mikemcgarry wrote:
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%
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 _________________
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