Given:

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

Find the value of *x*.

Question

Given:

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

Find the value of *x*.

Accepted Answer

**Solution:**

1. Let the work be 1 unit.

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

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

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

5. Work done by 8 women, 8 men, and 8 children in 1 day:
   $$
   8 \times \frac{1}{2yx} + 8 \times \frac{2}{3xy} + 8 \times \frac{1}{6xy} = \frac{8}{2yx} + \frac{16}{3xy} + \frac{8}{6xy}
   $$

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

Sum:
   $$
   \frac{4}{xy} + \frac{16}{3xy} + \frac{4}{3xy} = \frac{4}{xy} + \frac{20}{3xy} = \frac{12}{3xy} + \frac{20}{3xy} = \frac{32}{3xy}
   $$

6. Given that these 8 women, 8 men, and 8 children together complete the work in $22 \frac{1}{2} = \frac{45}{2}$ days.

So, work done in 1 day = $\frac{1}{\frac{45}{2}} = \frac{2}{45}$.

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

Cross-multiplied:
   $$
   32 \times 45 = 2 \times 3xy
   $$

$$
   1440 = 6xy \implies xy = \frac{1440}{6} = 240
   $$

7. Next, 9 men complete the work in $(y + 20)$ days.

Work done by 1 man in 1 day = $\frac{2}{3xy}$ (from step 3).

Work done by 9 men in 1 day = $9 \times \frac{2}{3xy} = \frac{18}{3xy} = \frac{6}{xy}$.

Given time = $(y + 20)$ days, so work done in 1 day = $\frac{1}{y + 20}$.

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

Cross-multiplied:
   $$
   6(y + 20) = xy
   $$

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

$$
   6y + 120 = 240 \implies 6y = 120 \implies y = 20
   $$

8. From $xy = 240$ and $y = 20$, find $x$:

$$
   x \times 20 = 240 \implies x = \frac{240}{20} = 12
   $$

**Answer:** $12$

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