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)

 

  1. Average of first n odd number= n

 

  1. Average of consecutive number= (Firtst number+Last number)/2

 

  1. Average of 1 to n odd numbers=  (Last odd number+1)/2

 

  1. Average of 1 to n even numbers= (Last even number+2)/2

 

  1. Average of squares of first n natural numbers=[(n+1)(2n+1)]/6

 

  1. Average of the cubes of first n natural number=[n(n+1)^2]/4

 

  1. 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+……)
Some Important Concepts:

1.
Average =total of data/No.of data

2.If the value of each item is increase by the same value a, then the average of the group or items will also increase by a.

3.If the value of each item is decreased by the same value a, then the aveage of the group of items will also decrease by a.

4.If the value of each item is multiplied by the same value a, then the average of the group or items will also get multiplied by a.

5.If the value of each item is multiplied by the same value a, then the average of the group or items will also get divided by a.

6.If we know only the average of the two groups individually, we cannot find out the average of the combined group of items.

7.Average of n natural no’s=(n+1)/2

8.Average of even No’=(n+1)

9.Average of odd No’= n

10.General Formula=(1st number +Last number)/2

Quant Quiz on Average

1.The average marks obtained by 120 cndidates in a certain examinationis 35.if the average marks of passed candidate is 39 and that of the failed candidates is 15,what is the number of candidates who passed the examination

(a)110
(b)115
(c)120
(d)100
(e)none of these

2.The average age of 8 persons in a committee is increased by 2 years when two men aged 35 years and 45 years are substituted by two women.Find the average age of these two women.
(a)48
(b)36
(c)42
(d)29
(e)none of these

3.In a class, there are 20 boys whose average age is decreased by 2 months, when one boy aged 18 years is replaced by a new boy.Find the age of new boy.
(a)19 years
(b)14 years 6 month
(c)15 years
(d)14 years 8 month
(e)12 years

4.One -third of certain journey is covered at the rate of 25kmph,one-fourth at the rate of 30kmph and the rest at 50 kmph.Find the average speed for the whole journey.

(a)33.33 kmph
(b)36
(c)42
(d)27
(e)none of these

5.The average salary of the entire staff in a office is Rs 120 per month.The average salary of officers is Rs 460 and of non officers is Rs 110. If the number of officers is 15,then find the No of non-officers in the office.
(a)480
(b)500
(c)600
(d)430
(e)510

6.There were 35 students in a hostel.If the number of students increased by 7, the expenses of the mess increases by Rs. 42 per day,while the average expenditure per head diminishes by Rs.1.Find the original expenditure of the mess.
(a)420
(b)400
(c)480
(d)460
(e)none of these

7.The average weight of A,B,and C is 84 kg. If D joins the group, the average weight of the group becomes 80 kg.If another man E who weights is 3 kg more than D Replaces A, Then the average of B,C,Dand E becomes 79 kg. What is the weight of A?
(a)64
(b)72
(c)75
(d)100
(e)80

8.In an exam, the average was found to be 50 marks. After deducting computational errors the marks of the 100 candidates had to be changed from 90 to 60 each and average came down  to 45 marks. Total No of candidates who took the exam were.
(a)300
(b)600
(c)200
(d)150
(e)none of these

9. Find the average of all even No’s up to 100 or average of first 50 even No’s.
(a)50
(b)50.5
(c)51
(d)49
(e)none of these

10. The average of 8 No’s is 20. The average of first two No’s is 15.5 and that of next three No’s is 64/3.if the 6th No be less than the seventh and eight No. by 4 and 7 respectively, then find eight No.
(a)26
(b)25.5
(c)27
(d)25
(e)none of these

 

Answers with Explanation:

  1. (d) Let No. of passed candidates are x

So, x * 39 + (120 – x) 15= 120 * 35

x = 100

2.(a) Total increase = 8 * 2 = 16 years

So, total age of two women = 35 + 45 +16= 96

Average age of two women=96/2=48 years

3(d) Total decrease = 20 * 2 = 40 month

= 3 years 4 month

So, age of new boy  = 18 years – 3 years 4 month

= 14 years 8 month


4.(a) Average speed = total distance / total time

LCM = 3, 4 = 12

Average speed = 12 / (1/3*12*1/25 + ¼ * 12*1/30 +5/50)

= 12/ (4/25 + 3/30+ 5/30

= 150*12/54 = 33.333 kmph


5.(e) Let no. of non- officers be x

15 * 460 + x * 110 = (x + 15) 120

x = 510


6.(a) Let average expenditure per head be = x

So, (35 + 7) (x -1) – 35 x = 42

42 (x -1) – 35x = 42

X =12

Original expenditure of mess = 12 * 35 = 420


7.A + B + C = 3 *84 = 252

A + B + C + D = 4 * 80 = 320 —- (i)

So, D =   68& E = 68 + 3 = 71

B + C + D +E = 79 * 4 = 316 — (ii)

From Eq. (i) & (ii)

A-E = 320 – 316 = 4

A = E +4 = 71 + 4 = 75


8.(b) Let the total no. of candidates =  x

So, (50x-100 (90-60)) / x = 45

x = 600


9.(c) 2, 4, 6, ………… 100

Average = (1st No. + Last No.) /2 = (2+100)/2 = 51


10.(c) Total Sum of 8 nos.  = 20 * 8 = 160

Sum of 1st two nos. = 2 * 15.5 = 31

Sum of next three nos. = 3 * 64/3 = 64

Let 6th no. be x 

31 +  64 + x + (x+4) + (x+7) = 160

x = 18

8th no. = 18 + 7 = 25

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