Difference between revisions of "2004 AMC 8 Problems/Problem 18"

Problem

Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers $1$ through $10$. Each throw hits the target in a region with a different value. The scores are: Alice $16$ points, Ben $4$ points, Cindy $7$ points, Dave $11$ points, and Ellen $17$ points. Who hits the region worth $6$ points? $\textbf{(A)}\ \text{Alice}\qquad \textbf{(B)}\ \text{Ben}\qquad \textbf{(C)}\ \text{Cindy}\qquad \textbf{(D)}\ \text{Dave} \qquad \textbf{(E)}\ \text{Ellen}$

Solution 1

The only way to get Ben's score is with $1+3=4$. Cindy's score can be made of $3+4$ or $2+5$, but since Ben already hit the $3$, Cindy hit $2+5=7$. Similarly, Dave's darts were in the region $4+7=11$. Lastly, because there is no $7$ left, $\boxed{\textbf{(A)}\ \text{Alice}}$ must have hit the regions $6+10=16$ and Ellen $8+9=17$.

Solution 2

Taking the 2 largest scores to narrow out the points we get

Ellen with either $10,7$ or $9,8$ and Alice with either $10,6$ or $9,7$, the 2 pairs that work are Alice= $10,6$ Ellen= $9,8$, therefore Alice scored the $6$.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 