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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?

A. 4 B. 5 C. 6 D. 8 Answer: Option A
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Solution(By Apex Team)

Let there be n numbers in the original list and let their mean be x. Then, sum of n numbers = nx $\begin{array}{l}\therefore\large\frac{nx+15}{n+1}=x+2\\ \Rightarrow nx+15=(n+1)(x+2)\\ \Rightarrow nx+15=nx+2n+x+2\\ \Rightarrow2n+x=13\ldots..(i)\\ \text{ And, }\\ \therefore\large\frac{nx+16}{n+2}=(x+2)-1\\ \Rightarrow nx+16=(n+2)(x+1)\\ \Rightarrow nx+16=nx+n+2x+2\\ \Rightarrow n+2x=14\ldots..(ii)\\ \text{Solving (i) and (ii), we get:}\\ \text{n = 4, x = 5}\end{array}$

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