Mathematical approach in the calculation of work

Question

I have a statement that says:

In doing a job, Emilio and Sebastián take 2 months, Emilio and Rolando take 3 months, Sebastián and Rolando take 6 months. So, how long will the three in doing this work, if they work together?

I tried to make a system of equations, but I came up with an erroneous result.

I tried to solve with reasons, but I did not get anywhere. It is also assumed, that I should solve this in less than 2 minutes, but I have been trying for 45 minutes and I can not find the form, how could I then solve it?

I also know the equation to state the time it will take, which is: $\frac{1}{t} = \frac{1}{s} + \frac{1}{r} + \frac {1}{e}$, where $ t = $ total time among those, $ s = $ Sebastian's time, $ e = $ Emilio's time, $ r = $ Rolando's time.

Answer 1

method 1:

let rates of working of Emilio , Sebastian and Rolando be E,S and R respectively

$E+S=\dfrac{1}{2} $

$S+R=\dfrac{1}{6}$

$R+E=\dfrac{1}{3}$

adding all of them

$2(E+S+R)=1$

$E+S+R=\dfrac{1}{2}$

so, if they (all three) work togeather the work will be finished in $2$ months

alternative method:

i am taking work in terms of "days " not in "months"

assume total units of work = $LCM (2\times 30,6\times30,3\times30)=6 \times 30$ units

since $E$ and $S$ together finish work in $60 $ days,work done by them in a day=$180/60=3 unit/day$

$S$ and $R$ together finish work in $180 $ days,work done by them in a day=$180/180=1 unit/day$

$R$ and $E$ together finish work in $90$ days,work done by them in a day=$180/90=2 unit/day$

If they all work togeather work done by them in a day =$3\dfrac{units} {day}$

so, all of them will together finish work of $180$ units in =$ \dfrac{180}{3} $days=$2$ months

Answer 2

You can write, $$\frac{1}{e}+\frac{1}{s}=\frac{1}{2}$$ $$\frac{1}{e}+\frac{1}{r}=\frac{1}{3}$$ $$\frac{1}{s}+\frac{1}{r}=\frac{1}{6}$$

From these series of simultaneous equations, you can easily get your desired answer.