2002 AMC 10B Problems/Problem 25

Problem

When $15$ is appended to a list of integers, the mean is increased by $2$ . When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1$ . How many integers were in the original list?

$\mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 6\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 8$

Solution

Let $x$ be the sum of the integers and $y$ be the number of elements in the list. Then we get the equations $\frac{x+15}{y+1}=\frac{x}{y}+2$ and $\frac{x+15+1}{y+1+1}=\frac{x+16}{y+2}=\frac{x}{y}+2-1=\frac{x}{y}+1$ . With a lot of algebra, the solution is found to be $y= \boxed{\textbf{(A)}\ 4}$ .

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2002 AMC 10B Problems/Problem 25

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