### Importance of Averages

In CAT average is a concept that is covered in Arithmetic topics like mixtures and allegations and it is an important concept since it forms the base of many other topics in the Quantitative ability section. Out of the total questions from arithmetic concepts, average concept forms around 10% of the questions. Other than the Quant section, this topic becomes of use in solving the Data interpretation and Logical reasoning questions also.

CAT

 Averages Year Noof Questions Goodattempt Difficulty 2019 Slot 1 1 1 Easy to Moderate Slot 2 1 1 Easy to Moderate 2018 Slot 1 2 1 to 2 Moderate Slot 2 2 1 Moderate to Difficult 2017 Slot 1 1 1 Moderate Slot 2 1 1 Moderate
CAT 2021 is expected to follow a similar difficulty level and paper pattern like previous years.

### List of Concepts in this Chapter

Averages:

Averages are a simple but effective way to determine the complete group by a single value. The formula for the calculation of averages is:

Average = (Sum of All Items in the group) / (Total number of items in the group)

Points to remember:

• If the value of each item in the group increases or decreases by the value same value ‘P’, the average of the group also increases or decreases by ‘P’ respectively
• If the value of each item in the group is multiplied or divided by the value same value ‘P’ where P is not equal to 0, the average of the group also gets multiplied or divided by ‘P’ respectively
• The average should always be more than the smallest value in the group and less than the largest value in the group
• If there are n items in the group and they are denoted by Q1, Q2, Q3…. Qn, the average of these n items is given by:
• Average = P + 1/n=∑_(i=1)^n(Qi-P)
Weighted Averages:
Out of the topics discussed in this document, this topic finds its way in most CAT questions as it forms the basics of mixtures and allegations along with several other topics
This method comes into use when the two groups are combined and the average that is required is of the two groups combined.
This cannot be done solely based on the average data available for the two groups because the weights in terms of the number of items in each group may vary. Here comes the usage of weighted average as the concept.
Example: If there are 3 sets of people who use the food delivery application Zomato, different times in a week.
40 people use Zomato 2 times in a week, 40 people use Zomato 3 times a week and 20 people use Zomato 4 times a week. If you have to calculate the average frequency of a Zomato user, you will use the weighted average formula. It will be as follows:
Weighted Average = (40*2 + 40*3 + 20*4) / 100
= 2.8
This signifies that on an average, a Zomato user uses the application 2.8 times a week.

### Some Questions from Previous Papers

Some important CAT questions on averages from previous year papers are given below which can help in the preparation of CAT 2021

Averages:

1. Ramesh and Gautam are among 22 students who write an examination. Ramesh scores 82.5. The average score of the 21 students other than Gautam is 62. The average score of all the 22 students is one more than the average score of the 21 students other than Ramesh. The score of Gautam is? (CAT 2019- Slot 1)

Let the score of Gautam =x
Total marks of all the students = 21*62 +x
As per the question,
The average score of all the 22 students is one more than the average score of the 21 students other than Ramesh,
Or (21*62 +x)/22 -1 = (21*62 +x – 82.5)/21
(21*62 + x -22)/22 = (21*62 +x – 82.5)/21 On solving x =51

2. The average of 30 integers is 5. Among these 30 integers, there are exactly 20 which do not exceed 5. What is the highest possible value of the average of these 20 integers? (CAT 2019- Slot 2)

Sum of all 30 integers =5*30 = 150
Minimum possible sum of remaining 10 integers = 10*6 = 60
(as all remaining 10 integers are greater than 5 so their sum will be minimum when they all are equal and each have value =6)
So, some of 20 numbers = 150 – 60 = 90
Thus, required average = 90/20 = 4.5

3. In an apartment complex, the number of people aged 51 years and above is 30 and there are at most 39 people whose ages are below 51 years. The average age of all the people in the apartment complex is 38 years. What is the largest possible average age, in years, of the people whose ages are below 51 years? (CAT 2018- Slot 1)

Let the average age of people aged 51 years and above be x years. Let the average age of people aged below 51 years be y years.

Let the number of people aged below 51 years be N.

Given, the average age of all the people in the apartment complex is 38 years.

Therefore,

(X*30 + y*N)/ 30*N = 38 ….(1)

We want to maximize y, which occurs when x is minimum i.e. for x=51. Substituting the value of x in (1) we get

390=N*(38-y)

Again, when y is maximum, N is also maximum i.e. 39

Therefore, the maximum value of y = 28.

4. A CAT aspirant appears for a certain number of tests. His average score increases by 1 if the first 10 tests are not considered, and decreases by 1 if the last 10 tests are not considered. If his average scores for the first 10 and the last 10 tests are 20 and 30, respectively, then the total number of tests taken by him is? (CAT 2018- Slot 1)

Let the average score of the aspirant in all the tests be x. Let the number of tests be n.
The aspirant’s average score for the first 10 tests and last 10 tests are 20 and 30 respectively.
(Nx-200)/(n-10) = x+1 and
(Nx-300)/(n-10) = x-1
Solving, we get n=60

### How to deal with that topic preparation

For the understanding of the candidate, we have divided the preparation into 3 levels for the upcoming CAT 2021.

Level 1

• If a candidate is in the initial stages of preparation, he should follow the following steps:
• Try and understand the basic concepts and formulas coming under averages and weighted averages as well
• Understand in detail, the differences between each concept and their efficient application
• Do not go to shortcut methods for solving the questions because that might hamper the understanding of the concept

Level 2

• This level is for the students who have cleared their concepts and are looking for faster methods to solve questions
• Try to understand the techniques always applicable in the questions of this concept and short cuts that can be used
• The questions on averages often use the “points to remember” mentioned above, hence, they should be memorized
• At this stage, the student should be comfortable in pointing out where to use average and weighted average respectively

Level 3

• After the student has a good grasp of the basics of the concept and understands under which scenarios the shortcuts can be applied, he reaches this level
• Try to solve questions from previous years CAT, papers and look for solutions available
• Look at the different approaches used to solve the same question and how can the time be effectively reduced on using a particular approach