Sets and Venn Diagrams. Some word problems concern sets and members of sets. So let's just say that the topic of sets overall is potentially complicated, because human beings can belong to several sets at once. Any one human being is a certain gender, they have a certain native language, a certain socioeconomic class, they have a favorite flavor of ice cream, etc. Read full transcript
There are hundreds of ways to classify human beings. It's a very complicated topic. Fortunately, the test focuses on a few relatively simple scenarios. The first set scenario, the easiest, is one in which there are two groups, and each member of the collection may belong to either group individually, or to both groups at once, or to neither.
For example, at a high school, the two groups might be on the baseball team and in the band. Obviously, some students will be on the baseball team, others will be on the band. There'll be a few talented students who are on both the baseball team and in the band, and, of course, there'll always be some who are not participating in either.
This scenario is often represented by a Venn Diagram. A basic Venn Diagram consists of two overlapping circles inside a rectangle. Occasionally, the test will give this diagram as part of the question. If not, students often find it helpful to sketch one for their calculations. Elements inside one circle are in one of the groups, and elements in the overlap region are in both groups.
Notice that this diagram consists of four distinct regions. So the red region, A, this is inside the left circle but not inside the right circle. So these are all the people in the green circle, inside the green circle, but outside the purple circle. So they're in the green circle only, not in the purple circle.
B is the overlap group. They're in both circles. C, these are all the people in the right circle, not inside the left circle, so these are in the purple circle only, and not in the green circle. And then D are all the people outside of both circles. So notice that these four regions, when we add them up, this adds up to the total.
Either the total number of students, or the total number of people that are in this particular study. Notice that the way information is given here, the verbal information, can be very tricky. Let's say for the sake of argument that the left circle means in the band, and the right circle means on the baseball team.
If the question says 35 students are in the band, well that means A + B = 35. Because when we say 35 are in the band, we're counting all the band members. Some of the band members are in the band only. Other band members happen also to be on the baseball team, but they're all in the band. And so both of them are counted together as in included when we say 35 are in the band.
But if we change the wording slightly, we say, 35 students are in the band only. Well, this means that A = 35. Now we're talking about the students that are not included in the purple circle, they're only in the green circle, and so that's region A alone. So it's very important to be extremely careful when you read the wording of these questions, because subtle changes in the wording can mean profound mathematical consequences.
A typical sets question of this variety almost always gives the size of the whole group, which would equal the sum of these four regions. The other information will provide some other regions, and from addition or subtraction, we can determine the rest. Here's a simple practice question. Pause the video and then we'll talk about this.
Okay, so we'll start out by drawing a Venn Diagram. And the green circle on the left, that's going to represent the band. The purple circle on the right, that's going to represent the baseball team. We know that 60 are in the band, so that's A + B = 60. 35 are on the baseball team, That's B + C = 35. Notice that the B's are counted twice.
They're counted once for their band membership, once for their baseball membership. 25, that's the region outside. Well that's interesting. So, if outside the circle there are 25, and there are 100 students all together, it means that all the people inside the two circles that must add up to 75.
Because 75 plus 25 equals 100, and so it means that A + B+ C = 75. Well it's interesting because B plus C by itself, we see at the top that's 35. So we can replace B plus C with 35, and that allows us to solve for A, A = 40. Now we can plug that into the first equation. A + B = 60, so 40 + B = 60, and B = 20, and that's our answer.
Notice, incidentally, if you're curious, A = 40, B = 20, C = 15, D = 25. And those together add up to 100. But our answer here is B = 20. Here's a slightly more complicated problem. Pause the video, and then we'll talk about this.
Okay, so, French and Spanish. Let the left circle equal French and the right circle equal Spanish. And what I'll say here, unfortunately I need a lot of work to work out the Algebra on this problem. So I had to go to a slide that didn't have text of the question on it, but if you look under this video you should see the text of the question printed, and so you can refer to that as I'm talking about this if you wanna read along with me.
So, the first thing it says, is just as many as study neither as study both. Well, the both region is B, the neither region is D, so B=D. Okay that's important. Second we're told one quarter of those who study Spanish also study French. Well those who study Spanish, that's B plus C so one quarter of B plus C Equals the ones who also study French, so those are the people in the overlap group.
So one quarter B + C = B. I'm just gonna clear the fractions by multiplying that equation by 4, and then it turns out I can subtract B from both sides, and I get C = 3B. And at this point, that gives me a bit of insight. This suggests a method of solution now, because we're able to express D in terms of B.
Now we can express C in terms of B. If we could express A in terms of B, then we could solve for the value of B, and that would allow us to solve for everything. Okay, so let's think about A. The final sentence is that the total number who study French is 10 fewer than those who study Spanish only.
So the total number who study French, A+B, is 10 fewer than Spanish only. Spanish only is C, so A + B = C- 10. So now we're just gonna plug in C = 3B and subtract B from both sides, so A = 2B- 10. So now we've expressed A, C, and D all in terms of B. This is important, because we know A + B + C + D = 200, and we can substitute those other letters for expressions involving B.
And so we'll just add 10 to both sides so we'll get 210 on the left, and then we have 2B + B + 3B + B, that adds up to 7b. So 7B = 210, divide by seven, we get B = 30 so now we have the value of B. Well, we're looking for how many students study French only so A, we're looking for, we'll just plug that into the expression for A. A = 2B- 10, or 2 * 30- 10.
60- 10 = 50, and that's our answer. In summary, Venn Diagrams can be helpful in problems with two overlapping sets. The problem may give a Venn Diagram, but if it doesn't, one if often helpful for solving this kind of question. It's good to sketch it on the scrap paper, and remember to be careful interpreting wording.
If it says all in X, like everyone in the band, or everyone studying French, or something like that, that always includes those who are also in the other group. It includes all the overlap, but if it says all those in X only, all those studying French only, all those on the baseball team only, then that excludes the overlap group. We have to be very careful interpreting the work.