# Machines/ work rate

• Jul 8th 2013, 07:03 PM
usre123
Machines/ work rate
Six machines, working at the same constant rate together can complete a certain job in 12 days. How many additional machines, each working at the same constant rate are needed to complete the job in 8 days?

What I did (and where am I wrong?):
each machine does "2" of the job in 6 days (since each machine does 2 parts of the job). so rate of each machine is 2/6 which is 1/3.
(1/3)*6 +(1/3) *x = 1/8.

I understand I'm taking out the rate wrong...but what else should it be? Rate is always 1/ work done in a day.... so 12/6=2, even if i use rate of 1/2, the answer is still wrong. Im very confused.

Thanks a lot!

• Jul 8th 2013, 08:48 PM
Soroban
Re: Machines/ work rate
Hello, usre123!

Your reasoning is faulty . . .

Quote:

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

I baby-talked my way through the problem like this . . .

$\begin{array}{cccc}\text{6 machines} & \text{in 12 days} & \text{can do 1 whole job.} \\ \text{6 machines} & \text{in 1 day} & \text{ can do }\frac{1}{12}\text{ of the job.} \\ \\[-4mm] \text{1 machine} & \text{in 1 day} & \text{can do }\frac{1}{72}\text{ of the job.} \\ \\[-4mm] x\text{ machines} & \text{in 1 day} & \text{can do }\frac{x}{72}\text{ of the job.} \\ \\[-4mm] x\text{ machines} & \text{in 8 days} & \text{can do }\frac{8x}{72}\text{ of the job.} \end{array}$

That final fraction $\frac{8x}{72} \,=\,\frac{x}{9}$ equals one whole job.
. . Hence:- $\frac{x}{9} \:=\:1 \quad\Rightarrow\quad x \,=\,9$

Nine machines are required.

Therefore, 3 additional machines are needed.
• Jul 8th 2013, 09:22 PM
ibdutt
Re: Machines/ work rate
if i was to do I would go like this, the method is called unitary method:
If the whole work is to completed in 12 days we need number of machines = 6
If the whole work is to completed in 1 day we need number of machines = 6 x 12 [ to do work in short time we need more machines ]
If the whole work is to completed in 8 day we need number of machines = (6 x 12)/8 [ to do work in more time we need less machines ]
= 9 machines
Thus we need to have one more machine to complete the work in 8 days.

ALTERNATELY
6 machines complete the work in 12 days
1 machine will complete the work in 12 x 6 = 72 days.
Thus 1 machine in one day can finish 1/72 of work
So i machine in 8 days would complete (1/72) x 8 = 1/9 work
Hence 1 work will be completed by 8 machines in 9 days
• Jul 9th 2013, 06:49 AM
HallsofIvy
Re: Machines/ work rate
If 6 machines do a certain job in 12 days then each machine does 1/6 of the job in 12 days and so each machine does (1/6)/12= 1/72 of the job in 1 day. If we have a total of X machines, working at that rate, they will do X/72 of the job each day. To finish the job in 8 days, we must have (X/72)(8)= X/9= 1 so X= 9. We need a total of 9 machines to do the job in 8 days. Since we already have 6 machines, we will need to add 9- 6= 3 new machines.