2012 CEMC Gauss (Grade 8) Problems/Problem 13

Problem

Three numbers have a mean (average) of $7$ . The mode of these three numbers is $9$ . What is the smallest of these three numbers?

$\text{ (A) }\ 1 \qquad\text{ (B) }\ 2 \qquad\text{ (C) }\ 3 \qquad\text{ (D) }\ 4 \qquad\text{ (E) }\ 5$

Since the mode of these three numbers is $9$ , we can infer that at least two of the numbers are $9$ .

Let $x$ be the number that we don't know. Setting up an equation involving the average, we get:

$\frac{x + 9 + 9}{3} = 7$

$\frac{x + 18}{3} = 7$

$x + 18 = 21$

$x = 3$

$3 < 9 = 9$ , so this is the smallest number. Thus, the answer is $\boxed {\textbf {(C) } 3}$ .