In a class of 80 students, each of them studies at least one language

Question

In a class of 80 students, each of them studies at least one language - English, French and German. It was found that 65 studied English,60 studied French and 55 Studied German. Find the Maximum number of people who studied all the 3 languages.

A. 40
B. 55
C. 50
D. 65
E. 45

I am baffled how the official answer is correct, I assume another answer.

A. 40
B. 55
C. 50
D. 65
E. 45

OhsostudiousMJ · Answer

Okay, here's my take.

Total = A+B+C-(Sum of exactly 2 overlaps)-2(All 3 overlaps) + Neither.

So, to maximize "All 3 overlaps", the other two terms in the equation must be 0. That is, the only overlaps must be that of all 3 languages.

Therefore, 80 = 65+60+55-2(All3)
Answer = 50.  C

GMATinsight · Answer

The answer is correct as maximum amount common to all three sets = 50

Please check the figure

Answer: Option C

btrg · Answer

Cant we have 55 as answer. If 55 students study all then 10 study only English 5 study only french and 0 study only german.

Posted from my mobile device

GMATinsight · Answer

btrg

No, we can't have that number 55 as the sum needs to be balanced 80 and if you make common intersection as 55 then sum won't be 80.

sidship21 · Answer

Hi Bunuel hiranmay VeritasKarishma ,

Could you please explain why the lines approach, as you used in this question (https://gmatclub.com/forum/in-a-village ... 98257.html) doesn't work here? I really liked the lines approach you used in the referred question, but somehow it doesn't work here, or else the answer would be 55.

Thanks a lot

nick1816 · Answer

Since we want to maximize the number of of people who studied all the 3 languages, minimize the number of people who studied only 2 languages. Untitled.png We get, (65-x)+(60-x)+(55-x)+x = 80 2x = 100 x = 50

KarishmaB · Answer

Here, each of them studies at least 1 language so if 55 overlap, you can account for only 5 + 10 = 15 more students. That gives 70 students, not 80.

Think in terms of instances and people instead. 
You have 65 + 60 + 55 = 180 instances.
You have 80 people such that each has at least 1 instance.

So of 180, distribute 80 away to 80 people. Now you are left with 100 instances. Give 2 each to 50 people so that 50 people have all three.
This is the maximum possible.

ThatDudeKnows · Answer

We can Plug In The Answers (PITA). GMAC would have put the answer choices in order, and I like trying the second and fourth answer choices.

40
 45 
50
 55 
65

45: All three = 45. We need 20 more in E, 15 more in F, and 10 more in G. That's 45+20+15+10 = 90. Too many.
55: All three = 55. We need 10 more in E, 5 more in F, and 0 more in G. That's 55+10+5+0 = 70. Too few.

We need something in the middle. 50.

Answer choice C.

ThatDudeKnowsPITA

ragnarok13 · Answer

@ KarishmaB  Ma'am, suppose French overlaps completely with German, so we have 20 remaining. And then for English, it will have to include that extra 20 and the rest of 45 it will overlap with the common French and German region. So 45 is the maximum overlap. 
Can you please explain how we are arriving at 50. I have also tried with English and French overlap (and in that case i was getting 40 which isnt the minimum) but i am not able to understand 50 with the diagrams. Kindly help. Thanks.

In a class of 80 students, each of them studies at least one language

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In a class of 80 students, each of them studies at least one language

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