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**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