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70 students are enrolled in maths, english and german.

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70 students are enrolled in maths, english and german.  [#permalink]

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New post 30 Dec 2011, 08:16
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70 students are enrolled in maths, english and german.
40 are in maths
35 are in english
30 are in german
15 students are involved in all three of the courses
How many are enrolled in exactly 2 of the 3 courses - maths, english and german.

The answer explanation given is
Let N stand for the number enrolled in exactly 2 courses
70=40+35+30-N-2(15)
N = 5

I don't understand why they do 2 times 15 :- 2(15)

Could someone please explain.
Thank you in advance.
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Magoosh GMAT Instructor
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Re: 70 students are enrolled in maths, english and german.  [#permalink]

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New post 30 Dec 2011, 11:21
Hi, there! I'm happy to help with this question. :)

The Barron's answer is correct, but they shot through some complicated stuff awfully quickly.

Fact #1 = There are 70 students total.

Fact #2 = There are 40 in maths, 35 in English, 30 in German, for a total of 105.

This 105 is, of course, more than the total number of students ---- this is because the folks taking exactly two courses (the "doublers") have been counted twice, and the folks taking three courses (the "triplers") have been counted three times.

Fact #3 = 15 students are taking all three, i.e. there are 15 triplers.

So, if we want to make the 105 number jive with the actual total of 70, we have do two things
a) subtract the doublers once -- they are counted twice, and we only want them counted once
b) subtract the triplers twice --- they are counted three times, and we only want them counted once, so we have to subtract twice that total. That's where the mysterious factor of 2 arises ------ it's what you have to subtract from how many times they've been counted (3x) so that you are left with only counting them once. (If we subtracted the number of triplers times 3, the result would be that they wouldn't be counted at all!)

If we call the number of doublers N, this logic leads us to the Barron's equation

70 = 105 - N - 2*15

which leads to the Barron's answer of N = 5.

Does that make sense? Please let me know if you have any further question.

Mike :-)
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Mike McGarry
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Re: 70 students are enrolled in maths, english and german.  [#permalink]

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New post 30 Dec 2011, 11:30
Bunuel did excellent discussion on the topic covering whole concept on the following link:

formulae-for-3-overlapping-sets-69014.html
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Re: 70 students are enrolled in maths, english and german.  [#permalink]

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New post 30 Dec 2011, 12:42
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Hi Mike
Thank you very much.

I worked through this as follows and took quiet some time. I am not very good at understanding and applying formulas...

Fact 1 :- 70 students are taking either 3 or 2 or 1 subject
Fact 2 :- 15 are definitely taking all 3

so now 70-15= 55 remaining are taking 2 or 1 subject

When you draw a Venn diagram ..it is seen..
Students taking Maths remaining are 40 - 15 = 25
Students taking English are 35-15=20
Students taking German are 30 - 15 = 15

So sum of these are total students taking only 1 subject = 60

Therefore 60-55=5 are students who take only 2 subjects.

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Re: 70 students are enrolled in maths, english and german.  [#permalink]

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New post 31 Jan 2019, 03:58
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: 70 students are enrolled in maths, english and german.   [#permalink] 31 Jan 2019, 03:58
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