+656 There are twice as many students in the cooking club as in the drama club. Suppose there are \(a \) students in the drama club and \(b \) students who are members of both clubs. Find an expression for the total number of students who are in the cooking club or the drama club but not both. Give your answer in simplest form.
Let c be the number of students in the cooking club and d be the number of students in the drama club but not the cooking club. Then we have the following equations:
c = 2d a = c + d
Solving for d, we get d=a/3. Therefore, the number of students who are in the cooking club or the drama club but not both is c−d = 2d−d = a/3.
+177 Sometimes, it is hard to wrap one's head around how to determine how one quantity is related to another without a visual picture. As a result, I decided to create a Venn diagram that should make the situation clearer. I have included all the given information in the Venn diagram.
The given information states that there are twice as many students in the cooking club as in the drama club. Since we are using the a-variable to represent the number of students in the drama club, this means that 2a students are in the cooking club. We are letting the b-variable be in the middle; this represents the students who are members of both clubs. Our objective is to find an expression that represents the students in the one of the two clubs but not both.
I will start with the cooking club. The cooking club has 2a members. With the help of the diagram, it is clear there are b students who are members of both. Therefore, 2a - b represents the number of students who are only apart of the cooking club.
Considering the drama club, there are a total members. There are b students who are members of both. Therefore, a - b represents the number of students only participating in the drama club.
Now, we just find the sum of these parts. \(2a - b + a - b = 3a - 2b\) students who are members of one of the clubs but not both.
