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?

Solution(By Examveda Team)

Let there be n numbers in the original list and let their mean be x.
Then, sum of n numbers = nx
$$\eqalign{ & \therefore \frac{{nx + 15}}{{n + 1}} = x + 2 \cr & \Rightarrow nx + 15 = \left( {n + 1} \right)\left( {x + 2} \right) \cr & \Rightarrow nx + 15 = nx + 2n + x + 2 \cr & \Rightarrow 2n + x = 13.....(i) \cr} $$
And,
$$\eqalign{ & \therefore \frac{{nx + 16}}{{n + 2}} = \left( {x + 2} \right) - 1 \cr & \Rightarrow nx + 16 = \left( {n + 2} \right)\left( {x + 1} \right) \cr & \Rightarrow nx + 16 = nx + n + 2x + 2 \cr & \Rightarrow n + 2x = 14.....(ii) \cr} $$
Solving (i) and (ii), we get:
n = 4, x = 5