A speaker truthfully made the following statements:
- "At a party there were 14 adults, 17 children, 12 males and 19 females."
- "Then I arrived at the party and the number of different man-woman couples possible became equal to the number of
different boy-girl couples possible. (For example, if there were 5 men and 7 women at the party, then there would have been 5*7 or, 35 possible man-woman couples.)"
Is the speaker a man, a woman, a boy, or a girl?
Let m = number of men before speaker arrives
Then w = number of women before speaker arrives = 14 - m
b = number of boys before speaker arrives = 12 - m
g = number of girls before speaker arrives = m + 5
If speaker is a man, then
(m+1)(14-m) = (12-m)(m+5)
14 + 13m - m^2 = 60 + 7m - m^2
6m = 46, so m is not an integer, so speaker is not a man
If speaker is a woman, then
(m)(15-m) = (12-m)(m+5)
15m - m^2 = 60 + 7m - m^2
8m = 60, so m is not an integer, so speaker is not a woman
If speaker is a boy, then
(m)(14-m) = (13-m)(m+5)
14m - m^2 = 65 + 8m - m^2
6m = 65, so m is not an integer, so speaker is not a boy
If speaker is a girl, then
(m)(14-m) = (12-m)(m+6)
14m - m^2 = 72 + 6m - m^2
8m = 72, so m = 9, and speaker can only be a girl.
Before she arrives, there are 9 men, 5 women, 14 girls and 3 boys. After she arrives, there are 45 adult couples and 45 younger couples. Only answer.
Edited on January 11, 2013, 2:51 pm
Algebraic solution (spoiler)
