At the Zamenhof Language School, at least 70% of the students take Eng

Question

At the Zamenhof Language School, at least 70% of the students take English each year, at least 40% take German each year, and between 30% and 60% take Italian each year. Every student must take at least one of these three languages, and no student is allowed to take more than two languages in the same year. What is the possible percentage range for students taking both English and German in the same year?

(A) 0% to 70%
(B) 0% to 100%
(C) 10% to 70%
(D) 10% to 100%
(E) 40% to 70%

(A) 0% to 70%
(B) 0% to 100%
(C) 10% to 70%
(D) 10% to 100%
(E) 40% to 70%

This is one of a set of 15 challenging GMAT Quant questions. For the whole set, including the OE to this question, see: 
 Challenging GMAT Math Practice Questions

pandeyashwin · Accepted Answer

In order to maximize English and Germany , we minimize Italian and keep the People who picked Italian didn't pick any other subject.

In order to minimize English and Germany , we maximize the people who took both English and Italian. (60%). Now , we could keep Germany separate with 40% but English needs to be 70%. That's why English and Germany share 10% , remaining (100-60-10 = 30) is Germany alone.

quantumliner · Answer

The answer is (C) 10% to 70%

Detailed solution in the attachment.

US09 · Answer

The maximum number of students who can take English and German will be when the number of students taking Italian is the least.
Therefore, % of students taking Italian = 30%, maximum % of students taking English = 70%. Hence maximum % of students taking both English and German = 70%.

And, minimum number of students taking both English and German will be when the two sets of students taking English and German overlap the least. Let this common group be x.
Therefore 70 + 40 - x = 100%, x = 10%. Hence minimum overlap between given two sets is = 10%.

Therefore, C.

sriramsundaram91 · Answer

Pretty Straight forward, though took 6 minutes to solve.

1. At least is used for German and English, so we have to be careful about that.
2. Students taking Italian is 30-60%

Worst Case Scenario: (Least German and English)
All German Students took Italian, If 60% Students took Italian, and all 60% took German, then 10% of German Students are left. One could argue that the rest 10% cannot take anything and just take only german but that is not possible since 40% are atleast Students taking German, which shall result in 110%(70% German Atleast(60%Italian+10%only German) +40% Italian), thus 10% should take German So that 100% remains.

Best Case:

30% Took only Italian and thus 70% Took German all the 70% German learning students took English.

ScottTargetTestPrep · Answer

Solution:

Let’s assume that there are 100 students at Zamenhof Language School. To determine the smallest number of students who take both English and German, notice that even if we choose the smallest possible values of 70 and 40 for the number of students who take English and German, respectively, the smallest value for the number of students who take both courses is 70 + 40 - 100 = 10. Indeed, if 10 students take both English and German, 30 students take German only, 30 students take both Italian and English and the remaining 30 students take English only, we see that all the percentage requirements in the question are satisfied. This shows that the smallest percentage for the students who take both English and German is 10%.

We only need to decide between C and D; in other words, we only need to decide whether the greatest percentage for the students who take both English and German is 70 or 100. Clearly, the percentage of the students who take both English and German cannot be 100 because otherwise, there can be no students who take Italian (since no students take more than two languages). To verify that a value of 70% is possible for “both English and German”, assume that all 100 students take English, 70 take German in addition to English and 30 take Italian in addition to English. Once again, every requirement in the question is satisfied, and this shows that 70 is the greatest possible percentage for the students who take both English and German.

Answer: C

KarishmaB · Answer

At least 70% English so it could be as high as 100%.
At least 40% for German so it could be as high as 100%.
30% - 60% Italian.

Maximize
We need to maximise the overlap of English and German while having 0 overlap among all three. This means we should minimise Italian so that the rest can be the overlap of English and German. So if 30% take Italian, the rest 70% could have taken both English and German.

Minimize
We should minimise both English and German and have them as far apart as possible. If 70% take English and 40% take German, there will necessarily be an overlap of 10% (since 70% + 40% = 110%)

So the range will be 10% to 70%.
Answer (C)

serenatnia · Answer

isn’t german maximum value is 40? therefore, max number that took both test is 40 too?

KarishmaB · Answer

We are given "at least 40%" take German. So German has to be 40 or more than 40. It cannot be less than 40, that is all.

At the Zamenhof Language School, at least 70% of the students take Eng

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