Question

Given, 2 men and 7 women can do a piece of work in 4 days. It is done by 4 men and 4 women in 3 days. How long will it take for 1 man to do the work?

Answer 1

Hint: In the question, we have been asked to find the work done by 1 man to do the complete work. In order to solve the question, we need to make a pair of linear equations in 2 variables i.e. one variable for the work done by man and the other variable taken for the work done by women. We will solve the pair of linear equations in two variables by substitution method.

Complete step by step solution:
Let the time taken by 1 man to do the work= \[x\] days
Let the time taken by woman to do the word= \[y\] days
Therefore,
Work done by 1 man in a day = \[\dfrac{1}{x}\]
Work done by 1 woman in a day = \[\dfrac{1}{y}\]
According to question,
Work done by 2 men in a day = \[\dfrac{2}{x}\]
Work done by 7 women in a day = \[\dfrac{7}{y}\]
Work done by 2 men and 7 women in 1 day = \[\dfrac{1}{4}\]
Thus,
\[\Rightarrow \dfrac{2}{x}+\dfrac{7}{y}=\dfrac{1}{4}\]------- (1)
Similarly,
Work done by 4 men in a day = \[\dfrac{4}{x}\]
Work done by 4 women in a day = \[\dfrac{4}{y}\]
Work done by 4 men and 4 women in a day = \[\dfrac{1}{3}\]
Thus,
\[\Rightarrow \dfrac{4}{x}+\dfrac{4}{y}=\dfrac{1}{3}\]------- (2)
Putting \[\dfrac{1}{x}=a\] and \[\dfrac{1}{y}=b\] in equation (1) and equation (2) respectively, we obtain
\[\Rightarrow 2a+7b=\dfrac{1}{4}\] and \[ 4a+4b=\dfrac{1}{3}\]
Simplifying the above equations, we get
\[\Rightarrow a=\dfrac{1-28b}{8}\]------ (3) and \[\Rightarrow 12a+12b=1\]------ (4)
Substituting the equation (3) in equation (4), we obtain
\[\Rightarrow 12\left( \dfrac{1-28b}{8} \right)+12b=1\]
Simplifying the above equation, we get
\[\Rightarrow 3\left( \dfrac{1-28b}{2} \right)+12b=1\]
Multiplying 3 by the bracket, we get
\[\Rightarrow \dfrac{3-84b}{2}+12b=1\]
Multiplying both the sides of the equation by 2, we obtain
\[\Rightarrow 3-84b+24b=2\]
Combining the like terms, we get
  \[\Rightarrow 3-60b=2\]
Simplifying the above equation for the value of \[b\], we get
\[\Rightarrow -60b=2-3\]
\[\Rightarrow -60b=-1\]
\[\Rightarrow b=\dfrac{1}{60}\]
Substitute the value of \[b=\dfrac{1}{60}\] in equation (4), we obtain
\[\Rightarrow 12a+12\times \left( \dfrac{1}{60} \right)=1\]
\[\Rightarrow 12a+\dfrac{1}{5}=1\]
Multiplying both the sides of the above equation by 5, we obtain
\[\Rightarrow 60a+1=5\]
Subtract 1 from both the sides of the equation, we get
\[\Rightarrow 60a+1-1=5-1\]
\[\Rightarrow 60a=4\]
Dividing both the sides of the above equation by 60, we obtain
\[\Rightarrow a=\dfrac{4}{60}=\dfrac{1}{15}\]
\[\Rightarrow a=\dfrac{1}{15}\]
Replace \[a\] by \[\dfrac{1}{x}\] and \[b\] by \[\dfrac{1}{y}\], we get
\[\Rightarrow a=\dfrac{1}{15}\] and \[ b=\dfrac{1}{60}\]
\[\Rightarrow \dfrac{1}{x}=\dfrac{1}{15}\] and \[ \dfrac{1}{y}=\dfrac{1}{60}\]
Simplifying the above equation for the value of \[x\] and \[y\], we get
\[\Rightarrow x=15\ and\ y=60\]
Therefore,
Time taken by 1 man alone to do the complete work = \[x=15\ days\]
Work done by 1 man in = \[\dfrac{1}{15}\]
Therefore, time taken by 1 man to complete the work = 15 days.

Note:
In these types of questions, we use substitution methods to solve a pair of linear equations in two variables. Instead of a substitution method, we could have solved these pairs of linear equations by graphical method. In graphical method, we plot the equation of pairs of lines and try to find the intersection point. The intersection point i.e. where two lines intersect each other is the solution of the pair of equations.