At a party, each man danced with exactly four women and each woman danced with exactly three men. Nine men attended the party. How many women attended the party?

Answer

To solve the problem step by step, we can follow these calculations:

1. **Identify the number of men and their dancing partners**:
- We know that there are 9 men at the party.
- Each man danced with exactly 4 women.

2. **Calculate the total number of dance pairs**:
- The total number of dance pairs can be calculated by multiplying the number of men by the number of women each man danced with.
- Total dance pairs = Number of men × Number of women each man danced with = 9 men × 4 women = 36 dance pairs.

3. **Set up the relationship for women**:
- Let the number of women be denoted as \( w \).
- Each woman danced with exactly 3 men.
- Therefore, the total number of dance pairs can also be expressed as the number of women multiplied by the number of men each woman danced with.
- Total dance pairs = Number of women × Number of men each woman danced with = \( w \times 3 \).

4. **Set the equations equal to each other**:
- Since both expressions represent the total number of dance pairs, we can set them equal to each other:
\[ 3w = 36 \]

5. **Solve for \( w \)**:
- To find the number of women, divide both sides of the equation by 3:
\[ w = \frac{36}{3} = 12 \]

6. **Conclusion**:
- Therefore, the number of women who attended the party is \( 12 \).