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A high school is made by 25 pupils, of which 13 cycle, 11 swim, and 8 go skiing. There is no one practising all three sports. In yesterday's test the sportsmen (those practising at least one sport) went pretty well: they all got 8 or 9 (out of 10) as a mark. There were 9 pupils who got less than 8. We can therefore deduce that :

(A) every swimmer can also ski

(B) one sportsman might have gotten a 10

(C) four swimmers can also ski

(D) someone in the class got a 10

(E) three swimmers only can ski as well

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(E) three swimmers only can ski as well.

If the sportsmen (those practising at least one sport) got 8 or 9 as a mark and there were 9 pupils who got less than 8, we have to deduce that the 9 do not practice sports at all.

Thus, 25 minus 9 equal 16: the sportsmen.

We have 13 plus 11 plus 8 equal 32 sports practiced overall with no one practising all three sports; thus, each sportman must practice exactly two sports.

Thus of the 16 sportsmen, we have that: 13 are cycling and 3 not-cycling, that implies that the three not-cycling must practice swim and ski.

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(A) every swimmer can also ski

This is quite evidently false. There are 11 swimmers, and only 8 skiers (if this is a word). So, at least three swimmers cannot ski (or may be they can, but don't actually practice the sport; in any case, we are not told about what people can do, just about what they do in a regular basis).

(B) one sportsman might have gotten a 10

That's also necessarily false. The text is clear: "they all got 8 or 9 (out of 10) as a mark". So no one got a 10, or an 8.5, or any other mark except precisely 8 or precisely 9.

(C) four swimmers can also ski

(E) three swimmers only can ski as well

C is false, and E is true. 16 pupils practice one of the given three sports (because 9 pupils got less than 8, and, as all practicioners of the given sports got either 8 or 9, those 9 pupils must not practice any of the given sports). But no pupil practices all of the three sports. In order to that be true, we need that only three people practice both swimming and ski, so that we would have:

  • 3 people who swim and practice ski;
  • 5 people who cycle and practice ski;
  • 8 people who cycle and swim.

If 4 swimmers also practiced ski, then there would be only 11 people who either swim or practice ski, but not both (7 who swim but not ski, and 4 who ski but do not swim), so, if there are 13 cyclists, two of them would have to practice all three sports.

(D) someone in the class got a 10

This is again false. The pupils who practice sports all got either 8 or 9. There were 9 people who got less than 8 (and so must not practice any sport). If anyone got a 10, then it must have been someone who also does not practice any sport. But, if so, there would be at least 10 people who do not practice any sport, and in this case it would be impossible, as discussed above, that we had 13 cyclists, 11 swimmers and 8 skiers without at least some pupils practicing all three sports.

  • "One sportsman might have gotten a 10" is actually quite meaningless. Was that someone who knew all the answers, but his pen broke when he tried to answer the last question? – gnasher729 Jun 27 '16 at 12:48
  • This is quite full answer – user22143 Jun 27 '16 at 17:35

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