The average weight of 15 men is 18 kg and the average weight of $Y$ women is 12 kg. If the average weight of all the men and women combined is 15 kg, find the value of $Y$.

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**Solution:**

1. Let the number of women be $Y$.

2. Total weight of 15 men = $$$15 \times 18 = 270 \text{ kg}$$$.

3. Total weight of $Y$ women = $$$12Y \text{ kg}$$$.

4. Total number of people = $15 + Y$.

5. Given the average weight of all men and women together is 15 kg, so:

$$\frac{270 + 12Y}{15 + Y} = 15$$

6. Multiply both sides by $15 + Y$:

$$270 + 12Y = 15(15 + Y)$$

$$270 + 12Y = 225 + 15Y$$

7. Rearranging terms:

$$270 - 225 = 15Y - 12Y$$

$$45 = 3Y$$

8. Solving for $Y$:

$$Y = \frac{45}{3} = 15$$

**Answer:** $15$

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