Let's assign variables to the daily work rates of men and women:

• Let the work rate of a man be $m$ (fraction of the work completed by one man in one day).
• Let the work rate of a woman be $w$ (fraction of the work completed by one woman in one day).

Step 1: Formulate two equations based on the problem's data.

• First condition: 3 men and 7 women complete the work in 10 days.

  In one day, the combined work done by 3 men and 7 women is $3m + 7w$. Since they complete the entire work in 10 days, the daily work rate must be $\frac{1}{10}$: $3m + 7w = \frac{1}{10}$

• Second condition: 4 men and 6 women complete the work in 8 days.

  Similarly, in one day, the combined work done by 4 men and 6 women is $4m + 6w$. Since they complete the entire work in 8 days, the daily work rate must be $\frac{1}{8}$: $4m + 6w = \frac{1}{8}$

Step 2: Solve the system of equations.

We now have the system: 3m + 7w = \frac{1}{10} \tag{1} 4m + 6w = \frac{1}{8} \tag{2}

We can solve this system by elimination or substitution. Let's first multiply equation (1) by 4 and equation (2) by 3 to align the coefficients of $m$:

12m + 28w = \frac{4}{10} = 0.4 \tag{3} 12m + 18w = \frac{3}{8} = 0.375 \tag{4}

Now subtract equation (4) from equation (3):

$(12m + 28w) - (12m + 18w) = 0.4 - 0.375$ $10w = 0.025$ $w = \frac{0.025}{10} = 0.0025$

Step 3: Substitute $w = 0.0025$ into one of the original equations to find $m$.

Substitute $w = 0.0025$ into equation (1): $3m + 7(0.0025) = \frac{1}{10}$ $3m + 0.0175 = 0.1$ $3m = 0.1 - 0.0175 = 0.0825$ $m = \frac{0.0825}{3} = 0.0275$

Step 4: Find how long it takes for 10 women to complete the work.

The work done by 10 women in one day is $10w$. Since $w = 0.0025$, the combined daily work rate of 10 women is: $10w = 10 \times 0.0025 = 0.025$

Thus, the number of days it will take for 10 women to complete the work is: $\text{Number of days} = \frac{1}{0.025} = 40$

Final Answer:

It will take 40 days for 10 women to complete the work.

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Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. How many days will 5 men take to complete the same work alone?
2. If 6 men and 4 women work together, how long will they take to complete the work?
3. What is the total work rate of 2 men and 8 women together?
4. How many men are needed to complete the work in 6 days if no women are involved?
5. What if 1 man and 10 women work together—how long will it take them to complete the work?

Tip: When dealing with work problems, always convert rates into fractions of the total work per day to solve efficiently.