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Hyderabad: These series of articles on the arithmetic will aid you in your preparation for government recruitment tests. In most competitive exams like SI, Constable, Groups, etc., there are direct questions from the Average topic. Sometimes, questions are application-based, which require a great amount of logical thinking.

The government job aspirants need to develop a logical approach to answer the problems in this area. It is better to avoid formula-based approach to save time in exam. Let us go into the topic and try to observe the logic in basic problems.

Average: We use the word average or arithmetic mean in our daily observation. Example: Average marks, average rainfall, batting and bowling averages.

‘Average’ is most likely to be the value around which a group of values show a tendency to concentrate. Thus, we can say that average represents on entire group by single value.

Important points:

1. Highest observation contributes lower observation to balance it.

Example: If A, B are two persons having Rs 2 and Rs 4 respectively
If B gives Rs 1 to A, then A has Rs 3
Now B has Rs 3
This is called average.

2. The average always lies between the lowest and highest number of the set.
Average of a given term always lies in the range at given data.
i.e., lowest quantity
The average of 2,4 is 3
2 < 3 < 4

Example 2: The average of 6, 6, 6 is 6.
6≤6≤6
Note: If quantities of the data are equal, then the average will also be the same as quantities

3. If ‘A’ be the average of x 1, x 2….. x m, y1 , y2….yn where x 1, x 2…. xm be below A and y1 and y2…. yn be above A, then
(A- x1) (A- x 2) …… (A- x n) = (y1-A) ( y2-A)… (yn-A)
i.e., the surplus above of the average is always equal the net deficit below average

Example: The average of 100, 200, 300, 400, 500 is 300.
Sum of the values of below 300 is 100 200 = 300
The average deficit = 300 – (300/2)
= 300 -150 = 150
Sum of the values of above 300 is 400 500 = 900
The average surplus = (900/2) – 300
= 450 – 300 = 150
Hence the surplus above the average = below the average.

4. If the value of each item in a group is increased / decreased by the same value, say x, then the average of the group also increases/decreases by x.

Example 1: The average of 5 numbers is 20. If 2 is added to each number, then the new average is 20 2 = 22.

Example 2: The average of 7 numbers is 37. If 3 is subtracted from each number then the new average is 37-3 = 34. This concept is useful while dealing with the problems on age.
5. If the value of each item in a group is multiplied by the same value x, then the average of the group is also multiplied by x.

Example: The average of 11 numbers is 17. If each number is multiplied by 5, then the new average is 17×5 = 85

6. If the value of each item in a group is divided by x where x≠0 then the average of the group is also divided by x.

Example: If the average of 10 numbers is 32, If each observation is divided by 4 then the average is 32/4 = 8

7. In the given series of numbers, if the gap is same between consecutive terms, then the average is
first number last number / 2

Example: The average of 3, 6, 9, 12, 15, 18, 21
Solution: Here the gap is same
(3 21)/2 = 24/2 = 12

To be continued…

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Author: Howard Caldwell