2024 CEMC Gauss (Grade 8) Problems/Problem 9

Problem

In a drawer, the ratio of the number of spoons to the number of forks is $1:2$ . The total number of spoons and forks in the drawer cannot be equal to

$\textbf{(A)}\ 12 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 3$

Solution

Let $s$ be the number of spoons. Since the ratio of the number of spoons to the number of forks is $1:2$ , we can multiply the left and right side of this ratio by $s$ to find that the number of forks is $2s$ .

We can now see that the total number of spoons and forks in the drawer is $s + 2s$ = $3s$ . Since $s$ is a positive integer, $3s$ must also be a positive integer, but must also be divisible by $3$ .

The only answer of the answer choices that is not divisible by $3$ is $\boxed {\textbf {(D) } 10}$ .

~anabel.disher

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2024 CEMC Gauss (Grade 8) Problems/Problem 9

Problem

Solution