Given:
- 4 men and 6 women complete a piece of work in 8 days
- 3 men and 7 women complete the same work in 10 days

To find:
- How many days will 10 women take to complete the work?

Assumptions:
- The amount of work done by each person is the same.
- The work is directly proportional to the number of people and inversely proportional to the number of days.

Solution:

Step 1: Find the work done by each person per day:
- Let the work done by 1 man per day be M
- Let the work done by 1 woman per day be W

Step 2: Calculate the total work:
- The total work is the product of the work done by each person per day and the number of days taken to complete the work.
- Work done by 4 men and 6 women in 8 days: 4M * 8 + 6W * 8 = 32M + 48W
- Work done by 3 men and 7 women in 10 days: 3M * 10 + 7W * 10 = 30M + 70W

Step 3: Set up equations:
- From the given information, we can set up two equations:
- 32M + 48W = Total work
- 30M + 70W = Total work

Step 4: Solve the equations:
- We have two equations and two variables (M and W). We can solve these equations to find the values of M and W.
- Subtracting the second equation from the first equation, we get:
- 32M + 48W - (30M + 70W) = 0
- 2M - 22W = 0
- M = 11W

Step 5: Find the work done by 10 women in 1 day:
- Substituting the value of M in terms of W in any of the equations, we get:
- 32(11W) + 48W = Total work
- 352W + 48W = Total work
- 400W = Total work
- Work done by 10 women in 1 day = Total work / 400

Step 6: Find the number of days taken by 10 women to complete the work:
- Let the number of days taken by 10 women be D
- Work done by 10 women in D days = (Work done by 10 women in 1 day) * D
- Work done by 10 women in D days = (Total work / 400) * D

Step 7: Substitute the values and solve for D:
- (Total work / 400) * D = Total work
- D = 400

Answer:
- 10 women will take 400 days to complete the work.