### Important Aptitude Formulas on Average:

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)/2
For 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 = 6
Using 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, x, x, x… 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 + 8
or,       10x + 80 = 9x + 100
or,       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 = 308
or,      2x = 48
or,        x = 24

Age of the captain = (x + 11) = (24 + 11) = 35 years.

SOLVED EXAMPLE
PART 1
Question No. 01
The average age of 30 boys of a class is equal to 14 years. When the age of the class teacher is included the average becomes 15 years. Find the age of the class teacher.
(A) 30 years
(B) 35 years
(C) 45 years
(D) 52 years

Explanation:
Total age of 30 boys = 14 × 30 = 420 years
Total age when the teacher is included = 15 × 31 = 465 years
∴ Age of the class teacher = 465 - 420 = 45 years
Alternate Method: Direct Formula
Age of new entrant = New average + (No. of old members × change in average)
= 15 + 30 (15 - 14) = 45 years.
Question No. 02
The captain of a cricket team of 11 members is 26 years old and the wicket keeper is 3 years older. If the ages of these two are excluded, the average age of the remaining players is one year less than the average age of the whole team. What is the average age of the team?
(A) 23 years
(B) 24 years
(C) 25 years
(D) None of these

Explanation:
Let the average age of the whole team by x years.
∴ 11x - (26 + 29) = 9(x -1)
=> 11x - 9x = 46
=> 2x = 46
=> x = 23.
So, average age of the team is 23 years.
Question No. 03
The average age of boys in the class is twice the number of girls in the class. If the ratio of boys and girls in the class of 36 be 5 : 1, what is the total of the ages (in years) of the boys in the class?
(A) 380
(B) 342
(C) 372
(D) 360

Explanation:
Number of boys = (36 × 5/6) = 30 years
Numbers of girls = 6
Average age of boys = (2 × 6) = 12 years
Total age of boys = (30 × 12) years = 360 years.
Question No. 04
A car owner buys petrol at Rs.7.50, Rs. 8 and Rs. 8.50 per litre for three successive years. What approximately is the average cost per litre of petrol if he spends Rs. 4000 each year?
(A) Rs. 7.98
(B) Rs. 8
(C) Rs. 8.50
(D) Rs. 9

Explanation:
Total quantity of petrol consumed in 3 years
= {(4000/7.50) + (4000/8) + (4000/8.50)} litres
= 4000 {(2/15) + (1/8) + (2/17)} litres
= (76,700/51) litres
Total amount spent = Rs. (3 × 4000) = Rs. 12,000
∴ Average cost = Rs. (12,000 × 51)/ 76,700
= Rs. 6120/767
= Rs. 7.98
Question No. 05
In Ankit's opinion, his weight is greater than 65 kg but less than 72 kg. His sister does not agree with Ankit and she thinks that Ankit's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 68 kg. If all are them are correct in their estimation, what is the average of different probable weights of Ankit?
(A) 67 kg.
(B) 68 kg.
(C) 69 kg.
(D) None of these

Explanation:
Let Ankit's weight be x kg.
According to Ankit, 65 < x < 72
According to Ankit's sister, 60 < x < 70
According to Ankit's mother, x ≤ 68
The values satisfying all the above conditions are 66, 67 and 68.
∴ Required average = (66 + 67 + 68)/3 = 201/3 = 67 kg.
Question No. 06
Indian shooting team of 8 persons joins in a shooting competition. One of them scored 85 points. If he had scored 92 points, the average score for the team would have been 84. The numbers of points, the team scored was
(A) 672
(B) 665
(C) 645
(D) 588

Explanation:
Let the total score be x.
Therefore, (x + 92 - 85)/8 = 84
=> (x + 7) = 672
=> x = 665
Question No. 07
If the average marks of three batches of 55, 60 and 45 students respectively is 50, 55, 60, then the average marks of all the students is:
(A) 53.33
(B) 54.68
(C) 55
(D) None of these

Explanation:
Required average = {(55 × 50) + (60 × 55) + (45 × 60)} / (55 + 60 + 45)
= (2750 + 3300 + 2700)/160
= 8750/160
= 54.68
Question No. 08
Three years ago, the average age of Raju, Ranjan and Ravi was 27 years and that of Ranjan and Ravi, 5 years ago was 20 years. Raju’s present age is
(A) 30 years
(B) 35 years
(C) 40 years
(D) 48 years

Explanation:
Present age of (Raju + Ranjan + Ravi) = {(27 × 3) + (3 × 3)} = 90 years.
Present age of (Ranjan + Ravi) = {(20 × 2) + (2 × 5)} = 50 years.
Therefore, Raju’s present age = (90 - 50) = 40 years.
Question No. 09
The average monthly income of ‘P’ and ‘Q’ is Rs. 5050. The average monthly income of ‘Q’ and ‘R’ is Rs. 6250 and the average monthly income of ‘P’ and ‘R’ is Rs. 5200. The monthly income of ‘P’ is:
(A) 3500
(B) 4000
(C) 4050
(D) 5000

Explanation:
Let P, Q and R represent their respective monthly incomes. Then, we have:
P + Q = (5050 × 2) = 10,100 …..... (i)
Q + R = (6250 × 2) = 12,500 …….. (ii)
P + R = (5200 × 2) = 10,400 …….. (iii)
Adding (i), (ii) and (iii), we get:  2(P + Q + R) = 33,000    or   P + Q + R = 16,500 …..... (iv)
Subtracting (ii) from (iv), we get P = 4000.
∴ P's monthly income = Rs. 4000
Question No. 10
The average of 5 consecutive numbers is ‘n’. If the next two numbers are also included, the average will
(A) Increase by 1
(B) Remain same
(C) Increase by 1.4
(D) Increase by 2

Explanation:
Let, 5 consecutive numbers be, x, x+1, x+2, x+3 and x+4
Their average is = (5x + 10)/5 = (x + 2)
The next two numbers are = (x + 5) and (x + 6)
Therefore, Average of the numbers = [(5x + 10) + (x + 5) + (x + 6)]/7 = (7x + 21)/7 = (x +3)
So, the average increased by 1.

