Study Notes for CAT 2021: Averages

Concept of Average

The average is defined as the sum of all the numbers divided by the total no. of numbers. Let there are n elements present from (x_{1 ………}Xn), then the average is calculated as –

Now, for smaller values, this formulae seems easy but when you come across a situation where you have to find out the Average of these numbers like -  78,84,68,95,75. Then, this formulae will be time-consuming for you. So , you need to find a short cut to solve such questions-

Suppose Numbers are  - 68, 75,78,84,95

Now, Average always lies between the highest and lowest number like in this case, it will be between 68 and 95.

First, choose any number between highest and lowest number say- 76, Then calculate the deviation of all the numbers from this number (76)

In this way, the Average of many numbers can be calculated faster than the traditional way.

So, the Average can also be defined as a number that can replace all the numbers present in a group so that the average of all the numbers cannot be altered.

Properties of Averages:

1. Average of any group always lies between the maximum and the minimum value of the group.
2. If there are odd numbers of terms in an Arithmetic progression then the average will always be the central value of the progression. ( Let there are 7 number of terms in a A.P then the average will be the 4^th term of the progression )
3. If all the number of the group increased or decreased by some values then the average will also be increased or decreased by that value.
4. If all the numbers are divided or multiplied by some constant number then the average will also be multiplied or divided by that number.

Some important  Formula related to Averages

Let there are n consecutive term say, X,X+1.X+2 ………. X+n

 Almost in all exams, the question asked on Averages is based on 4 - 5 types. These types are-

1. Problems related to Age.
2. Problems related to replacement.
3. Problems related to Batting or bowling
4. Problems related to Time and Speed
5. Problems related to numbers and some miscellaneous problems.

We will discuss all the types and some questions on these types one by one-

1. Problems related to Age 

Before discussing the question, there is some basics concept related to ages-

If the age of a person is X years then there are two possibilities in this case. One is the age before n years and age after n years.

Age before/ago N years =  (X-N) Years

Age after/later N years = (X+N) Years

Q.1.  In a family, there are 7 members whose current average age is 24 years and a child was born 6      years ago then what is the average age of the family at the time of birth?

Solution .-

As you know, Average = Sum/Number

Let the average age of 5 members of the family excluding the child is X years then,

• (6X + 6)/7 =24
• 6X+6= 168
• So, X= 27 Years

So, the age before 6 years ago was -27-6 =21 years.

2^nd Method

You can solve this question by this method also

This method is good to solve the problems is the faster way.

1. Problems Related to Replacement

 This types of problem frequently asked in all the exams. So, let’s discuss the problems à

Q.1 .- There are 15 members in a group whose average age is 25 years and one member left the group then the average increases by 1 year then the age of the member is?

Solution.-->

In this case, the average age of the overall group increases so it means that the age of the person that left the group must be less the average.

So, the average is increased by 1 year then the 1 year must be added to all the 14 members age then the age of the person who left the group is à 25-14 = 11 Years

Or we can also do by this method

So, if in the above question the average is decreased or a new member joins the group then the age of the new person can be calculated.

1. Problems related to Batting/ Bowling

Q.1. A batting average of a batsman after 15 innings is 45 runs and after 16^th innings his average increases by 2 runs then how much runs he scored in the last innings?

Solution.-

Runs after 15 innings = 15 x 45 = 675

Runs after 16 innings = 16 x 47 = 752

Runs in the 16^th innings = 752-675 =77 Runs

Q.2. A batsman has certain average after 15^th innings and in the 16^th innings, he scores 90 Runs and his average increases by 2 runs. Find the average after the 16^th innings?

Solution.-

     Let the average after 15 innings be X runs

1. Problems Related to Time & Speed

The basics formulae for the average speed is à

Q.1.  A person covers half of the distance at 40 kmph and the remainder half at 20 kmph and the rest at 10 kmph. Find the average speed during the journey?

Solution.-

Let the total distance be  = X Km

So, time take to complete the X/2 Km = X/80

Time take to complete the X/4 Km = X/80

Time taken to complete the rest X/4 = X/40

Average speed = 20 Kmph.

1. Problems related to Numbers

Q.1 – In a Company the average salary of the entire staff is Rs. 2000 and the average salary of officers is Rs.3000 and workers is Rs. 1500 and there are 4 officers in that company then find out the number of workers in the company.

Solution.- 

Numbers of workers in the company is 12 - 4 = 8.

So, these were the important questions of the topic Average. Practice them for upcoming banking exams.

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