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Re: GRE Quant - Overlapping Set Theory [#permalink]
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How to Use a Double Matrix on the GRE



Using a Venn diagram when dealing with most Sets questions is correct and effective, there are some questions in which a double-matrix is necessary (and much more powerful than a wimpy Venn). This little guy will make even the scariest-looking Sets question into a simple set of rows and columns, and its ability to help us determine whether a statement is sufficient in DS is unmatched!

To make a double-matrix, simply create a chart with rows and columns. The rows are assigned to one variable, and the columns to another. At the bottom of each column and at the far-right of each row, place a box for the column-total and row-total. Let’s check out a sample GMAT question to see what this might look like!

Question: 33 out of the 47 students in an advanced degree program have a higher than average GPA. How many students in the program are receiving some form of academic scholarship?

(1) More students do not have a scholarship than have a scholarship.
(2) The same number of students have a higher than average GPA and are receiving some form of academic scholarship as have neither a higher than average GPA nor an academic scholarship.

Solution: From the question-stem, we know 33 students have a high GPA, while 14 do not. We need more information about which of these students have scholarships to be able to answer this value question. Statement (1) is insufficient because it does not give us information to find the exact numerical value of the students receiving some form of scholarship.

Statement (2) tells us that the number of students who fit “both” is equal to the number of students who fit “neither.” Let’s set up a chart to visualize the four possible categories for the students. Since “both” = “neither,” let’s fill in “x” for those boxes.

Image

Since each column and row must total, if there are “x” students receiving no scholarship and not a higher GPA, and the total students who don’t have a higher GPA is 14, then 14-x students must not have a higher GPA and have a scholarship. The total we are looking for is represented by the red “?,” and we can set up an equation to solve: x + (14 – x) = 14. Sufficient.

The correct response is (B).


To wrap up, keep in mind that it’s possible (although highly unlikely) that you might see a Sets question on the GRE involving three variables instead of two. In that case, you’d have 5 columns instead of 4, and 5 rows instead of 4, but the same rules of totaling apply!
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Re: GRE Quant - Overlapping Set Theory [#permalink]
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How to draw a Venn Diagram




At a certain school, each of the 150 students takes between 1 and 3 classes. The 3 classes available are Math, Chemistry and English. 53 students study math, 88 study chemistry and 58 study english. If 6 students take all 3 classes, how many take exactly 2 classes?
A. 37
B. 43
C. 45
D. 60

Step 1: Deconstruction

This is where you extract the given information from a problem. Use a symbol for each section given to you. I usually use the first letter in caps. Please remember at this point that any number specified without explicitly mentioning that it's "exactly" or "only" for a certain subject is not to be taken so.

For instance, in this case, it says 53 math students. This doesn't mean students who are taking only 1 subject (Math). This could include students who are taking Math and Chemistry or Math and English or all three too. So now break down the numbers given to us.

Total = 150
M (Total) = 53
C (Total) = 88 [Why the hell are people studying more Chemistry than Math?]
E (Total) = 58
MCE = 6

We are asked to find MC + CE + EM.

Step 2: Drawing the diagram

ALWAYS start from the center of the Venn Diagram wherever information is available. This will make life infinitely easier.

In this case, the center is the intersection of all three circles, i.e MCE = 6

So fill that in to the diagram you've drawn.

Image

Now that you've gotten that, let's start filling in a variable for each section not known to us. Here, consider each letter to represent only that specific section and not the entire circle or a larger portion.

Image

So now that you have the parts filled in, what you need to do is write down what you have in the diagram in terms of numbers. So we are given the totals for each subject.

Look at math first. There are four types of people taking math (each group of these people mutually exclusive, and not in common with any other group)
1. Only math: a
2. Math and Chemistry: y
3. Math and English: x
4. All three: 6

So now represent this as a sum and you get

\(a+x+y+6 = 53\) and hence \(a+x+y = 47\)

Similarly for the other subjects you get:
\(b+x+z+6 = 58\) and hence \(b+x+z = 52\)

\(c+y+z+6\) = 88 and hence \(c+y+z = 82\)

And then you have the total:

\(x+y+z+a+b+c+6=150\) and hence \(x+y+z+a+b+c = 144\)

Step 3: Solution

This is perhaps the most intuitive part, but in my experience the first part of this step is the same in all overlapping sets problem. It's only what's asked for that's different.

Add all the individual equations together to get a combined equation with all the variables and a number.

So you get:

\(2(x+y+z) + a+b+c = 47+52+82 = 181\)

Rearranging this to get \(a+b+c\)we get \(a+b+c = 181 - 2(x+y+z)\)

Substitute this into the total equation we derived earlier saying \(x+y+z+a+b+c = 144\) so you get:

\(x+y+z+181 - 2(x+y+z)= 144\)

Upon rearranging this you get:

\(x+y+z = 181-144 = 37 \) which is option A the right answer.
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Re: GRE Quant - Overlapping Set Theory [#permalink]
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Re: GRE Quant - Overlapping Set Theory [#permalink]
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1) GRE Lesson: Double Matrix Method



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Re: GRE Quant - Overlapping Set Theory [#permalink]
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Re: GRE Quant - Overlapping Set Theory [#permalink]
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