PART 2
Question No. 01
A cricketer, whose bowling average is 12.4, takes 5 wickets for 26 runs and thereby decreases his average by 0.4. The number of wickets taken by him before his last match is
(A) 85
(B) 82
(C) 84
(D) 83

Explanation:
Let the number of wickets taken before the last match be x.
Then, (12.4x + 26)/(x + 5) = 12
=> 12.4x + 26 = 12x + 60
=> 0.4x = 34
=> x = 34/0.4 = 340/4 = 85
Question No. 02
The average weight of men ‘A’, ‘B’ and ‘C’ is 84 kg. Another man ‘D’ joints the group and the average now becomes 80 kg. If another man ‘E’, whose weight is 3 kg more than that of ‘D’, replaces ‘A’, then the average weight of ‘B’ ‘C’ ‘D’ and ‘E’ becomes 79 kg. The weight of ‘A’ is
(A) 70 kg
(B) 72 kg
(C) 75 kg
(D) 80 kg

Explanation:
A + b + C = (84 × 3) = 252 kg
A + b + C + D = (80 × 4) = 320 kg.
Therefore, D = (320 - 252) kg = 68 kg.
Given, E = (68 + 3) kg = 71 kg.
Therefore, B + C + D + E = (79 × 4) = 316 kg.
Now, (A + b + C + D) - (B + C + D + E) = (320 - 316) kg = 4 kg.
So, A - E = 4 kg.
=> A = 4 + E = 75 kg.
Question No. 03
The average weight of 16 boys in a class is 50.25 kg and that of the remaining 8 boys is 45.15 kg. Find the average weights of all the boys in the class.
(A) 47.55 kg
(B) 48 kg
(C) 48.55 kg
(D) 49.25 kg

Explanation:
Required average = {(50.25 × 16) + (45.15 × 8)} / (16 + 8)
= (804 + 361.20)/24
= 1165.20/24
= 48.55
Question No. 04
The average age of a class is 15.8 years. The average age of the boys in the class is 16.4 years while that of girls is 15.4 years. What is the ratio of boys to girls in the class?
(A) 1 : 2
(B) 3 : 4
(C) 3 : 5
(D) 2 : 3

Explanation:
Let the ratio be k : 1
Then, (k × 16.4) + (1 × 15.4) = (k + 1) × 15.8
Or, (16.4 - 15.8) k = (15.8 - 15.4)
Or, k = 0.4/0.6 = 2/3
Therefore, the required ratio = 2/3 : 1 = 2 : 3
Question No. 05
The average age of husband, wife and their child 3 years ago was 27 years and that of wife and the child 5 years ago was 20 years. The present age of the husband is:
(A) 35 years
(B) 40 years
(C) 50 years
(D) None of these

Explanation:
Sum of the present ages of husband, wife and child = {(27 × 3) + (3 × 3)} years = 90 years.
Sum of the present ages of wife and child = {(20 × 2) + (5 × 2)} years = 50 years.
∴ Husband's present age = (90 - 50) years = 40 years.
Question No. 06
The average weight of A, B and C is 45 kg. If the average weight of ‘A’ and ‘B’ be 40 kg and that of ‘B’ and ‘C’ be 43 kg, then the weight of ‘B’ is:
(A) 17 kg
(B) 20 kg
(C) 26 kg
(D) 31 kg

Explanation:
Let A, B, C represent their respective weights. Then, we have:
A + B + C = (45 × 3) = 135 …...... (i)
A + B = (40 × 2) = 80 ......... (ii)
B + C = (43 × 2) = 86 .........(iii)
Adding (ii) and (iii), we get: A + 2B + C = 166 ........ (iv)
Subtracting (i) from (iv), we get: B = 31.
∴ B's weight = 31 kg.
Question No. 07
Mr. 'A' travels to Mr. 'B' 150 km away at an average speed of 55 km per hour and returns at 45 km per hour. His average speed for the whole journey in km per hour is
(A) 52.5
(B) 48.5
(C) 49.5
(D) 47.5

Explanation:
Average speed = 2xy/(x + y)
= (2 × 55 × 45)/(55 + 45) km/hr
= 49.5 km/hr.
Question No. 08
The average of 20 numbers is zero. Of them, at the most, how many may be greater than zero?
(A) 0
(B) 1
(C) 10
(D) 19

Explanation:
Average of 20 numbers = 0.
∴ Sum of 20 numbers (0 × 20) = 0.
It is quite possible that 19 of these numbers may be positive and if their sum is a then 20th number is (-a).
Question No. 09
A pupil's marks were wrongly entered as 83 instead of 63. Due to that the average marks for the class got increased by half (1/2). The number of pupils in the class is:
(A) 10
(B) 20
(C) 40
(D) 73

Explanation:
Let there be x pupils in the class
Total increase in marks = (x × ½) = x/2
x/2= (83 - 63)
=> x/2= 20
=> x = 40
Question No. 10
A family consists of two grandparents, two parents and three grandchildren. The average age of the grandparents is 67 years, that of the parents is 35 years and that of the grandchildren is 6 years. What is the average age of the family?
(A) 28 4/7
(B) 31 5/7
(C) 32 1/7
(D) None of these