**Averages**

The average of a given number of quantities of the same kind is expressed as

Average = sum of quantities/Number of quantities

Average is also called the Arithmetic Mean.

Also, Sum of the quantities = Average × Number of the quantities

### TIPS:

[box type=”shadow” ]If all the given quantities have the same value, then the number itself is the average.

If all the given quantities are not all the same, then the average of the given quantities is always greater, then the smallest number and always less than largest number. Equivalently, at least one of the numbers is less than the average and at least one is greater than the average.

If each of the given quantities is increased by a constant p, then their average is also increased by p.

If each of given quantities is decreased by a constant p, then their average is also decreased by p.

If each of given quantities is multiplied by a constant p, then their average is also multiplied by p.

Whenever the given quantities form an arithmetic sequence and if the given quantities has odd terms, then the average is the middle term in the sequence and if the given quantities has even terms, then the average of the sequence is the average of the middle two terms.

In order to calculate the weighted average of a set of numbers, multiply each number in the set by the number of times it appears, add all the products and divide by the total number of numbers in the set.[/box]

[box type=”shadow” ]If the speed of an object from A to B is x km/h and from B to A is y km/h,then the average speed during the whole journey is 2xy/(x+y) km/h[/box]

[box type=”shadow” ]If the average of N1 quantities is and N2 quantities is y then the average of total (N1 + N2) quantities is given by (N1 x +N2 y)/(N1+N2)[/box]

### Examples :

[box type=”shadow” ]Question: Average weight of 32 students of a class is 30.5 kg. If weight of a teacher is also included then average weight is increased by 500 g. What is the weight of the teacher?

**Solution**: Total weight of 32 students = 30.5 × 32 = 976 kg

Average weight of (32 students + 1 teacher) = (30.5 + 0.5) = 31 kg

Total weight of (32 students + 1 teacher) = 31 × 33 = 1023 kg

Weight of teacher = (1023 – 976) kg = 47 kg[/box]

[box type=”shadow” ] Question: 12 men went to a restaurant. 11 of them spent 5 each and the 12th person spent 11 more than the average expenditure of all. Find the total money spent by them?

**Solution**: Let the average money spent by the 12 men =x

Money spent by the 12th man = (x + 11)

Money spent by the other 11 men = (11 × 5) =55

Total money spent by the 12 men = (55 + x + 11) =(x + 66)

x = x+66/12

12x = x + 66 ; x = 6 Total money spent by the 12 men = 6 × 12 = ` 72.[/box]