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### Solving for Set Sizes

###### Answered Questions

1.

On the floor of your residence hall, 2/5 of students plan to major in business, 3/10 plan to major in computer science, and 1/8 plan to major in both. What share of students plan to major in exactly one of either business or computer science?

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There are a couple of ways to solve this problem, but in both cases, we need to first visualize the problem as a Venn diagram. In one circle, we have the share of students planning to major in business; in the other, the share planning to major in computer science; and in the middle, the share planning to do both.

The first method is to find the union of the two majors (thus, the total number of students in the problem). To find the union, per the formula described in the tutorial for this topic, we add together the share in each major, then subtract the number in their intersection. This gives us 2/5 + 3/10 - 1/8 = 23/40.

Next, we subtract the number in the intersection again because the problem asks for only those who are majoring in exactly *one* major (and those in the intersection are majoring in two). Therefore, 23/40 - 1/8 = 9/20.

The other way to solve this would be to draw our Venn diagram, which has three sections (the intersection, the section for *only* business majors, and the section for *only* computer science majors), then fill in each section and add together the two sections of students with only one major.

The intersection, as we know, is 1/8. And to find the sections for students with only one major, we must take the share of students in each major and subtract the intersection. For students in business, we have 2/5 - 1/8 = 11/40. For students in computer science, we have 3/10 - 1/8 = 7/40.

Then, after adding together 11/40 and 7/40, we get 18/40, which reduces to 9/20, same as above.