AVERAGEAn average, or an arithmetic mean, is the sum of `n' different data divided by `n'Average = ( ∑ of Observation

Number of Observation), Sum = Average x No. of Observation

Points to remember:1. New value added = New average + No. of old values x change in average2. Old value removed = New average - No. of old values x change in average3. Age of new person = Age of the removed person + No. of members x change in averageIn all the above three cases, if there is a decrease in the average, the sign of change in average will be negative.

Formulae:Average of ‘n’ consecutive numbers = [x+(x+1)+(x+2)+(x+3)…]/nAverage of ‘n’ consecutive odd/even numbers = [x+(x+2)+(x+4)+(x+6)…]/nAverage of first ‘n’ natural numbers = (n+1)/2Average of the squares of the first ‘n’ natural numbers = [(n+1)(2n+1)]/6Average of the cube of the first ‘n’ natural numbers = [n(n+1)2]/4Average of first ‘n’ odd natural numbers = nAverage of first ‘n’ even natural numbers = n+1

SOLVED PROBLEMS

1. The average of 5 quantities is 10 and the average of 3 of them is 9. What is the average of the remaining 2?

The average of 5 quantities is 10. Therefore, the sum of all 5 quantities is 50.The average of 3 of them is 9.Therefore, the sum of the 3 quantities is 27.Therefore, the sum of the remaining two quantities = 50 - 27 = 23.Hence, the average of the 2 quantities = 23/2 = 11.5.

2. The average of 20 numbers is zero. Of them, at the most, how many may be greater than zero?

Average of 20 numbers = 0.Sum of 20 numbers (0 x 20) = 0.

It is quite possible that 19 of these numbers may be positive and if their sum is a then 20th number is (-a).

3. The average weight of 8 person's increases by 2.5 kg when a new person comes in place of one of them weighing 65 kg. What might be the weight of the new person?

Total weight increased = (8 x 2.5) kg = 20 kg.Weight of new person = (65 + 20) kg = 85 kg.

4. The captain of a cricket team of 11 members is 26 years old and the wicket keeper is 3 years older. If the ages of these two are excluded, the average age of

the remaining players is one year less than the average age of the whole team. What is the average age of the team?

Let the average age of the whole team by x years.11x - (26 + 29) = 9(x -1)11x - 9x = 462x = 46x = 23.

So, average age of the team is 23 years.

5. The average age of husband, wife and their child 3 years ago was 27 years and that of wife and the child 5 years ago was 20 years. The present age of the husband is:

Sum of the present ages of husband, wife and child = (27 x 3 + 3 x 3) years = 90 years.Sum of the present ages of wife and child = (20 x 2 + 5 x 2) years = 50

years.Husband's present age = (90 - 50) years = 40 years.

6. A Batsman makes a score of 87 runs in the 17th inning and thus increases his average by 3. Find his average after 17th inning.

Let the average after 17th innings = xThen average after 16th innings = (x-3)Therefore 16(x-3) + 87 = 17xTherefore x = 39 

7. Average cost of 5 apples and 4 mangoes is Rs. 36. The average cost of 7 apples and 8 mangoes is Rs. 48. Find the total cost of 24 apples and 24 mangoes.

Average cost of 5 apples and 4 mangoes = Rs. 36 Total cost = 36 * 9 = 324 Average cost of 7 apples and 8 mangoes = 48 Total cost = 48 * 15 = 720 Total cost of 12 apples and 12 mangoes = 324 + 720 = 1044 Therefore, cost of 24 apples and 24 mangoes = 1044 * 2 = 2088 

8. The average age of 36 students in a group is 14 years. When teacher’s age is included to it, the average increases by one. What is the teacher’s age in years?

Age of the teacher = (37 × 15 – 36 × 14) years = 51 years.

9. The average temperature on Wednesday, Thursday and Friday was 250. The average temperature on Thursday, Friday and Saturday was 240. If the temperature on Saturday was 270, what was the temperature on Wednesday?

Total temp on Wednesday, Thursday and Friday was 25 * 3 = 750

Total temp on Thursday, Friday and Saturday was 24 * 3 = 720

Hence, diff b/w the temp on Wednesday and Saturday = 30

If Saturday temp = 270, then Wednesday's temp = 27 + 3 = 300

10. In the first 10 over’s of a cricket game, the run rate was only 3.2. What should be the run rate in the remaining 40 0vers to reach the target of 282 runs?

Required run rate = [282 – (3.2 × 10)] / 40= 250 / 40 = 6.25.