To solve this problem, we need to determine the work rates of the men and women and then combine their rates to find out how long it will take for 1 man and 2 women to complete the task.

Step 1: Calculate the work rates of the men and women.

• Two men can complete the task in 8 days. So, the work rate of 2 men is: $\text{Work rate of 2 men} = \frac{1 \text{ task}}{8 \text{ days}} = \frac{1}{8} \text{ of the task per day}$ Therefore, the work rate of 1 man is: $\text{Work rate of 1 man} = \frac{1}{2} \times \frac{1}{8} = \frac{1}{16} \text{ of the task per day}$

• Three women can complete the task in 12 days. So, the work rate of 3 women is: $\text{Work rate of 3 women} = \frac{1 \text{ task}}{12 \text{ days}} = \frac{1}{12} \text{ of the task per day}$ Therefore, the work rate of 1 woman is: $\text{Work rate of 1 woman} = \frac{1}{3} \times \frac{1}{12} = \frac{1}{36} \text{ of the task per day}$

Step 2: Calculate the combined work rate of 1 man and 2 women.

• The work rate of 1 man and 2 women together is: $\text{Work rate of 1 man and 2 women} = \frac{1}{16} + 2 \times \frac{1}{36}$
• Simplifying this: $\frac{1}{16} + \frac{2}{36} = \frac{1}{16} + \frac{1}{18}$
• To add these fractions, find a common denominator: $\text{LCM of 16 and 18 is 144}$ $\frac{1}{16} = \frac{9}{144}, \quad \frac{1}{18} = \frac{8}{144}$ $\frac{9}{144} + \frac{8}{144} = \frac{17}{144}$

Step 3: Determine the time it takes to complete the task.

• The combined work rate is $\frac{17}{144}$ of the task per day.
• Time required to complete the task: $\text{Time} = \frac{1 \text{ task}}{\frac{17}{144} \text{ of the task per day}} = \frac{144}{17} \approx 8.47 \text{ days}$

Since this result is closest to 8 days, the correct answer is (b) 8 days.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. What if the time taken by 3 women were 10 days instead of 12 days?
2. How long would it take 2 men and 4 women to complete the task?
3. What if the work rate of women was twice as efficient as men?
4. Can you solve this problem using a different method, such as setting up equations?
5. What if the task were divided into two parts, with the man doing one part and the women doing the other?

Tip: When solving work problems, always break down the work rates individually before combining them. This simplifies the process.