AVERAGES

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GETTING STARTED

☛ An average is a central value for a set of numbers.

☛ It is calculated by adding all the numbers and dividing it by the total count of numbers.

Eg. Calcualate average of 85, 95 and 129.


(85+95+129)/3 = 309/3 = 103

Important Properties of Averages


☛ It can be observed that average value always lies between the smallest and the largest numbers of a given set.
     In the above example, the average is 103 and it lies between 85 and 129.

☛ If we have 5 numbers a, b, c,d,e whose average is n, it is to be noted that
(n-a)+(n-b)+(n-c)+(n-d)+(n-e) = 0

Let us understand with the help of an example:

Consider 5 nos. – 72, 98, 84,51,35

Average of 5 numbers = (72+98+84+51+35)/5 = 340/5 = 68

Now, ( 68-72)+ (68-98)+(68-84)+(68-51)+(68-35)

= (-4)+(-30)+(-16)+(17)+(33)

= 0

This property will help in solving many problems in short time.

☛ For n numbers in Arithmetic Progression with any difference,

Average of n numbers is the middle term of the series.

Eg. Consider a series 3,6,9,12,15.

Average of series = (3+6+9+12+15)/5 =9

It is same as the third term which also happens to be the middle term

For n numbers in Arithmetic Progression with any difference,

Average of n numbers =( [ First Term] + [Last Term])/2

Eg. Consider an AP, 7,14,21,28,35.

Common Difference = 7

Average of AP = Middle Term = 21

( First term + Last Term)/2 = (7+35)/2= 21

One of the most frequently asked type of question is:

☛ Average of a group changing due to addition/ removal of a person.

Eg. Average of a group of 5 boys is 20 years. On adding one more boy in the group, average age increases by 1 year. What is the average age of new person?

Conventional Method:

Average age of 5 boys = 20 yrs

Total age of 5 boys = 20 x 5 = 100 yrs

New average age of 6 boys = 21 yrs

Total new age of 6 boys = 126 yrs

Average of new boy = New Total age – Old Total Age = 126-100 = 26 years

A simpler method to solve the above example can be :

The average age of the group increases by 1.

So add increase of 1 year per person for 5 members to the new average ie. 21.

We get 5+21 = 26 years without much calculation and effort.

☛ Age of new member = New Average + [Increase in Average x Initial no. Of members]

Similarly, if average age of the group is decreased by 1 with 1 person adding to the group,

Again add new average to increase (-1 per person) for 5 persons.

 

PRACTICE PROBLEMS

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Created on By venkateshj

Averages - Concept Refresher

1 / 20

The average age of a group of students is 11 years. If a 46 year old teacher joins the group, the average increases by one year. What is the number of students in the group?

2 / 20

The average weight of 18 students of a class is 15 kg. If a student leaves the class, the average weight of the class drops by 0.6 kg. Find the weight of the student (in kg) who left the class.

Marks in sixth exam = Initial Average+( Decrease in marks x Total tests)
= 60 + (-5 x 6)
= 30 marks.
Hence (C)

3 / 20

Suresh wrote a total of six tests. His average mark in those was 55. His average mark in the first five tests was 60. Find his mark in the sixth test.

4 / 20

The average age of a class of 30 students is 14 years. The average increases by 2 years, if the age of a teacher is included. What is the age of the teacher?

5 / 20

A total of 65 chocolates were distributed among 19 children such that each girl gets 3 chocolates and each boy gets 5 chocolates. Which of the following is the number of girls?

6 / 20

Indian cricket team had to score 320 runs in 50 overs. The team had scored 120 runs at the end of 20 overs. How much minimum runs should the team score on an average in an over so as to win the match.

7 / 20

The average age of the boys in a class is 14 years. The average age of the girls in that class is 12 years. If the ratio of the numbers of
boys and girls in that class is 3 : 2, find the average age of the class.

8 / 20

There are seven people in a group. The average age of the oldest and the youngest is 13 years. If any one person leaves the group, the maximum
and the minimum average of the remaining are 15 years and 12 years respectively. Find the average age of the entire group, in years.

9 / 20

The average weight of four students A, B, C, D is 40 kg. The average weight of A, B, C, D, E is 37 kg. The weight of F is 10 kg more than that of E. The average weight of B, C, D, E, F is 38 kg. Find the weight of A (in kg).

10 / 20

The average of eleven distinct positive integers is 81. If the average of the first 6 is 83 and the average of the last six is 80, find the average of first 5 numbers.

11 / 20

Find the average of all the multiples of 6 less than 200.

12 / 20

Find the average of all even numbers less than 100.

13 / 20

Find the average of the numbers from 81 to 90

14 / 20

The average age of 6 men is 40 years. Their average age increased by 1 year when a new man joined them. Find the age of the new man.

15 / 20

Soham purchased four dozen bananas at Rs.10 per dozen, three dozen bananas at Rs.15 per dozen and six dozen bananas at Rs.16 per dozen. Find the approx. average cost per dozen of the bananas that he purchased.

16 / 20

The monthly incomes of three brothers Amar, Akbar and Anthony are 16000, 13000 and 11500 respectively. Find their average monthly income.

17 / 20

Find the average of the scores of the tests taken by Sumeet given that his scores were 47, 72, 81, 86,76, 54, 89, 93 and 95.

18 / 20

In a group of 13 people, x is 42 years old and y is 2 years younger than x. If x and y are replaced by two other people, the average age of the group drops by 2 years. Find the average age of the two people replacing x and y.

19 / 20

There are N students in a class. Their class teacher gave them a task of finding the average of the first N natural numbers and each of them left out a different number and found the average of the remaining numbers. The average of the averages obtained by all the students was 41. Find N.

20 / 20

In a class, there are 100 students. 40 of them are boys. Their average mark in a test is 70. The average mark of the girls in the test is 80. Find the average mark of the class in the test.

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