**Question 01**

**The average of 11 numbers is 10.9. If the average of the first six numbers is 10.5 and that of the last six numbers is 11.4, then the middle number is:**

Given: average of 11 numbers = 10.9

Sum of 11 numbers/ 11 = 10.9 (average = sum of data/ number of data)

Sum of 11 numbers = 10.9 x 11**Sum of 11 numbers = 119.9 ** ………Eq.1

Also given that average if first six numbers = 10.5

Sum of first six number/ 6 = 10.5

Sum of first six number = 10.5 x 6**Sum of first six number = 63 ** ……. Eq.2

Also given that average of last six numbers = 11.4

Sum of last six numbers/ 6 = 11.4**Sum of last six numbers = 11.4 x 6 = 68.4** ………Eq.3

The middle number has been added twice in the sum of first six and last six numbers

So, **sum of 11 numbers = sum of first six number + sum of last six numbers – middle number**

Middle number = (Sum of first six number + sum of last six number) – Sum of 11 numbers

= (63 + 68.4) – 119.9 (from Eq. 1, 2 and 3)

= 11.5

**Ans. Therefore, the middle number is 11.5**