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?
