X number of men can complete a work in $(X - 28)$ days, whereas $(X + 12)$ men can finish the same work in $(X - 32)$ days. If $\frac{3}{4}X$ women work together, in how many days will they complete the same work?

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**Solution:**

1. Let the total work be $W$. The rate of work done by one man per day is $\frac{W}{X(X-28)}$ because $X$ men complete the work in $(X-28)$ days.

2. Similarly, the rate of work done by one man per day is also $\frac{W}{(X+12)(X-32)}$ since $(X+12)$ men finish the work in $(X-32)$ days.

3. Since the work done by one man per day must be the same, equate the two expressions:
   $$\frac{W}{X(X-28)} = \frac{W}{(X+12)(X-32)}$$

4. Cancel $W$ and cross-multiply:
   $$ (X+12)(X-32) = X(X-28) $$

5. Expand both sides:
   $$ X^2 - 32X + 12X - 384 = X^2 - 28X $$
   $$ X^2 - 20X - 384 = X^2 - 28X $$

6. Subtract $X^2$ from both sides:
   $$ -20X - 384 = -28X $$

7. Rearranging:
   $$ -20X + 28X = 384 $$
   $$ 8X = 384 $$
   $$ X = 48 $$

8. Number of men = $48$

9. Rate of work of one man per day:
   $$ \frac{1}{48 	imes (48 - 28)} = \frac{1}{48 	imes 20} = \frac{1}{960} $$

10. Total work $W = 1$ unit.

11. Let the rate of work of one woman be $w$. Usually, women work at $\frac{3}{4}$ the rate of men (common assumption in such problems).

12. So, rate of one woman per day:
    $$ w = \frac{3}{4} 	imes \frac{1}{960} = \frac{3}{3840} = \frac{1}{1280} $$

13. Number of women working together = $\frac{3}{4} 	imes 48 = 36$

14. Combined rate of $36$ women:
    $$ 36 	imes \frac{1}{1280} = \frac{36}{1280} = \frac{9}{320} $$

15. Number of days taken by $36$ women to complete the work:
    $$ 	ext{Days} = \frac{1}{	ext{rate}} = \frac{1}{\frac{9}{320}} = \frac{320}{9} = \frac{80}{3} $$

**Answer:** $\frac{80}{3} 	ext{ days}$

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