Properties of Arithmetic Mean:

1. If each number is increased or decreased by a number \(k\), the mean is also increased or decreased by the same number \(k\).
2. If each number is multiplied or divided by a number \(k\), the mean is also multiplied or divided by the same number \(k\).
3. If a new data is inserted or deleted or replaced, the surplus or deficiency is uniformly distributed over the updated number of data. For example if there are 3 students in a class and their average weight is 50 kg. If a new student whose weight is 62 kg joins the group, then surplus weight is 12 kg and it should be distributed over 4 students uniformly. So the average of the class is increased by \(\frac{{12}}{4}\)= 3 kg. Hence new average is 53 kg. Conversely if there are 4 students whose average weight is 53 kg, if one student whose weight is 62 kg leaves the group, then total deficiency created is 9 kg which will be distributed over 3 students equally, hence the average will be reduced by 3 kg. 

4. Average of the numbers which are in Arithmetic progression is  \(\color{blue}{\frac{{{\rm{first}}\;{\rm{number}}\;{\rm{ + }}\;{\rm{last}}\;{\rm{number}}}}{{\rm{2}}}}\)

For example average of the numbers 11, 13, 15, 17, …….47, 49, 51 is \(\frac{{11 + 51}}{2} = 31\)

Example 01: In three numbers, the first is twice the second and thrice the third. If the average of these three numbers is 44, then the first number is:

Solution: Let the three numbers be \(x, y\) and \(z\).

Therefore, \(x = 2y = 3z\),

\(y = \frac{x}{2}\) and z =\(\frac{x}{3}\)

Now, \(\frac{{x + \frac{x}{2} + \frac{x}{3}}}{3} = 44.\) 

Or \(\frac{{11x}}{{18}} = 44\,or\,\,x = 72\)

Example 02: There are 30 students in a class. The average age of the first 10 students is 12.5 years. The average age of the next 20 students is 13.1 years. The average age of the whole class is:

Solutions: Total age of 10 students = \(12.5 \times 10\)

= 125 years

Total age of 20 students = \(13.1 \times 20\) = 262 years

Average age of 30 students = \(\frac{{125 + 262}}{{30}} = 12.9\) years

Example 03: The average age of students of a class is 15.8 years. The average age of boys in the class is 16.4 years and that of the girls is 15.4 years. Find the ratio of the number of boys to the number of girls in the class. 

Solution: Let the number of boys in a class be \(x\).

Let the number of girls in a class be \(y\).

Sum of the ages of the boys = \(16.4 x\)

Sum of the ages of the girls = \(15.4 y\)

\(\therefore \) \(15.8 (x + y) = 16.4x + 15.4y\)

\( \Rightarrow 0.6x = 0.4y \Rightarrow \frac{x}{y} = \frac{2}{3}\)

\(\therefore \) Required ratio = 2 : 3

Example 04: A teacher wrote consecutive natural numbers on a board starting from 1. A student erases one of the number and average of the remaining numbers is found to be \(35\frac{7}{{17}}\). Find which number is missing.

Example 05: Average weight of the five students is 50 kg. Now weights of all possible pairs of students are taken. If these weights are \({w_1},{\rm{ }}{w_2},{\rm{ }}{w_3},{\rm{ }} \ldots ..{w_{10}}\). Find average of \({w_1},{\rm{ }}{w_2},{\rm{ }} \ldots ..{w_{10}}\).

Example 06: The average (runs conceded/ number of wicket) of a bowler decreases by 1 after playing one more innings in which he took 3 wickets and conceded 27 runs. If before this inning he has conceded 1800 runs then total how many wickets he has taken including this innings?

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