If 8 men or 12 women can do a piece of work in 10 days, then what is the number of days required by 4 men and 4 women to finish the work ?

Options

A. 12 days
B. 22 days
C. 10 days
D. None of these

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Determine the total work done by men and women
We know that:
- 8 men can complete the work in 10 days.
- 12 women can complete the work in 10 days.

First, we calculate the total work in terms of "man-days" and "woman-days."

**Total work in man-days:**
\[ \text{Total work} = \text{Number of men} \times \text{Number of days} = 8 \text{ men} \times 10 \text{ days} = 80 \text{ man-days} \]

**Total work in woman-days:**
\[ \text{Total work} = \text{Number of women} \times \text{Number of days} = 12 \text{ women} \times 10 \text{ days} = 120 \text{ woman-days} \]

### Step 2: Calculate the work done by one man and one woman in one day
Now we find out how much work one man and one woman can do in one day.

**Work done by one man in one day:**
\[ \text{Work done by 1 man in 1 day} = \frac{1}{80} \text{ of the work} \]

**Work done by one woman in one day:**
\[ \text{Work done by 1 woman in 1 day} = \frac{1}{120} \text{ of the work} \]

### Step 3: Calculate the work done by 4 men and 4 women in one day
Now, we calculate the combined work done by 4 men and 4 women in one day.

**Work done by 4 men in one day:**
\[ \text{Work done by 4 men in 1 day} = 4 \times \frac{1}{80} = \frac{4}{80} = \frac{1}{20} \]

**Work done by 4 women in one day:**
\[ \text{Work done by 4 women in 1 day} = 4 \times \frac{1}{120} = \frac{4}{120} = \frac{1}{30} \]

### Step 4: Combine the work done by 4 men and 4 women
Now we add the work done by 4 men and 4 women in one day:
\[ \text{Total work done by 4 men and 4 women in 1 day} = \frac{1}{20} + \frac{1}{30} \]

### Step 5: Find the LCM and add the fractions
To add these fractions, we need to find the least common multiple (LCM) of 20 and 30, which is 60.

**Convert the fractions:**
\[ \frac{1}{20} = \frac{3}{60} \quad (\text{since } 20 \times 3 = 60) \]
\[ \frac{1}{30} = \frac{2}{60} \quad (\text{since } 30 \times 2 = 60) \]

**Add the fractions:**
\[ \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \]

### Step 6: Calculate the total days required to finish the work
If 4 men and 4 women together do \(\frac{1}{12}\) of the work in one day, then the total number of days required to finish the work is the reciprocal of \(\frac{1}{12}\):
\[ \text{Total days required} = \frac{1}{\frac{1}{12}} = 12 \text{ days} \]

### Final Answer:
The number of days required by 4 men and 4 women to finish the work is **12 days**.

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