Quantitative Question

[grad29]

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?

 

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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 = Group₁ + Group₂ + Neither – Both, and fill in what you know. If sailing certification is Group₁ and first aid certification is Group₂, then 22 = 7 + Group₂ + 4 – 2. So, 22 = Group₂ + 9, and you can solve for Group₂ = 13.

 

 

Where did I go wrong in my logic?

 

 

[OriginalDuck]

 

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.

[TakeruK]

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.

[grad29]

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

[grad29]

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.