Venn Diagram question

[bigguy_aug]

I found this question in #300 Quant questions.
I do not understand the solution.

It will be great if someone can solve and explain their thought process.

Thanks

Here is the question:

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 at least 2 classes?

[kvcpk]

Atleast 2 classes means 2 classes or 3 classes.

number of students who take 3 classes = 6
number of students who take 2 classes = X+Y+Z

P+X+Y+6 = 53
Q+X+6+Z = 88
R+Y+6+Z = 58
Adding these 3,
P+Q+R+X+Y+Z+X+Y+Z+18 = 199

But we know that
P+Q+R+X+Y+Z+6 = 150

Hence X+Y+Z+12 +150 = 199
X+Y+Z = 37

So number of students who took atleast 2 classes = 37+6=43

Hope this helps!!

[bigguy_aug]

Thanks so much for such a detailed explanation.!!

[bigguy_aug]

I agree that the "number of students who take 2 classes = X+Y+Z"

and the fact that X+Y+Z = 37

but I do not understand why u added 6 to 37. (So number of students who took atleast 2 classes = 37+6=43_

[kvcpk]

I agree that the "number of students who take 2 classes = X+Y+Z"

and the fact that X+Y+Z = 37

but I do not understand why u added 6 to 37. (So number of students who took atleast 2 classes = 37+6=43_

Hi,

The question is asking for the number of students who took ATLEAST 2 courses.
Which means, it is the sum of
Number of students who took exactly 2 courses (=x+y+z = 37)
and
Number of Students who took exactly 3 courses. (= 6)

Thats the reason we get 37+6 = 43.

Hope this helps!!

[GMATGuruNY]

I found this question in #300 Quant questions.
I do not understand the solution.

It will be great if someone can solve and explain their thought process.

Thanks

Here is the question:

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 at least 2 classes?

Here is the formula for 3 overlapping groups in which sometimes 2 of the groups overlap and sometimes all 3 groups overlap:

T = G1 + G2 + G3 - (those in 2 of the groups) - 2*(those in all 3 groups)

The trick with overlapping group problems is to subtract the overlap. When we add together all the students who study math, all who study chemistry, and all who study English, those who study 2 subjects will be counted twice, so they need to subtracted from the total once. Those who study all 3 subjects will be counted 3 times, so they need to be subtracted from the total twice.

In the problem above:
T = 150
G1+G2+G3 = Math + Chemistry + English = 53+88+58
Students who study 2 subjects = x
Students who study all 3 subjects = 6

Plugging into the formula, we get:

150 = 53 + 88 + 58 - x - 2*6
150 = 187 - x
x = 37 students who study exactly 2 subjects.

Since 37 students study exactly 2 subjects and 6 students study all 3, 37+6 = 43 students who study at least 2 subjects.