Team of Workers

Alice and Bob working together can complete a task in 2 hours.
If Alice works with Charlie they can do the same task in 3 hours.
If Bob works with Charlie they can complete it in 4 hours.

If Alice, Bob and Charlie were to work together on the task how long would it take to complete.

Assume the individuals work at the same rate regardless of who is working ( i.e. the work to complete the task is parallelizable )

@joe-shmo

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@joe-shmo said
Alice and Bob working together can complete a task in 2 hours.
If Alice works with Charlie they can do the same task in 3 hours.
If Bob works with Charlie they can complete it in 4 hours.

If Alice, Bob and Charlie were to work together on the task how long would it take to complete.

Assume the individuals work at the same rate regardless of who is working ( i.e. the work to complete the task is parallelizable )

@venda said
I tried this using simultaneous equations.
I got the answers that A's contribution to the task was half an hours work(.5)
B's contribution was one and a half hours(1.5)
C's contribution was two and a half hours(2.5)
.5 +1.5 +2.5/3 =1.5 hrs

@Ponderable

Ponderable is correct.

@venda said
I tried this using simultaneous equations.
I got the answers that A's contribution to the task was half an hours work(.5)
B's contribution was one and a half hours(1.5)
C's contribution was two and a half hours(2.5)
.5 +1.5 +2.5/3 =1.5 hrs

@athousandyoung said
That's something like what I got. Setting it up as a matrix and then solving gives numbers like that. Maybe that's not the way to do it though. EDIT I don't think I was doing it right anyway.

@joe-shmo said
It involves the solution to a system of equations, but a matrix is a bit overkill as the resulting equations are easily rearranged. The trick to this is getting the equations right.

@venda said
As is ever the case.
Perhaps Ponderable would oblige us with the equations and an explanation?

@ponderable said
He would.

So we assume that any Team of Workers make one piece yielding the following equations:

(I) 2A+2B=1
(II) 3A+3C=1
(III) 4B+4C=1

(I) rearranges to (IV) B=(1/2)-A
(II) rearranges to (V) C=(1/3)-A

Setting (IV) and (V) into (III) gives us 4*((1/2)-A)+4*((1/3)-A)=1
solving that for A yields (VI) A=7/24

We now can fill in A into (IV) having B=5/24 and ...[text shortened]... f an hour to create the one item.

Lets hope that A gets the sevenfold amount of Money per hour 😉

Explanation:

The units for the equations should be:
( tasks / hour ) * ( hours ) = ( tasks )

So, the variables a, b, and c are work rates, in units of tasks per hour. The equations should be:

(a+b) * 2 = 1
(a+c) * 3 = 1
(b+c) * 4 = 1

Solve the above system for a, b, and c, then plug them into this equation:

(a+b+c) * h = 1

...and you should get the answer.

@joe-shmo said
"Lets hope that A gets the sevenfold amount of Money per hour 😉"

Nonsense... from each according to his ability, to each according to his needs?

@BigDoggProblem

He brought it up!

@joe-shmo said
@BigDoggProblem

He brought it up!

Both of your meekness has appeased the Dogg, for now.

But remember that he is always watching. 🐶