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 )

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

@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

There seems to be an error in your set up. Your answer isn't terribly far off, but its hard to tell if it is just coincidence without seeing the set up.

@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

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.

@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.

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.

@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.

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

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

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 in (V) to obtain C=1/24

Coming back to the main question (7/24)+(5/24)+(1/24)= 13/24 is the fraction he Workers do in one hour and they thus Need 24/13 of an hour to create the one item.

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

@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 😉

"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?

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?

Heh, let's stick to questions that can be solved definitively, shall we?

@BigDoggProblem

He brought it up!

@joe-shmo said
@BigDoggProblem

He brought it up!

I am sorry for the inappropriate stab at a jocular comment 🙁

Mea culpa!

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

But remember that he is always watching. 🐶