Given that *x* women can complete a piece of work in $2y$ days, $1.5x$ men can complete the same work in $y$ days, and $2x$ children can complete the work in $3y$ days. If 8 women, 8 men, and 8 children together can complete the work in $22 \frac{1}{2}$ days, and 9 men can complete the work in $(y + 20)$ days, find the value of *y*.

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**Solution:**

1. Let the total work be $W$ units.

2. Work done by *x* women in 1 day = $\frac{W}{2y}$.
   So, work done by 1 woman in 1 day = $\frac{W}{2yx}$.

3. Work done by $1.5x$ men in 1 day = $\frac{W}{y}$.
   Thus, work done by 1 man in 1 day = $\frac{W}{1.5xy} = \frac{2W}{3xy}$.

4. Work done by $2x$ children in 1 day = $\frac{W}{3y}$.
   So, work done by 1 child in 1 day = $\frac{W}{6xy}$.

5. Work done by 8 women, 8 men, and 8 children in 1 day =
   $$8 	imes \frac{W}{2yx} + 8 	imes \frac{2W}{3xy} + 8 	imes \frac{W}{6xy}$$

Simplify each term:
   - Women: $\frac{8W}{2yx} = \frac{4W}{yx}$
   - Men: $\frac{16W}{3xy}$
   - Children: $\frac{8W}{6xy} = \frac{4W}{3xy}$

Total work per day = $\frac{4W}{yx} + \frac{16W}{3xy} + \frac{4W}{3xy}$
   Combine men and children terms:
   $$\frac{16W + 4W}{3xy} = \frac{20W}{3xy}$$

So total work per day = $\frac{4W}{yx} + \frac{20W}{3xy}$

6. Since $x$ and $y$ are variables, multiply numerator and denominator appropriately to combine:
   $$\frac{4W}{yx} = \frac{4W}{xy}$$

So total work per day = $\frac{4W}{xy} + \frac{20W}{3xy} = \frac{12W + 20W}{3xy} = \frac{32W}{3xy}$

7. Given that 8 women, 8 men, and 8 children complete the work in $22 \frac{1}{2} = \frac{45}{2}$ days,
   so work done per day = $\frac{W}{\frac{45}{2}} = \frac{2W}{45}$.

Equate:
   $$\frac{32W}{3xy} = \frac{2W}{45}$$

Cancel $W$:
   $$\frac{32}{3xy} = \frac{2}{45}$$

Cross multiply:
   $$32 	imes 45 = 2 	imes 3xy$$

$$1440 = 6xy$$

$$xy = 240$$

8. Next, 9 men can complete the work in $(y + 20)$ days.
   Work done by 1 man in 1 day = $\frac{W}{1.5xy}$ (from step 3)

Work done by 9 men in 1 day = $9 	imes \frac{W}{1.5xy} = \frac{9W}{1.5xy} = \frac{6W}{xy}$

Given time = $(y + 20)$ days, so work done per day = $\frac{W}{y + 20}$

Equate:
   $$\frac{6W}{xy} = \frac{W}{y + 20}$$

Cancel $W$:
   $$\frac{6}{xy} = \frac{1}{y + 20}$$

Cross multiply:
   $$6(y + 20) = xy$$

Substitute $xy = 240$ from step 7:
   $$6(y + 20) = 240$$

$$6y + 120 = 240$$

$$6y = 120$$

$$y = 20$$

**Answer:** $20$

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