Question

Raman alone can finish a piece of work in 15 days, and Suman alone can do it in 18 days. If both of them work together, how much time will they take to finish the work?

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 two workers be x and y respectively. So, the total work to be finished comes to be 15x, in terms of x and 18y, in terms of y. The work capacity when both works together is x+y, which you need to use in the final equation.

Complete step by step answer:
Let the work done by Raman per day be x.
Let the work done by Suman 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:
$\text{Total work that needs to be completed=15x}................\text{(i)}$ .
Also;
$\text{Total work that needs to be completed=18y}$ .
Total work is independent of the worker doing the work.
$\therefore 15x=18y$
$\Rightarrow 5x=6y$
$\Rightarrow \dfrac{5x}{6}=y............(ii)$
Let the number of days taken to finish the work when both workers work together be t days.
If both Suman and Raman work together.
$\text{Total work that needs to be completed = }n\times \sum{\left( \text{work capacity of each worker working} \right)\text{.}}$
$\Rightarrow \text{Total work that needs to be completed = t}(x+y)$ .
Substituting Total work from equation (i).
$\text{15x = t}(x+y)$
On substituting the value of y from equation (ii), we get:
$\text{15x = t}(x+\dfrac{5x}{6})$
$\Rightarrow \text{15 = t}(1+\dfrac{5}{6})$
$\Rightarrow \text{15 = }\left( \dfrac{6+5}{6} \right)t$
$\Rightarrow \text{15 = }\left( \dfrac{11}{6} \right)t$
$\therefore t=\dfrac{90}{11}=8.182\text{ Days}$ .
Converting 0.182 days to hours = $0.182\times 24=4.368\text{ hours}$ .
Converting 0.368 hours to minutes = $.368\times 60=22.08\text{ minutes}$ .
So, we can conclude that if both Raman and Suman work together, it will take them 8 complete days along with 4 hours and 22 minutes of the ${{9}^{th}}$ day.

Note: Questions 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.