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**Introduction**

This chapter forms the backbone concept of most questions in Quantitative Aptitude & Data Interpretation sections of CAT exam. This is a crucial chapter & quick solving methods in this concept will help you save time - which is an essential factor for your success in **CAT 2021**.

Let's try to have a quick look at how **CAT** asks question to its aspirants, through this **2019 question.**

Question 1:**PYQ (CAT 2019) 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?**

This question helps you get the idea of how the questions are asked from this chapter.

**Definition**

Simple Average (or Mean) is defined as the ratio of the sum of the quantities to the number of quantities.

Here x_{1}, x_{2}, x_{3}, ----------- x_{n} represent the n values of a quantity under consideration & is the mean. Average or mean is said to be a measure of central tendency. Example:

__Let us take a very simple example of the first five natural numbers 1, 2, 3, 4 & 5:__

__Now let’s add 2 more 3’s to these 5 numbers:__

**Example 1: If a person with age 45 joins a group of 5 persons with an average age of 39 years. What will be the new average age of the group?**

**Sol:** Total age will be 45 + 5× 39 = 240. And there will be 6 persons now.

► So, the average will be 240/6 = 40.**(or)**

► Since, 45 is 6 more than 39, by joining the new person, the total will increase by 6 and so the average will increase by 1.

► So, the average is 39 + 1 = 40.

**Example ****2: Two students with marks 50 and 54 leave class VIII A and move to class VIII B. As a result, the average marks of the class VIII A fall from 48 to 46. How many students were there initially in class VIII A?**

**Sol:** The average of all the students of class VIII A is 46, excluding these two students.

► They have 4 and 8 marks more than 46. So, with the addition of these two students, 12 marks are adding more, and hence the average is increasing 2.

► There should be **6 students** in that class, including these two. This is the initial number of students.

**Example ****3: The average of x successive natural numbers is N. If the next natural number is included in the group, the average increases by:**

**(a) Depends on x **

**(b) Depends on the starting number of the series****(c) Both (1) and (2) ****(d) ****(e) None of these**

**Ans:** (d)

**Sol. **The average of consecutive numbers is the middle number. If one more number is added to the list, the middle number moves 0.5 towards the right. So the **answer is (d).**

**Weighted Mean**

- If somebody asks you to calculate the combined average marks of both the sections of class X, A and B when both sections have 60% and 70% average marks respectively?
- Then your answer will be 65%, but this is wrong as you do not know the total number of students in each section. So to calculate the weighted average, we have to know the number of students in both sections.
- Let N
_{1}, N_{2}, N_{3}, …. Nn be the weights attached to variable values X_{1}, X_{2}, X_{3}, …….. X_{n}respectively. Then the weighted arithmetic mean, usually denoted by: - For any two different quantities taken in different ratios. The weighted average is just like a see-saw. More the ratio of a quantity more will be the inclination of the average from mid-value towards the value with more ratios.

**Example ****4: The average marks of 30 students in a section of class X are 20 while that of 20 students of the second section is 30. Find the average marks for the entire class X?****Sol. **We can do the question by using both the Simple average & weighted average method.

**Real Facts About Average**

1. If each number is increased/decreased by a certain quantity n, then the mean also increases or decreases by the same quantity.

2. If each number is multiplied/ divided by a certain quantity n, then the mean also gets multiplied or divided by the same quantity.

3. If the same value is added to half of the quantities and the same value is subtracted from the other half quantities, then there will not be any change in the final value of the average.

**Average Speed**

- It is the total distance traveled divided by the time it takes to travel the distance.

- If d
_{1}& d_{2}are the distances covered at speeds v_{1}& v_{2}respectively and the time taken are t_{1}& t_{2}respectively, then the average speed over the entire distance (x_{1 }+ x_{2}) is given by:

Tip:Average Speed can never be double or more than double of any of the two speeds.

- If both the distances are equal i.e. d
_{1}= d_{2}= d then, {i.e. Harmonic mean of two velocities} - But if both the time taken are equal i.e. t
_{1}= t_{2}= t then,**Average speed**= {i.e. Algebraic mean of two velocities}

**Example ****5: The average of 10 consecutive numbers starting from 21 is :**

**Sol: **The average is simply the middle number, which is the average of 5^{th} & 6^{th }no. i.e, 25 & 26 i.e. **25.5**

Question 2:**There are two classes A and B., each has 20 students. The average weight of class A is 38 and that of class B is 40. X and Y are two students of classes A and B respectively. If they interchange their classes, then the average weight of both the classes will be equal. If weight of x is 30 kg, what is the weight of Y?**

Question 3:**The average weight of 10 apples is 0.4 kg. If the heaviest and lightest apples are taken out, the**

average is 0.41 kg. If the lightest apple weights 0.2 kg, what is the weight of heaviest apple?

average is 0.41 kg. If the lightest apple weights 0.2 kg, what is the weight of heaviest apple?

Question 4:**While finding the average of ‘9’ consecutive numbers starting from X; a student interchanged the digits of second number by mistake and got the average which is 8 more than the actual. What is X?**

Question 5:**There are 30 consecutive numbers. What is the difference between the averages of first and last 10 numbers?**

**There are 60 students in a class. These students are divided into three groups A, B, C of 15, 20 & 25 students each. The groups A & C are combined to form group D.**

Question 6: If all the students of the class have the same weight, which of the following is false?

Question 7:If one student from group A is shifted to group B, which of the following is necessarily true?

Question 8:What is the average weight of the students in group D?

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