3 men and 7 women can complete a work in 10 days. But 4 men and 6 women need 8 days to complete the same work. In how many days will 10 women complete the same work?

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.

---

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.

Discover how to solve a work and time problem where 3 men and 7 women finish a task in 10 days, while 4 men and 6 women take 8 days. Learn how to calculate the time for 10 women to complete the same task using algebraic techniques. Suitable for Grades 9-12.

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