# Average - Aptitude test, questions, shortcuts, solved example videos

Video on Average - shortcuts, tips and tricks

## Average

Important terms:

Average is defined as the ratio of sum of all terms in a group to the number of items in the group.

Important Formulae:

1) If number of quantities/items/members etc. and their total quantity are given, then we can find the average by using formula shown below. Sometimes, averages and total quantities are mentioned in the question and we have to determine number of quantities.

Hence, remember this formula can be used in number of ways to solve the numericals on ages/weights/ marks/ etc.

 Average = Sum of Quantities Number of Quantities

2) A person travels a distance at a speed of V1 km/hr and same distance at a speed of V2 km/hr. His average speed in the whole journey can be determined using the formula shown below:
 Average Speed = (2 V1 V2) (V1 + V2)

3) There are two batches A and B in a class. If we have to find the average of the whole class use the formula shown below:

Batch A: Number of students = a
Average of batch A = x

Batch B: Number of Students = b
Average of batch B = y
 Average of whole class (Batch 1 and Batch 2) = (ax + by) (a + b)

Quick Tips and Tricks:

1) Average of n natural numbers = (n + 1) / 2

2) Average of even numbers = (n + 1)

3) If value of each term increases/decreases by x, then the average of the group also increases/decreases by a.

4) If we know average of two groups individually, then the average of combined group cannot be determined.

5) In Arithmetic Progression, if number of terms are

i) Odd – Average is the middle term.

ii) Even – Average is the average of two middle terms.

6) Numericals on Age:

i) When new member is added in the group/family.

Case A: If average age increases
Age of new member added = Given previous average + ( Increase in average after new member is added x Total members including new member)

Case B: If average age decreases
Age of new member added = Given previous average - ( Decrease in average after new member is added x Total members including new member)

ii) If a person is replaced by another person in a group

Case A: If average age is increased, then
Age of new member = Age of separated member + (Increase in average after new member is added x Total members including new member)

Case B: If average age is decreased, then
Age of new member added = Age of separated member – (Decrease in average after new member is added x Total members including new member)

Type 1 : Average of Numbers

Examples:

Q 1. Find the average of all numbers between 5 and 49 which are divisible by 5.

a. 20
b. 25
c. 30
d. 35
View solution

Correct Option: (b)

The numbers divisible by 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45.

 Average = Sum of Quantities = (5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45) = 225 = 25 Number of Quantities 9 9

Q 2. The average of 11 numbers is 30. If the average of first six numbers is 17.5 and that of last six is 42.5, then what is the sixth number?

a. 30
b. 36
c. 45
d. 47
View solution

Correct Option : (a)

Given: Average of 11 numbers = 30

Step 1: Calculate total of 11 numbers by multiplying it by average value 30 = 11 x 30 = 330
Step 2: Calculate total of first six members by multiplying it by average value 17.5 = 17.5 x 6 = 105
Step 3: Calculate total of last six members by multiplying it by average value 42.5 = 42.5 x 6 = 255

Therefore, we can find sixth number by adding value of first six and last six numbers and subtracting it from the total value of 11 numbers.

Sixth number =(105 + 255)- 330 = 30

Q 3. The average of 15 numbers is 15. If the average of first five numbers is 14 and that of other 9 numbers is 16, then find the middle number.

a. 12
b. 11
c. 10
d. 9
View solution

Correct Option: (b)

Given: Average of 15 numbers = 15, Average of 5 numbers = 14, Average of 9 numbers = 16

 Average = Total Numbers No. of Numbers

 15 = Total Numbers 15

Therefore, total numbers = 15 x 15 = 225

Middle number = (Total numbers) – [(Average of 5 num x no of num) + ( Average of 9 num x no of num)]
= (225) – [(14 x 5) + (16 x 9)]
= (225) – 
= 11

Therefore, the middle number is 11

Q 4. The average of four consecutive even numbers is 27. Find the largest of these numbers.

a. 28
b. 30
c. 32
d. 34
View solution

Correct Option: (b)

Consider the consecutive even numbers as : x, (x + 2), (x + 4) and (x+ 6)

 Average = Sum of Quantities Number of Quantities

 = x + (x + 2) + (x + 4) + (x + 6) = (4x + 12) = 27 4 4

Simplifying we get, x = 24

Therefore,
Largest number = (x + 6) = (24 + 6) = 30
Smallest number = 24

Type 2: Average of weight / age / marks of two or more groups / classes.

