Students in Class Question

[mjsobo]

A student of mine recently emailed me this question asking me how to solve it. It's a good GMAT question.

In a group of 68 students, each student is registered for at least one of three classes - History, Math and English. Twenty-five students are registered for History, twenty-five students are registered for Math, and thirty-four students are registered for English. If only three students are registered for all three classes, how many students are registered for exactly two classes?
A. 13 B. 10 C. 9 D.8 E. 7

This question can be sort of tricky. So we are given that there are 68 total students, but a total of 25+25+34=84 registrations for classes. We also know that there are 3 students registered for 3 classes, so these three students take up 3*3=9 registrations. This leaves us with with 84-9=75 registrations between 68-3=65 students. So we can set up an equation for this.

Let x be the number of students in 2 classes.

75=2x+65-x <<---Bc the students in 2 classes take up to registrations.
10=x

So we know there are 10 students in 2 classes, 65-10=55 students registered for 1 class and the 3 students enrolled in 3 classes for a total of 68 students.

As a check, we can multiply the registrations for each student and add them to make sure they total 84.

3 students enrolled in 3 classes = 9 registrations

10 students enrolled in 2 classes = 20 registrations

55 students enrolled in 1 class = 55 registrations
Total of 84 Registrations. Our answer of B, 10 students, checks.

[vivek.kapoor83]

pls explain my query as attached

[Stuart@KaplanGMAT]

pls explain my query as attached

Hi Vivek,

the problem is that you've over-subtracted.

By subtracting x, y and z twice each, you've completely removed them from the equation.

We want to ensure that we count each portion of the circles exactly once. Since x, y and z appear in 2 circles each, we're counting them twice; by subtracting each of them once, we avoid duplication.

Similarly, the very centre of the diagram is part of all 3 circles; to ensure that we only count it once, we need to subtract it twice.

So, your calculation should have been:

68 = 25 + 25 + 34 - x - y - z - 2(3)

68 = 84 - 6 - x - y - z

x + y + z = 78 - 68 = 10

If you want further info on the 3-set equation (and other explanations of this particular question), take a look at https://www.beatthegmat.com/mgmat-cat-2- ... tml#202418.

[vivek.kapoor83]

Thanks Stuart for such a great help.
but some ques are still there in my mind.
Why we have to subtract the overlapping only once as it is appearing twice in venn's diagram and similarly why we have to subtract twice instead of thrice in overlapping of 3 sets.Logic behind it.

However, I just realized and solved like this
only in region 1 = 25-x-y-3...eq(1)
only in region 2 = 25-y-z-3..eq(2)
only in region 3 = 34-x-z-3..eq (3)
and also adding x+y+z+3 ....eq(4)
adding eq 1 to eq 4

68 = 78-x-y-z
or x+y+z = 10
is this the correct approach ?

[Stuart@KaplanGMAT]

Thanks Stuart for such a great help.
but some ques are still there in my mind.
Why we have to subtract the overlapping only once as it is appearing twice in venn's diagram and similarly why we have to subtract twice instead of thrice in overlapping of 3 sets.Logic behind it.

However, I just realized and solved like this
only in region 1 = 25-x-y-3...eq(1)
only in region 2 = 25-y-z-3..eq(2)
only in region 3 = 34-x-z-3..eq (3)
and also adding x+y+z+3 ....eq(4)
adding eq 1 to eq 4

68 = 78-x-y-z
or x+y+z = 10
is this the correct approach ?

You can definitely add the equations like that; in fact, adding them together is how we derive the general equation.

We want to count each portion of the diagram exactly once.

We subtract the double sections once each because they appear in two circles and thus get counted twice; we subtract the triple section twice because it appears in three circles and thus gets counted three times.

If we were to subtract the double sections twice, then we wouldn't be counting them at all; similarly, if we were to subtract the triple section three times, we wouldn't be counting it at all.

[vivek.kapoor83]

ya true Stuart !!! thanks for the great help