Question

8 men and 12 boys can finish a piece of work in 10 days while 6 men and 8 boys can finish it in 14 days. Find the time taken by one man alone and that by one boy alone to finish that work.
$
  {\text{A}}{\text{. Man: 160 days, Boy: 190 days}} \\
  {\text{B}}{\text{. Man: 140 days, Boy: 280 days}} \\
  {\text{C}}{\text{. Man: 240 days, Boy: 150 days}} \\
  {\text{D}}{\text{. Man: 260 days, Boy: 200 days}} \\
$

Answer 1

Hint: In this question we need to find the time taken by one man alone and that by one boy alone to finish the given work. In order to solve the question, we will assume that the time taken by one man to complete the work will be x days and time taken by a boy to complete the work to be y days. This will help us proceed with the question.

Complete step-by-step answer:
Let the time taken by one man to complete the work be x days and the time taken by one boy to complete the work be y days.

So, if the time taken by one man to complete the work is x days, then in one day one man can complete $\dfrac{1}{x}$ part of the work.
Similarly, if the time taken by one boy to complete the work is y days , then in one day one boy can complete $\dfrac{1}{y}$ part of the work.
Now, we have been given that 8 men and 12 boys can finish the piece of work in 10 days.
$ \Rightarrow \left( {\dfrac{8}{x} + \dfrac{{12}}{y}} \right)10 = 1$……………………… Equation (1)
And 6 men and 8 boys can finish it in 14 days.
$ \Rightarrow \left( {\dfrac{6}{x} + \dfrac{8}{y}} \right)14 = 1$……………………… Equation (2)
To make the equations let us assume $\dfrac{1}{x} = u{\text{ and }}\dfrac{1}{y} = v$.
So, the equations (1) and (2) become,
$\left( {8u + 12v} \right)10 = 1$
$ \Rightarrow 80u + 120v = 1$……………………….. Equation (3)
And
$\left( {6u + 8v} \right)14 = 1$
$ \Rightarrow 84u + 112v = 1$ ……………………. Equation (4)
Subtracting Equation (3) from Equation (4), we get,
$4u - 8v = 0$
$ \Rightarrow u = 2v$ …………………Equation (5)
Now, putting Equation (5) in Equation (3), we get,
$160v + 120v = 1$
$ \Rightarrow 280v = 1$
$ \Rightarrow v = \dfrac{1}{{280}}$
And $u = \dfrac{1}{{140}}$
Therefore, x=140, y=280.
So, it takes 140 days to finish the work by one man alone and 280 days to finish the work by one boy alone.

Note: Whenever we face such types of problems the key point to remember is that we need to have a good grasp over linear equations in two variables. In these questions we should always proceed in the way as we have done above otherwise we might make an error. This helps in getting us the required expressions and gets us on the right track to reach the answer.