Work Rate Question

[kyle.steiner123]

Machines A and B together can complete a task in 6/5 of an hour. Machines A anc C together can complete the same task in 3/2 of an hour. Machines B and C together can complete the same task in 2 hours. How long will it take machines A, B, and C to complete the same task?

A - 1/3
B - 1/2
C - 2/3
D - 5/6
E - 1

I tried to set up a system of equations in the W = R x T format but was not able to solve them. I assume there is an easier way but I just can't see it.

The equations I was using are below:
(1) W = (Ar + Br)(6/5)
(2) W = (Ar + Cr)(3/2)
(3) W = (Br + Cr)(2)

Where Ar, Br, and Cr are = to the rate of each machine.

Thanks.

[RonPurewal]

Machines A and B together can complete a task in 6/5 of an hour. Machines A anc C together can complete the same task in 3/2 of an hour. Machines B and C together can complete the same task in 2 hours.

When you have the time for 1 job, just "flip it over" -- i.e., take the reciprocal -- and that's the rate.
E.g., if 1 job takes 5 hours, that's 1/5 job per hour. If 1 job takes 2/3 hour, that's 3/2 job per hour. Etc.

So, here:
Rate A + Rate B = 5/6 job/hour
Rate A + Rate C = 2/3 job/hour
Rate B + Rate C = 1/2 job/hour

We don't want the individual rates; we just want Rate A + Rate B + Rate C. No reason to waste time solving the whole system.
If we just add all that junk together, we get
2(Rate A) + 2(Rate B) + 2(Rate C) = 5/6 + 4/6 + 3/6 job/hour
2(Rate A) + 2(Rate B) + 2(Rate C) = 12/6 job/hour
2(Rate A) + 2(Rate B) + 2(Rate C) = 2 job/hour
Rate A + Rate B + Rate C = 1 job/hour

So, together, they'll take 1 hour.

[kyle.steiner123]

Thanks Ron, appreciate the help!

[RonPurewal]

Thanks Ron, appreciate the help!

Sure.

That should pretty much be an automatic conversion (time for 1 job --> rate). In other words, you should make that conversion before you even press the "on" switch on your brain.