Question

2 men and 7 boys can do a piece of work in 4 days. The same work is done in 3 days by 4 men and 4 boys. How long would it take one man and one boy to do it?

Answer 1

Hint: To solve this question, we will make a pair of linear equations in two variables and then we will solve those linear equations by substitution method.

Complete step-by-step answer:
According to the equation, we have to find the time taken by a boy and a man to complete work individually. Let us assume that a boy takes $'x'$ days to finish the work.
Thus, the work done by a boy in one day = $\dfrac{1}{x}$
Now, let us assume that a man takes $'y'$ days to finish the work.
Thus, the work done by a man in one day = $\dfrac{1}{y}$
It is given that 2 men and 7 boys complete a work in 4days. Thus work done by 2 men in one day = $\dfrac{2}{y}$ and the work done by 7 boys in one day = $\dfrac{7}{x}$ .
Therefore, we get:
$\dfrac{2}{y}+\dfrac{7}{x}=\dfrac{1}{4}$
Let us assume $\dfrac{1}{y}=b$ and $\dfrac{1}{x}=a$ in the above equation. After doing this we will get:
$2b+7a=\dfrac{1}{4}$
In another situation, it is given that 4 boys and 4 men do the same work in 3 days. Thus work done by 4 men in one day = $\dfrac{4}{y}$ and the work done by 4 boys in one day = $\dfrac{4}{x}$ . Therefore, we get
$\begin{align}
  & \dfrac{4}{y}+\dfrac{4}{x}=\dfrac{1}{3} \\
 & \Rightarrow 4a+4b=\dfrac{1}{3}...........\left( ii \right) \\
\end{align}$
Now, we have got a pair of linear equations in two variables:
$\begin{align}
  & 2b+7a=\dfrac{1}{4}..............\left( i \right) \\
 & 4a+4b=\dfrac{1}{3}..............\left( ii \right) \\
\end{align}$
Now, we are going to solve these questions by the substitution method. From equation (ii), we have:
$\begin{align}
  & 4a+4b=\dfrac{1}{3} \\
 & \Rightarrow 4\left( a+b \right)=\dfrac{1}{3} \\
 & \Rightarrow a+b=\dfrac{1}{12} \\
 & \Rightarrow a=\dfrac{1}{12}-b...........\left( iii \right) \\
\end{align}$
Now we will substitute the value of ‘a’ from (iii) into (i). After substitution, we will get:
$\begin{align}
  & 2b+7\left( \dfrac{1}{12}-b \right)=\dfrac{1}{4} \\
 & \Rightarrow 2b+\dfrac{7}{12}-7b=\dfrac{1}{4} \\
 & \Rightarrow -5b=\dfrac{1}{4}-\dfrac{7}{12} \\
 & \Rightarrow 5b=\dfrac{7}{12}-\dfrac{1}{4} \\
 & \Rightarrow 5b=\dfrac{4}{12} \\
 & \Rightarrow 5b=\dfrac{1}{3} \\
 & \Rightarrow b=\dfrac{1}{15}.................\left( iv \right) \\
\end{align}$
Now, we know that $b=\dfrac{1}{y}$ . Thus:
$\begin{align}
  & \dfrac{1}{y}=\dfrac{1}{15} \\
 & \Rightarrow y=15days \\
\end{align}$
Now, we will substitute the value of ‘b’ from (iv) to (ii)
$\begin{align}
  & \Rightarrow 4a+\dfrac{4}{15}=\dfrac{1}{3} \\
 & \Rightarrow 4a=\dfrac{1}{3}-\dfrac{4}{15} \\
 & \Rightarrow 4a=\dfrac{1}{15} \\
 & \Rightarrow a=\dfrac{1}{60} \\
\end{align}$
Now, we know that $a=\dfrac{1}{x}$ .Thus
$\dfrac{1}{x}=\dfrac{1}{60}$
$\Rightarrow x=60days$
Thus, a boy takes 60 days while a man takes 15 days to complete a work individually.

Note: Instead of substituting method, we could have also used the graphical method. In graphical methods, we plot the equations of pairs of lines and find the intersection point. The intersection point is the solution of the pair of equations.