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?

Answer: Option A

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