2000 CEMC Gauss (Grade 8) Problems/Problem 14

Problem

A set of five different positive integers has an average (arithmetic mean) of $11$ . What is the largest possible number in this set?

$\text{ (A) }\ 45 \qquad\text{ (B) }\ 40 \qquad\text{ (C) }\ 35 \qquad\text{ (D) }\ 44 \qquad\text{ (E) }\ 46$

Solution

Let $x$ be the largest number possible. Because the average is $11$ and there are $5$ numbers, the sum of the numbers must be $11 \times 5 = 55$ .

To maximize $x$ , we can make the first four numbers as small as possible. All of the numbers are distinct and positive, so the first four numbers are $1$ , $2$ , $3$ , and $4$ . This means that we can set up an equation with $x$ :

$1 + 2 + 3 + 4 + x = 55$

$10 + x = 55$

$x = \boxed {\textbf {(A) } 45}$

~anabel.disher

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2000 CEMC Gauss (Grade 8) Problems/Problem 14

Problem

Solution