24 men can complete a work in $X$ days, and 32 women can complete the same work in $(X + 8)$ days. The ratio of work done by 15 men and 12 women in the same time is $3 : 1$.
If 10 men and 24 women work together for 6 days on the same work, and the remaining work is completed by 18 boys in 18 days, then find the number of days in which 12 boys would complete the entire work.

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**Solution:**

1. Let the total work be 1 unit.

2. Given, 24 men can complete the work in $X$ days, so 1 man's 1 day work = $\frac{1}{24X}$.

3. 32 women can complete the work in $(X + 8)$ days, so 1 woman's 1 day work = $\frac{1}{32(X+8)}$.

4. The ratio of work done by 15 men and 12 women in the same time is $3 : 1$.

So, 
   $$15 	imes \frac{1}{24X} : 12 	imes \frac{1}{32(X+8)} = 3 : 1$$

Simplifying,
   $$\frac{15}{24X} : \frac{12}{32(X+8)} = 3 : 1$$

$$\Rightarrow \frac{15}{24X} 	imes \frac{32(X+8)}{12} = 3$$

$$\Rightarrow \frac{15 	imes 32 (X+8)}{24X 	imes 12} = 3$$

$$\Rightarrow \frac{15 	imes 32 (X+8)}{288 X} = 3$$

$$\Rightarrow 15 	imes 32 (X+8) = 864 X$$

$$\Rightarrow 480 (X+8) = 864 X$$

$$\Rightarrow 480 X + 3840 = 864 X$$

$$\Rightarrow 3840 = 864 X - 480 X = 384 X$$

$$\Rightarrow X = \frac{3840}{384} = 10$$

5. So, men complete the work in 10 days, women in $10 + 8 = 18$ days.

6. Work done by 10 men in 1 day = $\frac{10}{24 	imes 10} = \frac{10}{240} = \frac{1}{24}$.

7. Work done by 24 women in 1 day = $\frac{24}{32 	imes 18} = \frac{24}{576} = \frac{1}{24}$.

8. Total work done by 10 men and 24 women in 1 day = $\frac{1}{24} + \frac{1}{24} = \frac{1}{12}$.

9. Work done by them in 6 days = $6 	imes \frac{1}{12} = \frac{1}{2}$.

10. Remaining work = $1 - \frac{1}{2} = \frac{1}{2}$.

11. 18 boys complete the remaining $\frac{1}{2}$ work in 18 days.

So, work done by 18 boys in 1 day = $\frac{1/2}{18} = \frac{1}{36}$.

12. Work done by 1 boy in 1 day = $\frac{1}{36 	imes 18} = \frac{1}{648}$.

13. Number of days taken by 12 boys to complete the whole work:

$$= \frac{1}{	ext{work done by 12 boys in 1 day}} = \frac{1}{12 	imes \frac{1}{648}} = \frac{648}{12} = 54$$

**Answer:** 54 days

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