0Averages
Basic Formulae for average of n numbers x1, x2, x3....xn is given by
An=(x1+x2+x3+.........xn)/n=Total of n numbers/n
This also means that (An)*n =Total of the numbers
Average of sum of first n natural numbers = (n+1)/2
Average of sum of square of first n natural numbers is = (n+1)*(2n+1)/6
Weighted Average Concept
Aw=(n1A1+n2A2+n3A3.......nkAk)/(n1+n2+n3......nk)
Ages and Average
Average age of a group of person is x years today then after n years their age will be (x+n)
Average age of a group of person is x years today then n years ago their age will be (x+n)
Average Speed of Journey
Average Speed=Total Distance /Total Time
Average Speed=(S1+S2)/2 (If time between two journey is same)
Average Speed= (2S1S2)/(S1+S2)(If Distance between two journey is same)
Alligations
Introduction
Alligation is a faster technique to solve questions based on weighted average. Be careful to understand the concept of allegation in depth so that you are able to solve the question quite fast.
Theory
Please refer to the below table which represent the averages of respective groups along with number of elements.
| Serial Number | Average of Group | Number of elements |
| 1 | A1 | N1 |
| 2 | A2 | N2 |
| 3 | A3 | N3 |
| 4 | A4 | N4 |
| 5 | A5 | N5 |
Weighted Average is given by below formulae
(Aw)=(N1A1+N2A2+N3A3+N4A4+N5A5)/(N1+N2+N3+N4+N5)
When just two groups are mixed, we can write the above equation as
(Aw)= (N1A1+N2A2)/(N1+N2)
| Alligation Equation |
Rewriting the above equation we will get the following equation:
N1/N2= (A2-Aw)/ (Aw-A1)
When to use Alligation
When two groups of elements are mixed together to form a third group containing the elements of both the groups. A1, A2 is average of first group of N1, N2 elements. We take A1<A2, then by principal of average A1<Aw
Graphical Representation of Alligation
Straight Line Approach
We will now modify the graphical approach method shown in the above diagram to tackle all types of questions. Consider the diagram shown below which represent straight line method.
Note: The above two methods will be explained in detail through examples given in the subsequent section for allegation.
Solved Examples: Average
Example 1:
A man travels at 60km/hr on a journey from A to B and returns at 100km/hr. Find his average speed of journey.
Solution:
Average Speed when distance is same =2(S1*S2)/ (S1+S2) = (2*60*100)/ (160) =75
Short Cut Approach using concept of alligation in average speed
Ratio of speed is 60:100=3:5(say n1:n2)
Divide 100-60 i.e. 40 into n1+n2 parts so one part is 40/8=5
Required answer will be 3 parts away from lower speed
60+3*5=75
Example 2:
Average of a batsman after 25 innings was 56 runs per innings. If after 26th innings his average increased by 3 runs, then what was his score in the 26th innings.
Solution:
Runs in 25 innings = 25*56=1400
Runs in 26 innings =26*59=1534
Runs in 26th innings =1534-1400=134
Shortcut Approach:
Average after 26th innings =56+3=59
As 26th innings increased average of each innings by 3 so runs scored = 59 +3*25=59+75=134
Example 3:
Average age of a class of 30 students and a teacher reduces by 0.5 if we exclude the teacher.If initial average is 14 years, find the age of class teacher.
Solution:
Age of teacher=Total age of students and teacher - Total age of students
=31*14-30*13.5=434-405=29 years
Shortcut Approach:
Average age of teacher = 14+0.5*30=29
Example 4:
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