Who is Chris?
At a party, there were 19 females, 12 males, 14 adults and 17 children. Then, Chris arrived and the number of different man-woman couples became equal to the number of boy-girl couples.
Is Chris a man, a woman, a boy or a girl?
Note: If there were 9 boys and 8 girls at the party, then there would have been 72 (9x8) boy-girl couples possible.
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Answer: Chris is a Girl.
Note: In the solution below, I am using the name 'Chris' again and again because we don't know whether Chris is 'he' or 'she'.
Before Chris arrived, let M be the number of male adults (men) at the party.
Then, the number of female adults (women) = 14 - M
The number of boys = 12 - M
The number of girls = 5 + M
Now, Chris arrived at the party and Chris is either a man or a woman or a boy or a girl. Let's consider each case one-by-one.
Case I: Let's assume that Chris is a Man. It is given that after Chris arrived, the number of different man-woman couples possible became equal to the number of boy-girl couples possible. Hence,
(M + 1) * (14 - M) = (12 - M) * (5 + M)
which on simplifying gives,
6M = 46
This is impossible as the value of M must be an integer.
Case II: Let's assume that Chris is a woman, then the equation is
(M) * (15 - M) = (12 - M) * (5 + M)
which on simplifying gives,
8M = 60
This is also impossible as the value of M must be integer.
Case III: Let's assume that Chris is a boy, then the equation is
(M) * (14 - M) = (13 - M) * (5 + M)
which on simplifying gives,
8M = 65 This is also impossible as the value of M must be integer.
Case IV: Let's assume that Chris a girl, then the equation is
(M) * (14 - M) = (12 - M) * (6 + M)
which on simplifying gives,
8M = 72
So, M = 9 which is an integer.
Thus, Chris is a Girl. (not really though :p)