rate and work - confused!

[kunalkulkarni]

Six Machines,each working at same constant rate ,together can complete the job in 12 days.How many additional machines each working at same constant rate,will be needed to complete job in 8 days?

A 3
B 4
C 5
D 6
E 2

I read the explanation for this problem in one of the posts where it is said:
Machine and days are inversely proportional.
How?
R*T=W
Since, rate is constant, isn't time(days) directly proportional to work?

Please help with the concept and explanation for above.

Thanks in advance,
Kunal

[Brent@GMATPrepNow]

Six Machines,each working at same constant rate ,together can complete the job in 12 days.How many additional machines each working at same constant rate,will be needed to complete job in 8 days?

A 3
B 4
C 5
D 6
E 2

If 6 machines take 12 days to complete the job, then we can say that the job takes 72 "machine days" to complete the job (6x12=72)

To complete the job in 8 days would require 9 machines (72 machine days divided by 8)

So, we need an additional 3 machines.

Answer = A

Cheers,
Brent

[kunalkulkarni]

Thank you Brent. 'Machine-days' concept similar to 'Man-days' helped me to understand!

Cheers,
Kunal

[ceilidh.erickson]

Since, rate is constant, isn't time(days) directly proportional to work?

Yes, the time is directly proportional to the work, for a single machine. The longer that machine works, the more work is produced.

Here, though, we have the added element of multiple machines working. It makes sense that the more machines we have working, the less time it will take them to complete the same amount of work. The number of machines and the time are thus inversely proportional. In effect, the rate isn't constant - the rate for each machine is constant, but we're taking the sum of the rates of all the machines.

As Brent said, the rate of each machine is 1 job in 72 days, or 1/72 job/day. If the time is changed to 8 days, we need to determine how many machines each working at 1/72 job/day can do 1 job in 8 days. We can think of the question like this:
(n machines)(1/72 job/day)(8 days) = 1 job
(n)(1/9) = 1
n must equal 9.
9 - 6 = 3 additional machines.

[kunalkulkarni]

It just can't get better!
What a wonderful elucidation on the concept Ceilidh! I am now convinced these attributes are inversely proportional.

Cheers,
Kunal

[GMATGuruNY]

Six Machines,each working at same constant rate ,together can complete the job in 12 days.How many additional machines each working at same constant rate,will be needed to complete job in 8 days?

A 3
B 4
C 5
D 6
E 2

An alternate approach is to plug in a rate for each machine.

Let the rate for each machine = 1 unit per day.
Thus, the rate for 6 machines = 6 units per day.
Work produced in 12 days = r*t = 6*12 = 72 units.
To produce 72 units in 8 days, the required rate = w/t = 72/8 = 9 units per day.
To increase the rate from 6 units per day to 9 units per day, 3 more machines are needed.

The correct answer is A.

[kunalkulkarni]

Thank you so much GMATGuru!!

Cheers,
Kunal