Question

2 women and 5 men can complete a piece of work in 4 days if the same work is completed by 3 women and 6 men will take 3 days to complete and find the time taken by one man and one woman to do the same work?

Answer 1

Hint: Assume the number of days to finish the whole work by one man as a variable and similarly for one woman as well with another variable. Now, use the two conditions to form two equations. Find the part of the work by given numbers of men and women in the given number of days and equate them to 1 by adding them as the whole work will be taken as 1 unit of work. Now solve both equations.

Complete step-by-step solution:
Let us suppose one man can complete the whole work in ‘x’ days and one woman completes it in ‘y’ days. So, one man does 1 unit of work in x days.
Hence, one man can do $\dfrac{1}{x}$ of work in 1 day.
Similarly, one woman can do $\dfrac{1}{y}$ unit of work in 1 day.
Now, the first condition is given that 2 women and 5 men can complete 1 unit of work in 4 days. So, 2 women can do $\dfrac{2}{y}$ unit of work in one day, as 1 woman is doing $\dfrac{1}{y}$ unit of work. Hence, 2 women can do $\dfrac{2}{y}\times 4$ unit of work in 4 days.
Similarly, 5 men can do $\dfrac{5}{x}\times 4$ unit of work in 4 days as one man can do $\dfrac{1}{x}$ unit of work in one day.
Now, we can write equation as
Amount of work by men + Amount of work by women = 1
$\begin{align}
  & \dfrac{2}{y}\times 4+\dfrac{5}{x}\times 4=1 \\
 & \Rightarrow \dfrac{8}{y}+\dfrac{20}{x}=1............\left( i \right) \\
\end{align}$
Now, we can get another equation by using the second condition that is 3 women and 6 men will take 3 days to complete the work.
So, 3 women will do $\dfrac{3}{y}\times 3$ part of the work in 3 days. And 6 men will do $\dfrac{6}{x}\times 3$ part of the work in 3 days. So, we can write another equation as,
Amount of work by men + Amount of work by women = 1
$\begin{align}
  & \dfrac{3}{y}\times 3+\dfrac{6}{x}\times 3=1 \\
 & \Rightarrow \dfrac{9}{y}+\dfrac{18}{x}=1.................\left( ii \right) \\
\end{align}$
Now, put $\dfrac{1}{y}=u$ and $\dfrac{1}{y}=v$ to solve equations (i) and (ii). So, we get
$8v + 20u = 1$………………(iii)
$9v + 18u = 1$………………(iv)
Now equation (iii) can be written as,
$\begin{align}
  & 20u=1-8v \\
 & \Rightarrow u=\dfrac{1-8v}{20} \\
\end{align}$
Now, put the value of ‘u’ in equation (iv), we get
$9v+18\left( \dfrac{1-8v}{20} \right)=1$
$\Rightarrow 9v+\dfrac{9}{10}\left( 1-8v \right)=1$
$\begin{array}{*{35}{l}}
   \Rightarrow 90v+9- 72v=10 \\
   \begin{align}
  & \Rightarrow 18v=1 \\
 & \Rightarrow v=\dfrac{1}{18}..................\left( v \right) \\
\end{align} \\
\end{array}$
Now, put value of ‘v’ in equation (iv), we get
$\begin{align}
  & 9\left( \dfrac{1}{12} \right)+18u=1 \\
 &\Rightarrow \dfrac{1}{2}+18u=1 \\
 &\Rightarrow 18u=\dfrac{1}{2} \\
 &\Rightarrow u=\dfrac{1}{36}................\left( vi \right) \\
\end{align}$
So, we can get values of x and y by using the equation $u=\dfrac{1}{x}$ and $v=\dfrac{1}{y}$, we get
$\begin{align}
  & \dfrac{1}{x}=\dfrac{1}{36} \\
 & \Rightarrow x=36 \\
\end{align}$
$\begin{align}
  & \dfrac{1}{y}=\dfrac{1}{18} \\
 & \Rightarrow y=18 \\
\end{align}$
Hence, one man can complete the work in 36 days and one woman can complete the work in 18 days.

Note: One can solve the equation $\dfrac{8}{y}+\dfrac{20}{x}=1$ and $\dfrac{9}{y}+\dfrac{18}{x}=1$ by eliminating $\dfrac{1}{x}$ or $\dfrac{1}{y}$ or by substituting value of $\dfrac{1}{x}$ or $\dfrac{1}{y}$ from one equation to another. But don’t take LCM to get the term ‘xy’. It would make the equation complex and may be difficult to solve. Hence, be clear and careful while solving both the equations. One may equate the part of work in 4 days and 3 days, and can get the equation as,
$\dfrac{2}{y}+\dfrac{5}{x}=\dfrac{1}{4}$ and $\dfrac{3}{y}+\dfrac{6}{y}=\dfrac{1}{3}$
Always use the fundamental concepts with the chapter work and time. And take the whole work as 1 unit in these kinds of questions.