Date: 08/28/2002 at 10:00:12 From: Pierre Subject: Logical Riddle Dear Sirs, I got this challenging question in my Math course: In a poll of 37 students, 16 felt confident solving quantitative comparison questions, 20 felt confident solving multiple choice questions, and 18 felt confident solving gridded response questions. Of all the students, 4 were confident solving gridded response and quantitative comparison questions, 5 were confident solving multiple choice and quantitative comparison questions, and 6 were confident with gridded response and multiple choice questions. Of those students, only 1 felt confident with all three types of questions. What number of students felt confident solving only multiple choice questions and no others? I tried to solve this riddle, and I got 10. My teacher told me that the answer is actually 8. I just can't figure this out. Would you please tell me which one is the right answer, and explain to me how to solve such questions? Thanks for any help.
Date: 08/28/2002 at 12:31:49 From: Doctor Peterson Subject: Re: Logical Riddle Hi, Pierre. I am always uneasy with these puzzles, because they are never worded clearly enough for me. Which numbers are meant to be inclusive? The first three numbers are clearly inclusive (that is, 16 like AT LEAST QC, but possibly others), since otherwise the answer would be given directly. Then the next three numbers seem likely to be inclusive as well, so that 4 like both GR and QC, AND possibly also MC. That implication is further supported by the wording "of those students [the 4, 5, and 6?], only 1 ...." So my assumption, probably like yours, is that the 1 is included in each of those numbers, so that there are 3, 4, and 5 who like ONLY two types of questions. That makes it easy to fill in a Venn diagram, and we quickly see that the circle for MC, whose total size is 20, has 1 shared with both QC and GR, 4 with QC alone, and 5 with GR, leaving 20-(1+4+5) = 10 for MC alone. So I agree with your answer... ... except that if we then add up the total, we find that 40 students like something, out of 37 students in all! So our assumptions do not fit all the facts. Looking for something to reinterpret, I consider the possibility that the single student who liked everything might NOT be included in the 4, 5, and 6. Taking it that way, the number for MC alone is 20-(1+5+6) = 8, agreeing with your teacher's answer. I think the problem is very poorly stated. You might like to try rewording it to make everything clear. That's a good exercise in communication, which mathematicians, pollsters, and puzzle-makers should pay attention to as well! (Except when a puzzle is deliberately vague to mislead you, which this one should not be.) - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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