grad29 Posted July 6, 2015 Share Posted July 6, 2015 All but 4 of the counselors at a certain summer camp have a sailing certification, first aid certification, or both. If twice as many of the counselors have neither certification as have both certifications, 7 of the counselors have a sailing certification, and there are a total of 22 counselors on staff, then how many of the counselors have a first aid certification? ' I got this question wrong. The way I thought of it was: A. There are 18 counselors with either a sailing certification, first aid certification, or both. B. If 7 of the counselors have a sailing certification, that means that 11 of the counselors must have either both certifications or a first aid certification. Based on A, there must be 4 counselors that don't have any certification and since there are 1/2 as many counselors that have both certifications, that must mean 2 counselors have both. So 7 counselors have a sailing certification, 2 counselors have both certifications, and 9 counselors have a first aid certification. So the answer is 9. However they provided the following explanation: First, write down the group formula, Total = Group_{1} + Group_{2} + Neither – Both, and fill in what you know. If sailing certification is Group_{1} and first aid certification is Group_{2}, then 22 = 7 + Group_{2} + 4 – 2. So, 22 = Group_{2} + 9, and you can solve for Group_{2} = 13. Where did I go wrong in my logic? Link to comment Share on other sites More sharing options...

OriginalDuck Posted July 7, 2015 Share Posted July 7, 2015 B. If 7 of the counselors have a sailing certification, that means that 11 of the counselors must have either both certifications or a first aid certification. It doesn't say that 7 of the counselors only have a sailing certification. That's where you went wrong. Link to comment Share on other sites More sharing options...

TakeruK Posted July 7, 2015 Share Posted July 7, 2015 Another way to think about it is Venn Diagrams. The crude diagram below represents a circle for "sailing", a circle for "first aid" and of course an overlapping region for both. ( Sailing ( Both ) First Aid ) You can fill in "2" for "both" based on "A" as you wrote above. ( Sailing ( 2 ) First Aid ) The "Sailing" circle must add up to 7 so the non-overlapping part of the sailing circle must be 5 (as there is already 2 in the overlapping part. ( 5 ( 2 ) First Aid ) Now there is only one part of the Venn Diagram unfilled, those with First Aid only (i.e. the answer). The Venn Diagram must add up to 18 (as all but 4 out of 22 counselors have sailing, first aid, or both, i.e. inside the Venn Diagram). So the answer must be 13 so that 5 + 2 + 13 = 18. Of course, this is exactly the same approach as the equation given in your answer guide. But I'm more a visual person and I prefer to "divide up" the people into diagrams like this instead of memorize a formula. I also think Venn Diagrams, with its visual boundaries, helps me not make the exact logic mistake that OriginalDuck pointed out. So I thought I might share this approach in case it helps. Link to comment Share on other sites More sharing options...

grad29 Posted July 7, 2015 Author Share Posted July 7, 2015 (edited) Another way to think about it is Venn Diagrams. The crude diagram below represents a circle for "sailing", a circle for "first aid" and of course an overlapping region for both. ( Sailing ( Both ) First Aid ) You can fill in "2" for "both" based on "A" as you wrote above. ( Sailing ( 2 ) First Aid ) The "Sailing" circle must add up to 7 so the non-overlapping part of the sailing circle must be 5 (as there is already 2 in the overlapping part. ( 5 ( 2 ) First Aid ) Now there is only one part of the Venn Diagram unfilled, those with First Aid only (i.e. the answer). The Venn Diagram must add up to 18 (as all but 4 out of 22 counselors have sailing, first aid, or both, i.e. inside the Venn Diagram). So the answer must be 13 so that 5 + 2 + 13 = 18. Of course, this is exactly the same approach as the equation given in your answer guide. But I'm more a visual person and I prefer to "divide up" the people into diagrams like this instead of memorize a formula. I also think Venn Diagrams, with its visual boundaries, helps me not make the exact logic mistake that OriginalDuck pointed out. So I thought I might share this approach in case it helps. Thanks. I'll start using Venn Diagrams for these types of problems since I'm not good at creating the formulas. Makes a lot more sense that way. Edited July 7, 2015 by westy3789 Link to comment Share on other sites More sharing options...

grad29 Posted July 7, 2015 Author Share Posted July 7, 2015 It doesn't say that 7 of the counselors only have a sailing certification. That's where you went wrong. Yes, it was. I misread it. Link to comment Share on other sites More sharing options...

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