Dear Aspirants,
Welcome to D2G’s Quantitative Aptitude concept on Average. Math is fun when you really know it. When it comes to average, especially in Bank exams mostly we get questions based on average in DI section and in miscellany part. Nothing is specific except concepts. If are strong in basic concepts we will find numerous ways to solve a single problem.
Now, what is Average?? Yes, an average is the sum of numbers divided by total numbers given. Its not just division but more than that. Lets see how average plays in our daily life. For example you may say ‘I’m having an average day today’ meaning your day is neither particularly good nor bad, it is about normal. The term ‘average’ refers to the ‘middle’ or ‘central’ point; when used in mathematics the term average refers to a number that is a typical representation of a group of numbers (or data set).
Basic Concepts:-
1) Average = Sum of observations/Number of observations
2) If average of a observations is x and average of b observations is y, then the average of total observations(a+b) will be (ax + by)/(a + b)
3) The average of odd numbers from 1 to n is (last odd number + 1)/2
4) The average of even numbers from 1 to n is (last even number + 2)/2
5) Average of natural number = n(n+1)/2
6) No. of passed students = total students × (total average – failed average)/(passed average – failed average)
Points to remember with examples:-
1) If all the given quantities have the same value, then the number itself is the average.
Given, Find the average of given numbers 25, 25, 25, 25.
By remembering the point above you can easily say the average is 25.
2) If all the given quantities are not same, then the average of the given quantities is always greater than the smallest number and always less than the largest number.
Given, The average of five consecutive even numbers such as 46, 48, 50, 52, 54.
46+48+50+52+52/5
250/5
50
so 50 is greater than 46 and lesser than 54.
3) If each of the given quantities is increased by a constant a, then their average is also increased by a and vice versa.
For example given the ages as 20, 44, 56 and 60 and their average is 45. What will be the average age of the family when the ages of each person is increased by 10 years?
Now instead of calculating like 30 + 54 + 66 + 70/4 = 55. We can say it as 45 + 10 = 55
4) If each of the given quantities is multiplied by a constant a, then their average is also multiplies by a.
Given, Set of numbers 22, 30, 41, 35 and their average is 32. Now, if each number in the given set is multiplied by 6. Then the average is?
Instead of calculating by our traditional method we can easily say it is 32 × 6 = 192
5) Whenever the given quantities form an arithmetic sequence in odd terms, then their average is the middle number in the given sequence.
For example, given 10, 12, 14, 16, 18. What is the average?
It is 14.
6) If the given quantities form an arithmetic sequence in even terms, then their average is the sum of the middle two terms.
Given, 20, 22, 24, 26, 28, 30. What is the average?
It is 24 + 26/2
50/2
25
Here we have a look on few frequently asked type of questions in exams:-
Classroom concept:-
1) When teachers age added,subtracted what is new average or
2) If teachers age added,subtracted what will be the change in average or
3) If a new boy comes, goes then what will be change in average or find the age of new boy etc..
Based on marks:-
1) Increase or decrease in average when marks of a new student added or vice versa
Based on Cricketer and Innings:-
1) Increase or decrease in average score of a batsman, or individual score of a cricketer when we are given averages
Lets learn through examples,
Q) The average age of 30 students in a class is 10. When teachers age is added the average age becomes 11. Find out the age of the teacher?
Traditional method:-
Total number of students = 30
Average age of students = 10
Therefore the sum of ages of all 30 students = 30 x 10 = 300
When teachers age is added the total number becomes = 30 + 1 = 31
The new average = 11
Therefore the sum of ages of students and the teacher = 31 x 11 = 341
Teachers age = (Sum of ages of teacher + students) – (Sum of ages of students)
= 341- 300
= 41
Now Use Your Commonsense:-
Let us assume that the teachers age was also 10. In such case the average of all 31 people ( 30 students + 1 teacher) remains same which is 10.
But this is not the case. On arrival of teacher the average has increased by 1. So 1 has been added to all the 31 people of the class.
It means teachers age should be more than 10. How much more ? It is 31 more
10 + 31 = 41
Formula method:-
TEACHERS AGE = NEW AVERAGE + (NUMBER OF STUDENTS X DIFFERENCE OF AVERAGES)
In above case
Teachers age = 11 + ( 30 X 1) = 11+ 30 = 41
Q) The average marks obtained by 10 students in an exam is 55. When a boy who scored 65 marks left the class and another one joined the average reduced by 3. What was the marks of new boy?
Traditional method:-
Average marks of 10 students = 55
Total marks = 55 x 10 = 550
Marks of 9 student after boy scoring 65 left = 550-65 = 485
New average for 10 students = 55 – 3 = 52
Total marks of 10 students now = 52 x 10 = 520
Therefore the marks of new boy = 520 – 485 = 35
Now use your commonsense:-
If the boy who came also scored 65 then no problem at all, the average would have remained same that is 55. But the average reduced by 3. So from each person he reduced 3.
Total persons are 10
So total marks he reduced is 10 x 3 = 30
So from 65 we can reduce this 30. We get 65-30= 35
Formula method:-
NEW PERSON’S AGE/WEIGHT/MARKS = AGE/WEIGHT/MARKS OF THE PERSON WHO LEFT + (TOTAL PERSONS X DIFFERENCE IN AGE/MARKS/WEIGHT)
New person’s mark = 65 + (10 X -3) = 65 -30 = 35
In our next post we will provide you with questions. Try to solve it in a simple way. It saves time. Make it as simple as possible. Formula does matter. Everything is easy if we love to do it!!
Happy Reading!!
