Know the GMAT Code: Overlapping Sets - Part 1
GMAT story problems can be pretty frustrating. First, you have to turn the problem into math, and then you still have to do the mathall in 2 minutes!
Its important to be able to recognize (quickly!) the kind of story problem you have and to have a plan for tackling that type of story. Thats exactly what were going to talk about today, in our next installment of the Know the Code series.
Try this problem from the GMATPrep free exams and then well talk!
*Of the students who eat in a certain cafeteria, each student either likes or dislikes lima beans and each student either likes or dislikes brussels sprouts. Of these students, [pmath]2/3[/pmath] dislike lima beans; and of those who dislike lima beans, [pmath]3/5[/pmath]also dislike brussels sprouts. How many of the students like brussels sprouts but dislike lima beans?(1) 120 students eat in the cafeteria.
(2) 40 of the students like lima beans.
(Havent done Data Sufficiency before? Or are you new enough to DS that youre wondering where the answer choices are? Start here and come back to this article later.)
Okay, Im going to take you through my thought process as I did this problem.
1-second Glance. Wall of text. DS. Fractions. (Is this going to be pure fractions? Or fractions layered into something else?)
Read to find out. Theres this group of students and some like this one thing or dislike it and then some others like something else or dislike it This is overlapping sets!
Im going to jump ahead in my process and Reflect a bit, even though I havent Jotted anything down yet. Why? Because jotting on a sets problem can be quite involved. I need to make sure that Im on the right path.
What type of set is this one?
We call this particular type of sets problem a double-set matrix. Theres a whole group of something (often people, but not always) and that group gets cut in two different ways. In this problem, the first cut is like or dislikes lima beans. You can be in one sub-group or the other sub-group, but you cant be in both cuts at the same time.
The second cut is likes or dislikes brussels sprouts. Again, everyone in the overall group is in one sub-group or the other.
Each of those sub-groups, or cuts, is considered a set, hence the name double-set matrix.
Now, what happens when you combine these two sub-groups together? Heres where the matrix part comes in. Check this out:
L stands for likes lima beans and ~L stands for the opposite (dislikes lima beans). In general, you can use that ~ symbol to mean the opposite of what that letter stands for.
Likewise, B means likes brussels sprouts and ~B means dislikes brussels sprouts.
All of the boxes stand for different cuts of the data. Theres a box for likes both L and B, a box for Total likes B (regardless of whether they like L), a box for likes neither L nor B, and even a box for Total Total: all people who are getting cut up into all of these smaller sub-groups.
And the problem is talking about various cuts of the data that are represented by this matrix, so yes, this is the path I want to go down. Time to start filling stuff into the matrix.
Of the students who eat in a certain cafeteria, each student either likes or dislikes lima beans and each student either likes or dislikes brussels sprouts. Of these students, [pmath]2/3[/pmath] dislike lima beans; and of those who dislike lima beans, [pmath]3/5[/pmath]also dislike brussels sprouts. How many of the students like brussels sprouts but dislike lima beans?
The first sentence is already taken care of just by constructing the matrix in the first place. Next, 2/3 of these students dislike lima beans. Start with of these studentswhat box in the matrix does this represent?
These students means all students: the Total Total. But we dont know how many students there are overallhmm.
Hey, the problem has fractions in it. Could we just pick a Smart Number for the total, like 100?
Thats a good thought to have, but we cant on this problem. Take a look at the full problem again to try to figure out why.
The two statements are part of the problem and they contain real numerical values for the various cuts of the number of students. If a problem contains a real number for something, then you cant choose a smart number for that thing.
Instead, use variables:
The first piece of info says nothing about whether these people like B, just that they like L, so that is the Total likes L box.
What about the second bit?
Of these students, [pmath]2/3[/pmath] dislike lima beans; and of those who dislike lima beans, [pmath]3/5[/pmath]also dislike brussels sprouts.
Careful! Theres a trap here. The single biggest, most consistent trap on overlapping sets problems comes down to this one question: which group are we talking about now? The first half of the info was talking about 2/3 of total students. The second half, however, is no longer talking about total students.
Rather, the second half is talking about only those who dislike L. Of the Total dislike L people, 3/5 also dislike B. What do we do with that?
Heres a tidbit we havent talked about before: in the double-set matrix, all of the rows add to the final box. And all of the columns add to the final box. So if 2/3 of all the people like L, then 1/3 of all of the people must dislike L.
Now, how to show that 3/5 of those Total dislike L people also dislike B?
3/5 of means 3/5 multiplied by
So multiply the Total dislike L people, which is (1/3)T, by 3/5.
Finally, what do they want to know?
How many of the students like brussels sprouts but dislike lima beans?
Mark this clearly on your diagram. Dont do any more math at this point (for instance, I wouldnt even multiply out 3/5 and 1/3). This is DS; I don't need to do the math. I just need to know that it can be done.
Time to look at the statements.
(1) 120 students eat in the cafeteria.
This is telling us the total number of students, T. If you know T, can you find the desired purple box?
Yes! First, you can figure out all of the boxes that have Ts in them. Next, remember that all columns add down (and all rows add across). So if you know two of the three entries in column ~L, then you can figure out the third one, the purple one. Sufficient! Cross off answers BCE.
(2) 40 of the students like lima beans.
Which box is this? The statement refers to all students who like L, regardless of how they feel about B, so this it the Total likes L box. If you know that 40 = (2/3)T, then you can find T. And if you can find T, then you can find the purple box, as we saw for statement (1). Sufficient again!
The correct answer is (D).
How do you feel about that? Want to try another? Here you go (also from the free GMATPrep tests).
*A certain one-day seminar consisted of a morning session and an afternoon session. If each of the 128 people attending the seminar attended at least one of the two sessions, how many of the people attended the morning session only?(1) [pmath]3/4[/pmath]of the people attended both sessions.
(2) [pmath]7/8[/pmath]of the people attended the afternoon session.
Join us next time, when well discuss the answer to the above problem. And see below for our Know the Code takeaways!
Key Takeaways for Knowing the Code:
(1) When youre studying, dont just stop when you see the textbook answer to a problem. Push yourself to find the patterns or to articulate the underlying principles behind what youre seeing.
(2) Then turn that knowledge into Know the Code flash cards:
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.