This chapter is one of the most important chapters of Basic Numeracy for CSAT exams. From the analysis of Previous Years’ Papers, it is clear that direct questions are not asked from this chapter, but its basics are necessary for Data Interpretation. In the year 2023, 2 questions and in 2022, 1 question was asked and in the years 20192011 approximately 12 questions were asked on the direct application of averages.
Averages can be defined as the central value in a set of data.
Example:
What is the average of the first five consecutive odd numbers?
Solution:
The first five consecutive odd numbers are: 1, 3, 5, 7, 9.
Here, the number of data or observations is 5 and the sum of these 5 numbers is 25.
So, average = 25 / 5 = 5.
When a person leaves a group and another person joins the group at the place of the person left, then:
(i) In case of an increase in average age,
Age of newcomer = Age of person left + (Number of persons in the group × Increase in average age)
(ii) In case of a decrease in average age,
Age of newcomer = Age of person left  (Number of persons in the group × Decrease in average age)
Example: The average age of a group of 10 friends is 10 years. If one of the friends leaves the group but at his place another friend whose age is 22 years joins the group, the average age becomes 8 years. Find the age of the friend who left the group.
(a) 40 years
(b) 42 years
(c) 44 years
(d) 46 years
Solution: (b)
Explanation:
Age of new friend = Age of friend left  [Number of persons in the group × Decrease in average age]
⇒ 22 = Age of friend left  (10 × (10  8))
∴ Age of friend left = 22 + (10 × 2)
= 22 + 20
= 42 years
1. Average Related to Natural Numbers
(i) Average of consecutive natural numbers till n
= (n + 1) / 2
(ii) Average of squares of consecutive natural numbers till n
=
(iii) Average of cubes of consecutive natural numbers till n
Example: Find the average of squares of the first five natural numbers.
(a) 10
(b) 11
(c) 3
(d) 5
Solution: (b)
Explanation:
Average of squares of the first five natural numbers
2. Average Related to Even Numbers
(i) Average of n consecutive even numbers = (n + 1)
(ii) Average of squares of n consecutive even numbers =
(iii) Average of squares of consecutive even numbers till n =
Example: Find the average of squares of consecutive even numbers till 10.
Solution: (b)
Explanation:
Average of squares of consecutive even numbers till 10
3. Average Related to Odd Numbers
(i) Average of n consecutive odd numbers = n
(ii) Average of consecutive odd numbers till n = (n + 1) / 2
Example: Find the average of consecutive odd numbers till 9.
(a) 4
(b) 5
(c) 7
(d)9
Solution: (b)
Explanation:
Average of consecutive odd numbers till 9
= (n + 1) / 2
= (9 + 1) / 2
= 10 / 2 = 5
Example: If half of the journey is traveled at a speed of 15 km/h and the next half at a speed of 12 km/h, find the average speed during the entire journey.
Solution:
Average speed=
Here, x= 15 and y= 12
2. If a person or a motor car covers three equal distances at the speed of x km/h, y km/h, and z km/h respectively, then for the entire journey, the average speed of the person or motor car is
Example: A train covers the first 160 km at a speed of 120 km/h, another 160 km at 140 km/h, and the last 160 km at 80 km/h. Find the average speed of the train for the entire journey.
Solution:
Average speed=
Here, x= 120, y= 140 and z= 80
3. If a person covers A km at a speed of x km/h, B km at a speed of y km/h, and C km at a speed of z km/h, then the average speed during the entire journey is
Example: A person covers 9 km at a speed of 3 km/h, 25 km at a speed of 5 km/h, and 30 km at a speed of 10 km/h. Find the average speed for the entire journey.
Solution:
Average speed=
Here, A= 9, B= 25, C=30, x= 3, y=5, z=10
4. If a person covers Ath part of the distance at x km/h, Bth part of the distance at y km/h, and the remaining Cth part at z km/h, then the average speed during the entire journey is
Example: A train covers 50% of the journey at 30 km/h, 25% of the journey at 25 km/h, and the remaining at 20 km/h. Find the average speed of the train during the entire journey.
Solution:
Average Speed=
Here, A=50, B=25, C=25, x=30, y=25 and z= 20
Note:
Sum of 1st n consecutive natural numbers = [n(n+1)]/2
Average of 1st n consecutive natural numbers = (n+1)/2
Example 1:
The average of a batsman in 16 innings is 36. In the next innings, he is scoring 70 runs. What will be his new average?
Solution:
Conventionally solving:
New average = (old sum+ new score)/(total number of innings) = ((16 ×36)+70)/((16+1)) = 38
Shortcut technique:
Example 2:
The average mark of 19 children in a particular school is 50. When a new student with marks 75 joins the class, what will be the new average of the class?
Solution:
Step 1) Take the difference between the old average and the new marks = 7550=25
Step 2) This score of 25 is distributed over 20 students => 25/20 = 1.25
Step 3) Hence, the average increases by 1.25=> 50+1.25 = 51.25.
