## Averages

1. Average:

If there are N numbers, then their Average, A is In short, A = Sum/Num

i.e. A = S/N

Note: Also remember that Example 1:

The average of 2, 4, 6, 8, and 10 is

(2 + 4 + 6 + 8 + 10)/5 = 30/5 = 6

Example 2:

The average of 10 numbers is 16. Find their sum.

From the above note, S = A × N

Therefore, the sum is: 16 × 10 = 160

Average of numbers in Arithmetic Series:

Numbers are said to be in arithmetic series  if there is a common difference between any two successive numbers.

For example, in 2, 4, 6, 8, 10 the common difference between any two consecutive numbers is 2.

Therefore, 2, 4, 6, 8, and 10 are said to be in arithmetic series.

Short-cut for finding Average of numbers in Arithmetic Series: Now, in the arithmetic series:

2, 4, 6, 8, 10

The smallest number is 2 and the largest number is 10.

Therefore, average is

(2 + 10)/2 = 6

Example 3:

The average of 10 numbers is 65. After 3 numbers are removed, the average changes to 55. What is the average of the 3 removed numbers?

Solution:

We know that S = A × N

So, sum of the 10 numbers in the given question is:

S = 65 × 10 = 650, and

Sum of the remaining 7 numbers (after 3 numbers are removed) is

S’ = 55 × 7 = 385

Now sum of the 3 removed numbers is S – S’, i.e.

S – S’ = 650 – 385 = 275

Therefore, average of the 3 removed numbers is 275/3 = 91.67

Example 4:

The average of 11 numbers is 45. The average of the first six is 40 and that of the last six is 50. Find the sixth number?

Solution:

Recall that S = A × N

So, S = 45 × 11 = 495, S’ = 40 × 6 = 240, and S” = 50 × 6 = 300

Now, the sixth number is: (S’ + S”) – S

i.e., (240 + 300) – 495 = 540 – 495 = 45

Example 5:

The average weight of group of 10 boys is increased by 5 pounds, when one of them who weighs 100 pounds is replaced by another boy. What is the weight of the new boy?

Solution:

Weight of the new boy = weight of the replaced boy + increase in total weight of the group

Increase in total weight of the group = increase in average × number of boys

i.e.  10 × 5 = 50

Therefore, Weight of the new boy = 100 + 50 = 150

Example 6:

What is the average of all the odd numbers from 1 to 100?

Solution:

The odd numbers from 1 to 100 are 1, 3, 5, 7, 9… 99.

You can clearly see the odd numbers are in arithmetic series with a common difference of 2.

Now, the short-cut to find average for numbers in arithmetic series is
(Lowest term + greatest term)/2.

Therefore, average of 1, 3, 5, 7, 9 ...99 is

(1 + 99)/2 = 50