Calculation of the Mean of Eleven Consecutive Natural Numbers
To find the mean of eleven consecutive natural numbers, we need to first determine the middle number in the series. Since the numbers are consecutive, the middle number will be the average of the first and last numbers in the series.
Let's assume the first number in the series is x. Therefore, the eleventh number will be x + 10. The middle number will then be (x + x + 10) / 2 = 2x + 5.
The mean of the eleven numbers will be the sum of all numbers divided by 11. Since the numbers are consecutive, the sum can be calculated by multiplying the middle number by 11.
So, the mean (m) of the eleven consecutive natural numbers will be m = 11(2x + 5) / 11 = 2x + 5.

Percentage Change in Mean when Next Six Numbers are Included
When the next six consecutive natural numbers are included in the group, the new mean will be the sum of all 17 numbers divided by 17.
The sum of the next six consecutive numbers can be calculated as the sum of the current 11 numbers plus the sum of the next six numbers. Since the numbers are consecutive, the sum of the next six numbers can be calculated by multiplying the middle number of the next six numbers by 6.
Let's denote the middle number of the next six numbers as y. Therefore, the new mean will be (11(2x + 5) + 6(y + y + 5)) / 17.
To find the percentage change in the mean, we can use the formula:
Percentage Change = ((New Mean - Old Mean) / Old Mean) * 100.
Substitute the expressions for the old mean (m) and the new mean into the formula and simplify to find the percentage change in the mean when the next six consecutive numbers are included.