To solve the problem, we need to determine how long it will take for 7 men and 5 women to plough the field, given that 3 men or 4 women can do it in 43 days.
### Step 1: Determine the work done by men and women
First, we need to find out how much work is done by 1 man and 1 woman in a day.
- If 3 men can complete the work in 43 days, the total work can be represented as:
\[
\text{Total work} = 3 \text{ men} \times 43 \text{ days} = 129 \text{ man-days}
\]
- Similarly, if 4 women can complete the work in 43 days, the total work can also be represented as:
\[
\text{Total work} = 4 \text{ women} \times 43 \text{ days} = 172 \text{ woman-days}
\]
### Step 2: Establish the relationship between men and women
From the above calculations, we can set up the equation:
\[
3 \text{ men} = 4 \text{ women}
\]
This means that the work done by men and women can be compared. To find the equivalent work done by one woman in terms of men, we can rearrange this to find:
\[
1 \text{ woman} = \frac{3}{4} \text{ men}
\]
### Step 3: Calculate the total work done by 7 men and 5 women
Now, we need to find out how much work 7 men and 5 women can do together in one day.
- The work done by 7 men in one day:
\[
\text{Work by 7 men} = 7 \text{ men} \times \frac{1}{43} \text{ (work done by 3 men in a day)} = \frac{7}{3 \times 43} = \frac{7}{129} \text{ of the work}
\]
- The work done by 5 women in one day:
\[
\text{Work by 5 women} = 5 \text{ women} \times \frac{1}{43} \text{ (work done by 4 women in a day)} = \frac{5}{4 \times 43} = \frac{5}{172} \text{ of the work}
\]
### Step 4: Combine the work done by men and women
Now, we can add the work done by 7 men and 5 women together:
\[
\text{Total work done in one day} = \frac{7}{129} + \frac{5}{172}
\]
To add these fractions, we need a common denominator. The least common multiple of 129 and 172 is 22164.
Calculating:
\[
\frac{7 \times 172}{22164} + \frac{5 \times 129}{22164} = \frac{1204 + 645}{22164} = \frac{1849}{22164}
\]
### Step 5: Calculate the total days required
Now, to find the total number of days required to complete the work, we take the reciprocal of the total work done in one day:
\[
\text{Days required} = \frac{Total \, work}{Work \, done \, in \, one \, day} = \frac{1}{\frac{1849}{22164}} = \frac{22164}{1849}
\]
Calculating this gives us approximately:
\[
\text{Days required} \approx 12
\]
### Final Answer
Thus, 7 men and 5 women will take approximately **12 days** to plough the field.
