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Committees

Every member of the club is a member of two committees. How many members are there in the club?

Dance Party

Age 11 to 14 Short Challenge Level:

Using the total number of dances
If each of the 12 men danced with 3 women, then there must have been 12$\times$3 = 36 dances altogether.

Since each woman danced with two men, each woman danced two dances - so there must have been 36$\div$2 = 18 women.


Using a simpler situation to help
If each of the women had only danced with one man, then 36 women would have been needed for each man to dance with 3 women.

But each woman danced with 2 men, so only half as many women were needed. So there were 18 women at the party.

Alternatively...

If each man danced with 3 women, and there were just 3 women at the party, each woman would have had to dance with all 12 men.

But each woman danced with just 2 men, that is, $\frac{1}{6}$ of 12, so there must have been 6 times as many women:
3 women $\times$ 6 = 18 women


Using ratio
If each man danced with 3 women but each woman only danced with 2 men, there must be more women than men, and the ratio of men : women must be 2 : 3.

Since there are 12 men, there must be 18 women, since the ratio 12 : 18 is equivalent to the ratio 2 : 3.


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