4 men and 6 women finish a job in 8 days, while 3 men and 7 women finish in 10 days. In how many days will 10 women finish it?

Options

A. 20 days
B. 30 days
C. 40 days
D. 50 days

Answer

To solve the problem step by step, we will first determine the work done by men and women based on the information given.

### Step 1: Establish the work done by the groups
We know that:
- 4 men and 6 women finish a job in 8 days.
- 3 men and 7 women finish the same job in 10 days.

Let the work done by one man in one day be \( M \) and the work done by one woman in one day be \( W \).

The total work done can be expressed as:
1. For the first group (4 men and 6 women):
\[ \text{Total work} = (4M + 6W) \times 8 \]
2. For the second group (3 men and 7 women):
\[ \text{Total work} = (3M + 7W) \times 10 \]

Since both expressions represent the same total work, we can set them equal to each other:
\[ (4M + 6W) \times 8 = (3M + 7W) \times 10 \]

### Step 2: Simplify the equation
Expanding both sides gives:
\[ 32M + 48W = 30M + 70W \]

Now, rearranging the equation:
\[ 32M - 30M = 70W - 48W \]
\[ 2M = 22W \]
\[ M = 11W \]

### Step 3: Substitute back to find total work
Now that we have \( M \) in terms of \( W \), we can substitute \( M \) back into one of the work equations to find the total work.

Using \( M = 11W \) in the first equation:
\[ \text{Total work} = (4(11W) + 6W) \times 8 \]
\[ = (44W + 6W) \times 8 \]
\[ = 50W \times 8 = 400W \]

### Step 4: Determine how many days 10 women will take to finish the work
Now, we need to find out how many days it will take for 10 women to finish the total work of \( 400W \).

The work done by 10 women in one day is:
\[ 10W \]

Let \( D \) be the number of days required for 10 women to complete the work:
\[ 10W \times D = 400W \]
Dividing both sides by \( 10W \):
\[ D = \frac{400W}{10W} = 40 \]

### Final Answer
Thus, it will take 10 women **40 days** to finish the job.

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