Average

 Average

Average = sum of observations ÷ number of observations. 

It is also known as Arithmetic Mean. 

• If the average of 'n₁' numbers is a₁, and the average of 'n₂'numbers is a₂, then average of total numbers n₁ and n₂ is 

                  (n₁a₁ + n₂a₂)/(n₁ + n₂)

• The average of 'n' consecutive natural numbers = n(n+1)/2n = (n + 1)/2
• The average of squares of 'n' consecutive natural numbers = n(n + 1)(2n + 1)/6n = (n + 1)(2n + 1)/6
• The average of cubes of 'n' consecutive natural numbers = n²(n+1)²/4n = n(n + 1)²/4

Examples 

1. Find the average of first 'n' natural numbers?
Solution:

         Sum of 'n' natural numbers = n(n + 1)/2
         Then, average = n(n + 1)/2n = (n + 1)/2

2. The average of four consecutive even numbers is 27. Find the largest of these numbers?
Solution:

     Let the numbers are x, x + 2, x + 4, x + 6

      Then,  x + (x + 2) + (x + 4) + (x + 6) / 4 = 27

        So, x = 24, largest number = x + 6 = 30

3. The average weight of 4 men is increased by 3 kg, when one of them who weighs 120 kg is replaced by another man. What is the weight of new man?

Solution:

      Increased weight = 4 × 3 = 12

This is increased due to the extra weight included due to the inclusion of new person.

So, weight of new man = 120 + 12 = 132 kg

4. 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?

Solution:

   Let average after 17th inning = x

    Average after 16th inning = x - 3

    Then, 16 (x - 3) + 87 = 17x

     So, x = 39

5. In an examination, a batch of 60 students made an average score of 55 and another batch of 40 made it only 45. What is the overall average score?

Solution:

     Average of combined group = (60 × 55 + 40 × 45) / (60 + 40) = 51 

6. What is the difference between the average of first 148 even positive numbers and the average of first 129 odd positive numbers? 

Solution:

   Average of first n positive even numbers = n + 1

   Average of first n positive odd numbers = n

So, required difference = (148 + 1) - 129 = 20

7. The average marks of a class of 54 students is 45. Out of them, two students score zero, and out of the rest, the first 34 students scored an average marks of 45, the next 16 students scored an average marks of 35. If out of the remaining two students, one has scored 20 marks more than the other then, find the ratio of their marks (higher to lower marks)?

Solution:

Total mark = 54 × 45 = 2430

Let highest mark among two students = x

Other mark = x - 20

then, 

2 × 0 + 34 × 45 + 16 × 35 + x + x - 20 = 2430

On solving,  x = 180 & other = x - 20 = 160

Required ratio = 180 : 160 = 9 : 8

Do it yourself 

1. Find average of first 60 natural numbers?

2. Find average of all prime numbers between 30 and 50?

3. Find average for the following readings : - 

         12, -2, 350%, 9, -80% (Ans. 4.34)

4. In a class of 10 students where the average weight is 50 kg. If the weight of each student is increased by 5 kg. What will be the new average? (Ans. 55)

5. The average weight of A, B, C is 45 kg. If the average weight of  A and B be 40 kg and that of B and C be 43 kg, find the weight of B? ( Ans. 31 kg)

6. The average age of 30 boys of a class is equal to 14 years. When the age of class teacher is included,  the average becomes 15 years. Find the age of class teacher? ( Ans. 45 year)

Previous Years Questions

1. The average of six numbers is 9. If the average of first five of these numbers is 8, what is the sixth number?

2. The average expenditure of a man for the first five months is Rs.1200 and for the next seven months is Rs.1300. If he saves Rs.2900 in that year,  his monthly average income is : ________.

3. The average of 12 numbers is 15 and the average of the first two is 14. What is the average of the rest?

4. The average of five numbers is 27. If one number is excluded,  the average becomes 25. The excluded number is: ________. (Ans. 35)

5. Find the average of all numbers between 6 and 34 which are divisible by 5. (Ans.20)

6. The difference between the average of first ten prime numbers and the first ten prime numbers of two digits is :-

7. The sum of 10 numbers is 686. What is their average?

8. The average of 8 consecutive even numbers written in ascending order is 17. What is the average of the last three numbers, 36 and 53 ? (Ans.31)