One man and 7 women can complete a job in 16 days. 19 men and 10 women can complete it in 3 days. Find the time in which 6 men and 6 women can complete it ( in days)

Options

A. 8
B. 6
C. 9
D. 12

Explanation

To solve the problem step by step, we will first define the work done by one man and one woman in a day, and then use the information given to find the time taken by 6 men and 6 women to complete the job.

### Step 1: Define Work Units
Let:
- \( M \) = work done by one man in one day
- \( W \) = work done by one woman in one day

### Step 2: Set Up the Equations
From the first part of the problem:
- One man and 7 women can complete the job in 16 days.

This means the total work done can be expressed as:
\[ 16 \times (M + 7W) = \text{Total Work} \]

From the second part of the problem:
- 19 men and 10 women can complete the job in 3 days.

This means the total work done can be expressed as:
\[ 3 \times (19M + 10W) = \text{Total Work} \]

### Step 3: Equate the Total Work
Since both expressions represent the same total work, we can set them equal to each other:
\[ 16(M + 7W) = 3(19M + 10W) \]

### Step 4: Expand and Rearrange
Expanding both sides gives:
\[ 16M + 112W = 57M + 30W \]

Rearranging the equation:
\[ 16M - 57M + 112W - 30W = 0 \]
\[ -41M + 82W = 0 \]
\[ 41M = 82W \]
\[ \frac{M}{W} = \frac{82}{41} = 2 \]

This means:
\[ M = 2W \]

### Step 5: Substitute Back to Find Work
Now we can substitute \( M \) in terms of \( W \) back into one of the original equations to find the total work. Using the first equation:
\[ 16(M + 7W) = 16(2W + 7W) = 16 \times 9W = 144W \]

### Step 6: Work Done by 6 Men and 6 Women
Now we need to find the time taken by 6 men and 6 women to complete the same work:
\[ \text{Work done by 6 men and 6 women in one day} = 6M + 6W = 6(2W) + 6W = 12W + 6W = 18W \]

### Step 7: Calculate Time Taken
To find the time taken \( T \):
\[ T = \frac{\text{Total Work}}{\text{Work done in one day}} = \frac{144W}{18W} = 8 \text{ days} \]

### Final Answer
Thus, the time in which 6 men and 6 women can complete the job is **8 days**.

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