14 men can complete a work in 18 days, and 15 women can complete the same work in 24 days. If 14 men work for the first 3 days and then 10 women work for the next 3 days, what fraction of the work remains unfinished after these 6 days?

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Accepted Answer

**Solution:**

1. Calculate the work done by 14 men in one day:
   - Since 14 men complete the work in 18 days, their one day work = $\frac{1}{18}$ of the work.

2. Calculate the work done by 15 women in one day:
   - 15 women complete the work in 24 days, so one day work of 15 women = $\frac{1}{24}$ of the work.

3. Find the one day work of 1 woman:
   - $\frac{1}{24} \div 15 = \frac{1}{360}$ of the work.

4. Calculate the one day work of 10 women:
   - $10 	imes \frac{1}{360} = \frac{10}{360} = \frac{1}{36}$ of the work.

5. Calculate work done by 14 men in 3 days:
   - $3 	imes \frac{1}{18} = \frac{3}{18} = \frac{1}{6}$ of the work.

6. Calculate work done by 10 women in 3 days:
   - $3 	imes \frac{1}{36} = \frac{3}{36} = \frac{1}{12}$ of the work.

7. Total work done in 6 days:
   - $\frac{1}{6} + \frac{1}{12} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}$ of the work.

8. Work left after 6 days:
   - $1 - \frac{1}{4} = \frac{3}{4}$ of the work.

**Note:** The above calculation shows $\frac{3}{4}$ work left, but this contradicts the options. Let's re-check the calculations carefully.

**Recalculation:**

- One day work of 14 men = $\frac{1}{18}$ (since 14 men finish in 18 days)
- Work done by 14 men in 3 days = $3 	imes \frac{1}{18} = \frac{1}{6}$

- One day work of 15 women = $\frac{1}{24}$
- One day work of 1 woman = $\frac{1}{24} \div 15 = \frac{1}{360}$
- One day work of 10 women = $10 	imes \frac{1}{360} = \frac{10}{360} = \frac{1}{36}$
- Work done by 10 women in 3 days = $3 	imes \frac{1}{36} = \frac{3}{36} = \frac{1}{12}$

- Total work done in 6 days = $\frac{1}{6} + \frac{1}{12} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}$

- Work left = $1 - \frac{1}{4} = \frac{3}{4}$

Since $\frac{3}{4}$ is not among the options, let's check if the problem expects the fraction of work left after the 6 days or if there is a misinterpretation.

**Alternative interpretation:**

- The problem states 14 men can do the work in 18 days, so one man’s one day work = $\frac{1}{14 	imes 18} = \frac{1}{252}$
- Work done by 14 men in one day = $14 	imes \frac{1}{252} = \frac{14}{252} = \frac{1}{18}$ (consistent)

- Similarly, one woman’s one day work = $\frac{1}{15 	imes 24} = \frac{1}{360}$
- Work done by 10 women in one day = $10 	imes \frac{1}{360} = \frac{10}{360} = \frac{1}{36}$

- Work done by 14 men in 3 days = $3 	imes \frac{1}{18} = \frac{1}{6}$
- Work done by 10 women in 3 days = $3 	imes \frac{1}{36} = \frac{1}{12}$

- Total work done = $\frac{1}{6} + \frac{1}{12} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}$

- Work left = $1 - \frac{1}{4} = \frac{3}{4}$

This confirms the work left is $\frac{3}{4}$, which is not in the options.

**Check if the question wants the part of work left after 3 days of men and 3 days of women working simultaneously or sequentially.**

Since the question states men work first 3 days, then women work next 3 days, the calculation is correct.

**Hence, the options might be incorrect or the question expects the fraction of work done, not left.**

- Work done = $\frac{1}{4}$
- Work left = $\frac{3}{4}$

**If we consider the fraction of work done by 14 men in 3 days and 10 women in 3 days, the total work done is $\frac{1}{4}$.**

**Answer:** $\frac{3}{4}$ work is left after 6 days.

If the question is asking for the part of work left, the correct answer is $\frac{3}{4}$. Since this is not in the options, the closest fraction from the options is $\frac{7}{30}$ which is approximately 0.233, which is less than $\frac{1}{4} = 0.25$. So the options might be incorrect or the question might have a typo.

**Final conclusion:**
The correct fraction of work left after 14 men work for 3 days and 10 women work for 3 days is $\boxed{\frac{3}{4}}$.

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