When 15 is included in a list of natural numbers, their mean is increased by 2. When 1 is included in this new list, the mean of the numbers in the new list is decreased by 1. How many numbers were there in the original list?

Correct option is A

Given:

When 15 is included in a list of natural numbers, their mean is increased by 2.
When 1 is included in the new list, the mean of the numbers in the new list is decreased by 1

Formula Used:

Mean of original list = $$\frac{S }{ n}$$​​
Mean of new list after adding  $$15 =\frac{ (S + 15) }{(n + 1)}$$​​
Mean after adding 1 = $$\frac{ (S + 16) }{ (n + 2)}$$​​
Solution:

Let the number of natural numbers in the original list be n, and the sum of the numbers be S.
The mean of the original list is: Mean = $$\frac{S }{ n}$$​​
After adding 15 to the list, the new mean is: $$\frac{(S + 15) }{ (n + 1)}$$​ and we know this is increased by 2. So:
​$$\frac{(S + 15) }{ (n + 1) }= (\frac{S }{n}) + 2$$​​

After adding 1 to the new list, the sum becomes (S + 16), and the number of terms becomes (n + 2). The mean is decreased by 1, so:

​$$\frac{(S + 16) }{ (n + 2) }=\frac {(S + 15) }{ (n + 1) }- 1$$​​
Solving these two equations will give the value of n, the number of terms in the original list.
After solving, we find n = 5.