Question

24 men can complete a piece of work in 16 days and 18 women can complete the same work in 32 days. 12 men and 6 women work together for 16 days. If the remaining work was to be completed in 2 days, how many additional men would be required besides 12 men and 6 women?

Answer 1

Hint: The total amount of work to be done remains constant irrespective of the factor of who is doing the work. Let the work capacities of the male workers and female workers be x and y, respectively. So, the total work to be finished comes to be $24\times 16x$ , in terms of x and $18y\times 32$ , in terms of y.

Complete step-by-step solution -
To start with the solution, we let the work done by a male worker per day be x, and the work done by a female worker per day be y.
Total work done in n number days is given by:
$n\times \sum{\left( \text{work capacity of each worker working} \right)\text{.}}$
Now, try to interpret the statement given in the question in mathematical terms using the above formula, we get:
$\text{Total work that needs to be completed=24}\times \text{16x}$ .
Also;
$\text{Total work that needs to be completed=18}\times \text{32y}$ .
Total work is independent of the number of workers and the type of worker doing the work.
$\therefore 24\times 16x=18\times 32y$
$\Rightarrow 2x=3y$
$\Rightarrow \dfrac{2x}{3}=y............(i)$
Now we let the number of additional male workers required to be p to finish the work in the next two days after 12 men and 6 women have worked for 16 days.

$\text{Total work that needs to be completed = }n\times \sum{\left( \text{work capacity of each worker working} \right)\text{.}}$
$\Rightarrow \text{Total work that needs to be completed = 16}\left( 12x+6y \right)+2\left( 12x+px \right)$ .
Substituting Total work from equation (i).
$\text{24}\times \text{16x = 16}\left( 12x+6y \right)+2\left( 12x+px \right)$
On substituting the value of y from equation (i), we get:
$\text{24}\times \text{16x = 16}\left( 12x+\dfrac{2}{3}\times 6x \right)+2\left( 12x+px \right)$
$\Rightarrow \text{24}\times \text{16x = 16}\left( 12x+4x \right)+2\left( 12x+px \right)$
$\Rightarrow \text{24}\times \text{16x = 16}\times \text{16x}+2\left( 12x+px \right)$
$\Rightarrow 8\times \text{16x = }2\left( 12x+px \right)$
$\Rightarrow 64\text{x =}\left( 12+p \right)x$
$\therefore p=52\text{ male workers}$ .
Therefore, we can say that the answer to the above question is 52 men.

Note: In this question, including work, have two things to be wisely selected. One is the elements of the problem that you are treating as variables, and the other is the unit of work. You can either let work done per unit time of each worker as variables or the total work to be a variable. The choice of unit and element for variable decides the complexity of solving.