3 men and 7 women can do a job in 5 days, while 4 men and 6 women can do it in 4 days. The number of days required for a group of 10 women working together, at the same rate as before, to finish the same job is :

Options

A. 30 days
B. 36 days
C. 40 days
D. 20 days

Answer

To solve the problem step by step, we need to find out how many days it will take for 10 women to finish the job, given the information about the work done by men and women together.

### Step 1: Establish the work done by men and women
Let's denote the work done by one man in one day as \( M \) and the work done by one woman in one day as \( W \).

From the problem, we have two scenarios:
1. **3 men and 7 women can complete the job in 5 days.**
- Total work done = (3M + 7W) * 5 days
2. **4 men and 6 women can complete the job in 4 days.**
- Total work done = (4M + 6W) * 4 days

### Step 2: Set up the equations
From the first scenario:
\[ 3M + 7W = \frac{Total \ Work}{5} \]
From the second scenario:
\[ 4M + 6W = \frac{Total \ Work}{4} \]

Since both expressions equal the total work, we can set them equal to each other:
\[ (3M + 7W) * 5 = (4M + 6W) * 4 \]

### Step 3: Simplify the equation
Expanding both sides gives:
\[ 15M + 35W = 16M + 24W \]

Rearranging the equation:
\[ 15M - 16M + 35W - 24W = 0 \]
\[ -M + 11W = 0 \]
\[ M = 11W \]

### Step 4: Substitute M in terms of W
Now we know that one man is equivalent to 11 women.

### Step 5: Calculate the equivalent work for 10 women
We need to find out how many days it will take for 10 women to complete the job.
Using the relation \( M = 11W \), we can convert men to women:
- 1 man = 11 women
- Therefore, 3 men = 33 women
- 4 men = 44 women

Now substituting into the first equation:
\[ (33W + 7W) * 5 = Total \ Work \]
\[ 40W * 5 = Total \ Work \]
\[ Total \ Work = 200W \]

### Step 6: Determine how long it takes for 10 women
Now we need to find out how many days \( D \) it will take for 10 women to complete the same amount of work:
\[ (10W) * D = 200W \]

### Step 7: Solve for D
Dividing both sides by \( 10W \):
\[ D = \frac{200W}{10W} = 20 \]

### Conclusion
Thus, it will take **20 days** for 10 women to complete the job.

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