Examples:

Q 5. There are two batches A and B of a class. Batch A consists of 36 students and batch B consists of 44 students. Find the average weight of whole class, if average weight of batch A is 40 kg and that of batch B is 35 kg.

a. 29.23 kg
b. 32.56 kg
c. 35.66 kg
d. 37.25 kg
View solution

Correct Option : (d)

Given: Average weight of batch A = 40 kg , average weight of batch B = 35 kg

1) First find the total weight of all students
- Weight of batch A = (36 x 40) = 1440
- Weight of batch B = (44 x 35) = 1540

Total weight of all students = (1440 + 1540) = 2980 kg

2) Find average weight of whole class
(Batch A + Batch B) students = (36 + 44) = 80 students

 Average Weight = Total weight of all the students = 2980 = 37.25 kg No. of Students 80

Q 6. In a school, average marks of three batches of 40, 50 and 60 students respectively is 45, 55 and 70. Find the average marks of all the students.

a. 54.78
b. 55.23
c. 50.36
d. 58.33
View solution

Correct Option: (d)

We know,

 Average = Sum of Quantities Number of Quantities

Here,
Number of quantities = Number of students in each batch

As average marks of students are given, calculate total marks of each batch first. So total marks for
Batch 1 = (40 x 45) = 1800
Batch 2 = (50 x 55) = 2750
Batch 3 = (60 x 70) = 4200

Sum of marks = (1800 + 2750 + 4200) = 8750

Therefore,
 Required Average = (Sum of Works) = (8750) = 58.33 (Total No. of Students in each batch) (40 + 50 + 60)

Type 3: Change in average when one entry is added/replaced.

Examples:

Q 7. The average age of a class of 29 students is 20 years. If the age of teacher is included, then the average increases by 3 months. Find the age of the teacher.

a. 25. 2 years
b. 27.5 years
c. 29 years
d. 31.5 years

View solution

Correct Option: (b)

 Average = Sum of Quantities Number of Quantities

1) First calculate total age of 40 students

Total age of 29 students = ( Average age x No. of students)
= (20 x 29) = 580 years
 2) Average age of 29 students + 1 teacher = 20 years + 3 months = 81 years 4

 3) Finally, total age of 29 students + 1 teacher = 81 x 30 = 607.5 years 4

Therefore, age of teacher = (Total age of 30 members - Total age of 29 students) = (607.5 – 580) = 27.5 years

Q 8. 2 years ago, the average age of a family of 5 members was 16 years. After a baby is born, the average age of family is the same today. Find the present age of the baby.

a. 4 years
b. 6 years
c. 8 years
d. 8 ½ years
View solution

Correct Option: (b)

Hint: (1) First find total age of 5 members 2 years ago (2) Present age of 5 members (3) Total age of 6 members (4) Age of baby = Total age of 6 members - Present age of 5 members

We know that,

 Average = Sum of Quantities Number of Quantities

1) First calculate total age of 5 members 2 years ago = (Average age of 5 members x number of members)
First calculate total age of 5 members 2 years ago = (16 x 5) = 80 years

2) Calculate the present age of 5 members
2 years ago, their total age was 80 years. Present age can be calculated as follows:
Present age of 5 members = [80 + (2 x 5)] = 90 years

3) Calculate total age of 6 members considering baby = (16 x 6 ) = 96 years

4) Age of baby = (96 – 90) = 6 years

Type 4 : Change in average when one entry is entered wrong.