Example 3:
The average age of Mr. Mark’s 3 children is 8 years. A new baby is born. Find the average age of all his children?
Solution:
The new age will be 0 years. The difference between the old average and the new age = 08= 8
This age of 8 years is spread over 4 children => (8/4= 2) Hence, the average reduces to 82= 6 years.
Example 1:
The average age of 29 students is 18. If the age of the teacher is also included the average age of the class becomes 18.2. Find the age of the teacher?
Solution:
Conventionally solving:
Let the average age of the teacher = x(29 × 18 + x × 1)/30Solving for x, we get x = 24.
Shortcut Technique:
Using the shortcut, based on the same method used previously:
Step 1: Calculate the change in average = 18.2 – 18 = 0.2.This change in 0.2 is reflected over a sample size of 30.
The new age is increased by 30 × 0.2 = 6 years above the average i.e. 18 + 6 = 24; which is the age of the teacher.
The concept of an assumed mean is not new. It is widely used to reduce the calculation in finding the average in statistics where the data is huge.
Here, We will demonstrate the application of the assumed mean to solve some aptitude questions based on averages and weighted averages.
Let us take an example to understand the concept
Example: In a class of 30 students, the average age is 12 years. If the age of the class teacher is included, the new average age of the class becomes 13 years. Find the age of the class teacher.
Solution:
Standard Approach
Applying the standard approach, the total age of the 30 students = years. When the class teacher is included, the new total age of the class = years. Note that the increase in the total age is because of the class teacher only. Hence the age of the class teacher = 403 – 360 = 43 years
Deviation method
To understand the deviation method, let us simulate the problem. In the case of average age, assume that each student has 12 chocolates with them.
Therefore, he came with 30 + 13 = 43 chocolates. Or the age of the class teacher is 43 years.
The weighted arithmetic mean, usually denoted by
,
where x1, x2, x3, …, xn are averages and w1, w2, w3, .., wn are their respective weightages
Example: In a class of 25 boys and 15 girls, the average heights of the two groups of boys and girls are 150 cm and 140 cm respectively. Find the average height of the class.
Solution:
Standard approach:
Total weight of the group of boys = and for the group of girls, the total weight =Therefore, the average height of the class =
Deviation Approach:
Assume that each boy and each girl is carrying 150 and 140 chocolates respectively.
[Question: 571173]
Q1: The average goal scored by 15 selected players in EPL is 16. The maximum number of goals scored by a player is 20 and the minimum is 12. The goals scored by players are between 12 and 20. What can be the maximum number of players who scored at least 18 goals?
a) 10
b) 5
c) 9
d) 6
e) None of these
Solution: Option (c)
To maximize the number of players who scored 18 and above number of goals, one should assume that only one person has scored 20. To counter him, there will be one person who will score 12 goals.
i.e. 15 – 2 = 13 players left.
Now to maximize the 18 and above goals for every two players who are scoring 18, there will be one player scoring 12. This is done, to arrive at an average of 16. We will have 8 players with a score of 18 and 4 players with a score of 12. The last player will have a score of 16 Thus, the maximum number of people with 18 and more goals = 9.
Q2: The average weight of a group of 8 girls is 50 kg. If 2 girls R and S replace P and Q, the new average weight becomes 48 kg. The weight of P= Weight of Q and the weight of R = Weight of another girl T is included in the group and the new average weight becomes 48 kg. Weight of T= Weight of R. Find the weight of P?
a) 48 kgs
b) 52 kgs
c) 46 kgs
d) 56 kgs
Solution: Option (d)
8 x 50 +R+SPQ= 48×8 R+SPQ=16
P+QRS= 16 R=S and P=Q
PR=8
One more person is included and the weight = 48 kg
Let the weight be a = (48 × 8 + a)9/9 = 48
A = 48 kg= weight of R
=> Weight of P= 48+8= 56 kg.
Q3: The average number of goals scored by 15 selected players in EPL is 16. The maximum number of goals scored by a player is 20 and the minimum is 12. The goals scored by players is between 12 and 20. What can be the maximum number of players who scored at least 18 goals?
a) 10
b) 5
c) 9
d) 6
Solution: Option (c)
To maximize the number of players who scored 18 goals or more, one should assume that only one person has scored 20. To counter him, there will be one person who will score 12 goals.
i.e. 15 – 2 = 13 players left.
Now to maximize the 18 and above goals for every two players who are scoring 18, there will be one player scoring 12. This is done, to arrive at an average of 16. We will have 8 players with a score of 18 and 4 players with a score of 12. The last player will have a score of 16 Thus, the maximum number of people with 18 and more goals = 9.
214 videos139 docs151 tests

1. What is the importance of averages in CSAT? 
2. What are some applications of averages? 
3. How can one find the median and mode of a set of numbers? 
4. What are some important facts to remember about averages? 
5. How can one use the method of deviation to find the average of a set of numbers? 

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