And the average is (16+15)/2 = 15.5

Rule 2: The average of the series which is in A.P. can be calculated by ½(first + last term) Example 1:  What will be the average of 216, 217 , 218?
Solution: So the answer would be = ½ (216 + 218) = 217
(Which is also the middle term of the series)

Example 2: 
What will be the average of first 10 natural numbers?
Solution: The first 10 natural numbers are 1,2,3,4,5,6,7,8,9,10
So the average will be ½ (1 + 10 ) = ½ (11)  = 5.5

Rule 3: If the average of n numbers is A and if we add x to each term then the new average will be = (A+ x).

For example: The average of 5 numbers is18. If 4 is added to each of the number then the average would be equal to __?
Solution: Old average = 18
New average will be = 4 + old average = 22
This is because each term is increased by 4 so the average would also be increased by 4 so the new average will be 22

Rule 4: If the average of n numbers is A and if we multiply p with each term then the new average will be = (A x p). For Example: The average of 5 numbers is 18. If 4 is multiplied to each of the number then the average would be equal to __?  
Solution: Old average = 18
New average will be = 4 x 18= 72

There are two more operation which can also be applied on the same principle as the above, i.e. subtraction and division.

Rule 5 : In some cases, if a number is included in the series of numbers then the average will change and the value of the newly added term will be = Given average + (number of new terms  x increase in average).

This value will also same as the New average + (number of previous terms  x increase in average ) .

For example: The average age of 12 students is 40. If the age of the teacher also included then the average becomes 44. Then what will be the age of the teacher?

Solution: Average given = 40
Number of students = 12
Therefore the age of the teacher = 40 + (12 + 1) x 4 = 40 + 52 = 92
And this is also calculated as 44 + (12 x 4)= 92
Therefore the average age of the teacher is 92 yrs

Alternatively 
The average of 12 = 40 that means the total number of units are 12 x 40 = 480
Now the new average is 44 and the number of terms are 13 so therefore the total number of units are = 44 x 13 = 572
So the included units would be equal to 572 – 480 = 92

Rule 6:  In some cases  a number is excluded and one more number is added in the series of the number then the average will change by q and the value of the newly added term will be = Replaced Term + (increased in average x number of terms ).

For example: The average age of 6 students is increased by 2years when one student whose age was 13 years replaced by a new boy then find the age of the new boy

Solution: The age of the boy will be = Age of the replaced boy +increase in average x number of terms
i.e. the age of the newly added boy = 13 + 2 x 6 = 25

Rule 7: There are two more cases when the series is divided into two parts and one of the terms is either included or excluded, then the middle term can be calculated by following methods.

Case 1 : When the term is excluded.
Average(total ) + number of terms in first part x {average (total) – average (first part)} + number of terms in second part x {average (total) – average (second part)}

Case 2: When the term is included.
Average (total) + number of terms in first part x {average (first part) – average(total) }+ x number of terms in second part x {average (second part) – average (total)}

For Example: The average of 20 numbers is 12 .The averages of the first 12 is 11 and the average of next 7 numbers is 10. The last number will be?
Solution:
Here in this case one number is excluded so the number would be =
Average(total ) + number of terms in first part x {average (total) – average (first part)} + number of terms in second part x {average (total) – average (second part)} i.e. =  12 + 12 x (12-11)+(12-10) x 7 = 38.