Examples:

Q 9. John's marks were wrongly entered as 83 instead of 63. If the average marks calculated for the whole class increased by half, then find the number of students in the class.

a. 30
b. 35
c. 40
d. 45
View solution

Correct Option: (c)

Assume number of students in the class be x

As the average increases by half, find the total increase in marks for x students.
Total increase in marks = (x) x (1/2) = x/2

Therefore,
Total increase in marks = False value – true value
x/2 = 83 – 63
x = 40 students

Alternate solution:

Let A be average and x be number of students

1st entry

A+ 0.5= 83 / x ------- (1)

2nd entry

A = 63 / x --------- (2)

From (1) and (2), we get

 A + 0.5 – A = 83 – 63 x x

 0.5 = 83 – 63 x

Therefore, x = 40

Q 10. The mean of 40 observations was 46. Later on it was found that an observation 38 was wrongly taken as 33. find the corrected value of mean.

a. 40.23
b. 42.36
c. 46.12
d. 51.23
View solution

Correct Option: (c)

 Average = Sum of Quantities Number of Quantities

1) Sum of observations = Average x No. of observations

= 46 x 40 = 1840

2) Correct sum = Sum of observations + (38 – 33)

= 1840 + (5)
= 1845
 Corrected Mean Value = Corrected Sum = 1845 = 46.125 No. of Observations 40

Type 5 : Average speed

Examples:

Q 11. A person covers a distance of 60 km from P to Q at a speed of 20 km/hr and returns from Q to P at a speed of 30 km/hr. Find the average speed of person.

a. 22 km/hr
b. 24 km/hr
c. 26 km/hr
d. 28.2 km/hr
View solution

Correct Option: (b)

Hint:

 Average Speed = (2 V1 V2) (V1 + V2)

V1 and V2 are the speeds at which the person travels.

We are given, that person travels P to Q at a speed of 20 km/hr and Q to P at a speed of 30 km/hr.
V1 = 20 km/hr and V2 = 30 km/hr

Therefore,
 Average Speed = (2 x 20 x 30) = 1200 = 24 km/hr (20 + 30) 50

Q 12. An express train runs at an average speed of 27 km/hr including the time of stoppage at stations. Another train runs at an average speed of 41 km/hr excluding the stoppage time at stations. Find how many minutes does a train stop in 1 hour.

a. 20.52 min
b. 15.23 min
c. 12.50 min
d. 10.75 min
View solution

Correct Option: (a)

Train 1: Travels at an average speed of 27 km/hr
Train 2: Travels at an average speed of 41 km/hr

Therefore, train 1 lags train 2 by (41 – 27) km i.e. 14 km.

Now, we have to find the time, train 2 stops in 1 hour.

We know, Speed = Distance/ Time

We know, Distance = 14 km, speed = 41 km/hr

Therefore, Time = Distance / Speed

= 14 / 41 = 0.342 hr

Answer is in minutes, hence multiply by 60
0.342 hr = 0.342 x 60 = 20.52 min

Type 6: Cricket/Scores/Innings

Examples:

Q 13. A batsman makes a score of 80 runs in the 16th inning and increases average by 3. What is his average after 16th inning?

a. 35
b. 32
c. 29
d. 25
View solution

Correct Option:(a)

Lets assume that the average after 16th inning be X.
The average after 15th inning = ( X – 3)

Therefore,

 Average = Sum of Scores Number of Innings

Sum of scores = Average x No. of innings

Total runs made after 15th inning at average (X – 3) = 15 (X – 3)
Total runs made in 16th inning = 80
Total runs made after 16th inning at average X = 16X

Therefore,
(Total runs after 15th inning) + (Total runs in 16th inning) = (Total runs after 16th inning)

15 (X – 3) + 80 = 16 X
Solving we get,
X = 35

Q 14. In a cricket match, 6 players had an average X of their runs. Average increases by 10 runs, if seventh player makes a score of 112 runs. What is the average of first 6 players.

a. 36
b. 39
c. 40
d. 42
View solution

Correct Option: (d)

The average of 6 players = X

Average increases by 10, when seventh player makes a score of 112 runs.

Therefore, average of 7 players = X + 10

 Average = Sum of Scores ------- (1) Number of Players

Here, average = X, number of players = 6
Hence,
Sum of scores = 6X

Score of 7 players = (Score of 6 players + score of 7 player) = (6X + 112) ------- (2)
Total average = (X + 10) --------- (3)

Substitute (2) and (3), in (1)
 (X + 10) (6X + 112) 7

Solving we get,

X = 42

Average of first 6 players = 42

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