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Party People II (Posted on 2013-01-11) Difficulty: 3 of 5
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?

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (2 votes)

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Solution Algebraic solution (spoiler) Comment 2 of 2 |
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
  Posted by Steve Herman on 2013-01-11 12:39:15

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