Expert: Josh - 2/27/2006

Question
Hello:

How is the answer determined for the following?
If possible, use a simple calculation.

One worker can perform a certain job in 8 days, another worker in 10 days and a third worker in 12 days. In what time can all three perform it working together?

I thank you for your reply.

Answer
Let J be the amount of work done per job.
Let R(n) be the rate at which work is carried out by worker "n". [Treat R(n) as an unknown]
Let T(n) be the time (in days) taken to finish J.

We have the general equation:
J(n)=R(n)*T(n).

For worker n=1, T(1)=8 [days], so we have
J=R(1)*8 ...[#1]
For worker n=2, T(2)=10 [days], so we have
J=R(2)*10...[#2]
For worker n=2, T(2)=12 [days], so we have
J=R(3)*12...[#3]

From [#1]-[#3], we deduce the rate of work as
R(1)=J/8, R(2)=J/10, R(3)=J/12. ...(*)

Combining the efforts of workers n=1,2,3, the job J has to be finished in less time, T.

Using the same ideas,
J=R(1)*T+R(2)*T+R(3)*T
=[R(1)+R(2)+R(3)]*T
...next, express R(n) in terms of J, see above (*)
...can you finish from here. Have a go,
...see the working below if you're not sure.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(continued)
J =[(J/8)+(J/10)+(J/12)]*T

This simplifies to 1=[(1/8)+(1/10)+(1/12)]*T,
T=1/[(1/8)+(1/10)+(1/12)]
=1/[(30/240)+(24/240)+(20/240)]
=1/[(30+24+20)/240]
=240/74  (takes just over 3 days)