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**