24 men can complete a work in *X* days, while 32 women can complete the same work in *(X + 8)* days. The ratio of work done by 15 men to 12 women in the same time period is 3 : 1.
What is the value of *X*?

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**Solution:**

1. Let the total work be 1 unit.

2. The work done by 24 men in *X* days is 1 unit, so the work done by 1 man in 1 day is:
   $$\frac{1}{24X}$$

3. Similarly, 32 women complete the work in *(X + 8)* days, so the work done by 1 woman in 1 day is:
   $$\frac{1}{32(X + 8)}$$

4. Work done by 15 men in 1 day:
   $$15 	imes \frac{1}{24X} = \frac{15}{24X} = \frac{5}{8X}$$

5. Work done by 12 women in 1 day:
   $$12 	imes \frac{1}{32(X + 8)} = \frac{12}{32(X + 8)} = \frac{3}{8(X + 8)}$$

6. Given the ratio of work done by 15 men to 12 women in the same time is 3 : 1, so:
   $$\frac{\frac{5}{8X}}{\frac{3}{8(X + 8)}} = \frac{3}{1}$$

7. Simplify the ratio:
   $$\frac{5}{8X} 	imes \frac{8(X + 8)}{3} = 3$$

$$\frac{5(X + 8)}{3X} = 3$$

8. Multiply both sides by *3X*:
   $$5(X + 8) = 9X$$

9. Expand and simplify:
   $$5X + 40 = 9X$$

$$40 = 9X - 5X = 4X$$

$$X = \frac{40}{4} = 10$$

**Answer:** $10$

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