Given:
\(3\) men and \(4\) women can do a piece of work in \(7\) days and, \(2\) men and \(1\) woman can do it in \(14\) days
As we know,
\(M_{1} D_{1}=M_{2} D_{2}\), where \(M_{1}\) and \(M_{2}=\) Number of person, \(D_{1}\) and \(D_{2}=\) Number of days
Total Work \(=\) Efficiency \(\times\) Time
\(M_{1}D_{1}=M_{2} D \)
\(\Rightarrow(3 \) men \(+4 \) women \() \times 7=(2\) men \(+1\) women \() \times 14 \)
\(\Rightarrow(3\)men \(+4\) women \()=(2 \) men \(+1\) women \() \times 2 \)
\(\Rightarrow 3 \) men \(+4 \) women \(=4 \) men \(+2\) women
\(\Rightarrow 2\) women \(=\) men
\(\Rightarrow\) (women : men) \(=1: 2\)
\(\Rightarrow\) Efficiency Ratio of (women : men) \(=1: 2\)
\(\Rightarrow\) Total Work \(=\) Efficiency \(\times\) Time
\(\Rightarrow(3\) men \(+4\) women \() \times 7=(2\) men \(+1\) women \() \times 14\)
\(\Rightarrow(3 \times 2+4 \times 1) \times 7=(2 \times 2+1 \times 1) \times 14\)
\(=70\)
\(\Rightarrow\) Total Work \(=70\)
\(\Rightarrow\) number of days to complete the work by \(7\) women \(=\frac{\text{ Total work}}{\text{Efficiency}}\)
\(= \frac{70 }{(7 \times 1)}\)
\(= 10\) days
\(\therefore\) Required number of days to complete the work by \(7\) women is \(10\) days.