Question

4 men and 4 boys can do a piece of work in 3 days. While 2 men and 5 boys can finish in 4 days. How long would it take for 1 boy to do it? How long would it take 1 man to do it?

Answer 1

Hint: At first we will assume the time taken by man and boy in one day for the completion of work to be x and y respectively, from these and the given data we will form to the equation to solve for the time taken by man and boy in one day.
Once we get the time taken by man and boy in one day, we can easily find the time taken by the individuals to complete the work.

Complete step-by-step answer:
Given data: 4 men and 4 boys can do a piece of work in 3 days
2 men and 5 boys can finish the same work in 4 days
Now let one man take x time per day and one boy takes y time per day
Now it is given that 4 men and 4 boys take 3 days to do the work
Therefore, time taken by 4 men and 4 boys in one day will be $\dfrac{1}{3}$
i.e. $4x + 4y = \dfrac{1}{3}........(i)$
Similarly, it is given that 2 men and 5 boys take 4 days to do the work
Therefore, time taken by 2 men and 5 boys in one day will be $\dfrac{1}{4}$
i.e. $2x + 5y = \dfrac{1}{4}...........(ii)$
Now, solving for equations(i) and (ii)
Substituting the value of 2x from equation(ii) to the equation(i), we get,
 $ \Rightarrow 2\left( {\dfrac{1}{4} - 5y} \right) + 4y = \dfrac{1}{3}$
Simplifying the brackets, we get,
 $ \Rightarrow \dfrac{1}{2} - 10y + 4y = \dfrac{1}{3}$
On simplifying the like terms, we get,
 $ \Rightarrow \dfrac{1}{2} - \dfrac{1}{3} = 6y$
Simplifying the left-hand side by taking LCM, we get,
 $ \Rightarrow \dfrac{1}{6} = 6y$
 $\therefore y = \dfrac{1}{{36}}$
Substituting the value of y in equation(i), we get,
 $ \Rightarrow 4x + 4\left( {\dfrac{1}{{36}}} \right) = \dfrac{1}{3}$
Multiplying the whole equation with 3, we get,
 \[ \Rightarrow 12x + 4\left( {\dfrac{1}{{12}}} \right) = 1\]
On simplifying the bracket term
 \[ \Rightarrow 12x + \dfrac{1}{3} = 1\]
On simplifying the like terms
 \[ \Rightarrow 12x = \dfrac{2}{3}\]
Dividing both sides by 12
 \[ \Rightarrow x = \dfrac{1}{{18}}\]
Since one man take x time per day and one boy takes y time per day
Therefore one man takes $\dfrac{1}{x}$ and one boy takes $\dfrac{1}{y}$ days to complete the work
Therefore one man takes 18 and one boy takes 36 days to complete the work

Note: Most of the students multiply the per day work of the man and the boy with the number of days given in the question, which is not correct as we have time taken in a day to do the work, hence its reciprocal will be the total time take to one complete work.