Browse · MATH
Printjmc
counting and probability intermediate
Problem
At the Gooddog Obedience School, dogs can learn to do three tricks: sit, stay, and roll over. Of the dogs at the school: \begin{array}{l@{\qquad}l}
50 dogs can sit & 17 dogs can sit and stay \\
29 dogs can stay & 12 dogs can stay and roll over \\
34 dogs can roll over & 18 dogs can sit and roll over \\
9 dogs can do all three & 9 dogs can do none
\end{array} How many dogs are in the school?
Solution
We draw a Venn Diagram with three circles, and fill it in starting with the center and proceeding outwards. There are dogs that can do all three tricks. Since dogs can sit and roll over (and possibly stay) and dogs can sit, roll over, and stay, there are dogs that can sit, roll over, but not stay. Using the same reasoning, there are dogs that can stay, rollover, but not sit, and dogs that can sit, stay, but not rollover.
So now we know how many dogs can do multiple tricks, and exactly what tricks they can do. Since dogs can sit, dogs can sit and rollover only, dogs can sit and stay only, and dogs can do all three tricks, the remaining dogs that can't do multiple tricks can only sit, and there are of these. Using the same reasoning, we find that dogs can only stay and dogs can only roll over.
Since dogs can do no tricks, we can add that to each category in the Venn Diagram to find that there are a total of dogs.
So now we know how many dogs can do multiple tricks, and exactly what tricks they can do. Since dogs can sit, dogs can sit and rollover only, dogs can sit and stay only, and dogs can do all three tricks, the remaining dogs that can't do multiple tricks can only sit, and there are of these. Using the same reasoning, we find that dogs can only stay and dogs can only roll over.
Since dogs can do no tricks, we can add that to each category in the Venn Diagram to find that there are a total of dogs.
Final answer
84