At a party each man dances with 4 women and each woman dances with 3 men. If 9 men attended the party, how many women attended the party? At a party each man dances with 4 women and each woman dances with 3 men. If 9 men attended the party, how many women attended the party?
I have no idea how to approach this problem.
 A: If you assume that a man dances with a woman, how many dances will be performed?

9 men -> dance 9 * 4 times with a woman, so 36 dances
Each woman performs 3 dances, so 12 women.
A: Count dance pairs . . .

Since there are $9$ men and each man dances with $4$ women (presumably this means each man dances exactly $4$ times), there are exactly $36$ dance pairs. 

Let $w$ be the number of women.

Since each woman dances with $3$ men (presumably this means each woman dances exactly $3$ times), there are exactly $3w$ dance pairs.

Thus, $3w=36$, so $w=12$.
A: Divide the $9$ men into $3$ groups of $3$.  Assign $4$ women to each group.  That'll satisfy the conditions of the problem, with a total of $12$ women.  
A: First naive thought: each of the $9$ men dances with $4$ women, so we need $36$ women for that.
No, wait... Every woman danced with $3$ men, so stands in for $3$ of those $36$ women. 
Then exactly $12$ women are needed.
A: I have a different view.
It is given that one man danced with 4 women.
W¹,W²,W³,W⁴-M¹
W²,W³,W⁴, W5-M²
'
'
' 
W1,W²,W5,W6-M9
This gives us a total of 6 women.
This also satisfies that each woman danced with 3 men.
How is it wrong?
