Let's solve this using the concept of work and rates of working.

Step 1: Set up the problem

Let:

• The work done by one man in one day be $M$.
• The work done by one woman in one day be $W$.

We have two situations:

1. 3 men and 7 women complete the work in 10 days.

   • The total work done by 3 men and 7 women in one day is: $3M + 7W$
   • Since they finish the work in 10 days, their combined work rate in one day is: $3M + 7W = \frac{1}{10} \quad \text{(1)}$

2. 4 men and 6 women complete the work in 8 days.

   • The total work done by 4 men and 6 women in one day is: $4M + 6W$
   • Since they finish the work in 8 days, their combined work rate in one day is: $4M + 6W = \frac{1}{8} \quad \text{(2)}$

Step 2: Solve the system of equations

We have the two equations:

1. $3M + 7W = \frac{1}{10}$
2. $4M + 6W = \frac{1}{8}$

Let's solve these equations simultaneously to find $M$ and $W$.

Multiply equation (1) by 4 and equation (2) by 3 to eliminate $M$:

$4(3M + 7W) = 4 \times \frac{1}{10} \quad \Rightarrow \quad 12M + 28W = \frac{4}{10}$ $3(4M + 6W) = 3 \times \frac{1}{8} \quad \Rightarrow \quad 12M + 18W = \frac{3}{8}$

Now, subtract the second equation from the first: $(12M + 28W) - (12M + 18W) = \frac{4}{10} - \frac{3}{8}$ $10W = \frac{32}{80} - \frac{30}{80} = \frac{2}{80} = \frac{1}{40}$ $W = \frac{1}{400}$

Step 3: Find $M$

Now that we have $W = \frac{1}{400}$, substitute it into one of the original equations, say equation (1): $3M + 7 \times \frac{1}{400} = \frac{1}{10}$ $3M + \frac{7}{400} = \frac{1}{10}$ Subtract $\frac{7}{400}$ from both sides: $3M = \frac{1}{10} - \frac{7}{400}$ Convert $\frac{1}{10}$ to have a denominator of 400: $\frac{1}{10} = \frac{40}{400}$ Now: $3M = \frac{40}{400} - \frac{7}{400} = \frac{33}{400}$ $M = \frac{33}{3 \times 400} = \frac{33}{1200} = \frac{11}{400}$

Step 4: Find how many days 10 women will take

We know the work rate of one woman is $W = \frac{1}{400}$. So, the work rate of 10 women is: $10W = 10 \times \frac{1}{400} = \frac{10}{400} = \frac{1}{40}$ This means 10 women can complete $\frac{1}{40}$ of the work in one day. Therefore, the number of days 10 women will take to complete the work is: $\frac{1}{\frac{1}{40}} = 40 \text{ days}.$

Final Answer:

10 women will complete the work in 40 days.

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Related Questions:

1. How long will it take for 2 men and 8 women to complete the same work?
2. If one man and one woman work together, how many days will they take to finish the work?
3. What would be the combined rate of 5 men and 5 women working together?
4. If the number of men is doubled and the number of women is halved, how will the work duration change?
5. How many men alone would be required to finish the work in 10 days?

Tip: When solving work problems, focus on expressing the rate of work (work done per day) for each individual and use it to set up equations.