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At a certain school with 200 students, all children must

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At a certain school with 200 students, all children must [#permalink] New post 05 Jul 2012, 15:15
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At a certain school with 200 students, all children must take at least one of three language classes: German, French, and Spanish. If 100 students take German and none of the students who take French also take Spanish, then how many students take exactly two of the three language classes?

(1) 80 of the students study only German.
(2) 120 students study French or Spanish
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Re: At a certain school with 200 students, all children must [#permalink] New post 06 Jul 2012, 14:02
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Refer to the pic: G=students taking German; F= students taking French; S= students taking Spanish

According to the problem:
total = 200 students
G=100
students outside G,F,S = 0
F and S have no intersection (as drawn on the pic) -> there are no students taking all 3 classes.

1) 80 students are only G -> G = (only G) + GF + GS, the intersections GF + GS = 20. Sufficient.

2) F + S = 120 (they have no intersection) -> Total = G + F + S - (GF + GS) (compensating for double counting). Everything in that equation is given except the intersections GF+GS which can be determined. Sufficient.
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Re: At a certain school with 200 students, all children must [#permalink] New post 31 Jan 2014, 04:51
The key information in the question is none of the students who take French also take Spanish This means there is no overlap between French and Spanish at all. It also means 0 people take all the three languages and 0 people take Spanish and French together.

With this information, exactly two languages can only be {German+french} or {German+Spanish}

Stmt 1: 80 take only German. Given 100 take German. So 100-80 =20 take German+one other language Sufficient.
Stmt2: 120 taken French or spanish. That is {French}+{Spanish} = 120
We can use this formula:

{German}+{French}+{Spanish}-{Exactly two languages}-2*{all the three} = total
100+120-{Exactly two languages}-2*0=200
{Exactly two languages}=220-200 = 20.

Stmt2 is also Sufficient.

Hence D.
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Re: At a certain school with 200 students, all children must [#permalink] New post 15 Apr 2014, 13:40
I would just use two simple venn diagrams for this problem. Let's proceed

Statement 1 tells us that German only is 80. Therefore, since we know that non of the students speak 3 languages or both french and spanish we can infer that German+French and German+Spanish = 20 students. Thus, we have a total of 20 students that speak exactly two languages.

Sufficient

Statement 2, we would have the same diagram knowing that Spanish of French only represent 120 total. Therefore, since we know the grand total is 200 if we add all German + (Spanish or French) we get 220. Now, since our total is 200 and none of them speaks 3 languages them we have that exactly 2 must be equal to 20.

Sufficient

D
Re: At a certain school with 200 students, all children must   [#permalink] 15 Apr 2014, 13:40
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