Six machines, each working at the same constant rate

Six machines, each 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, will be needed to complete the job in 8 days?
A. 2
B. 3
C. 4
D. 6
E. 8

Let \(x\) be the time needed for 1 machine to complete the job, so rate of one machine is \(\frac{1}{x}\) (rate is the reciprocal of time) --> rate of 6 machines would be \(\frac{6}{x}\).

As \(job=time*rate\) --> \(1=12*\frac{6}{x}\) --> \(x=72\) days needed for 1 machine to complete the job.

To complete the job in 8 days \(\frac{72}{8}=9\) machines are needed.

Difference: 9-6=3.

Answer: B.
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6 machines in 12 hours can make 6*12=72 units.
x machines in 8 hours can make x*8=72 units x=9

12-9=3 more machines are needed
hope it helps
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let each machine work at rate of x days
1/x+1/x+1/x+1/x+1/x+1/x = 1/12
6/x = 1/12
x = 72
now y is number of machines needed to complete job in 8 days so,
y/72 = 1/8
y = 9
required = y-x = 3 hence b.

Another solution which is faster is Since each machine works at a constant rate. The time needs to bought down from 12 to 8. So the new time is 2/3 of the original time. Thus to achieve this we need the rate to be 3/2 of original.

So 3/2* 6 = 9

So we need 9-6 = 3 more machines.

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Actually, this was more simple than I thought. When I first saw this, I started to panic thinking it's a rate question. But then I took a step back, and it was the simplest of calculations.

So here's the deal:

12 days -> 6 machines: 8 days -> ?

8 days means you definitely need more machines than the 6 and in such case the equation looks like-

12 * 6 = 8 * x

Therefore, x = 72 / 8 = 9

So 9-6=3 more machines

6 machines require 12 days = 72 machine days
how many machines to do the job in 8 days = \(\frac{72 machine-days}{8 days} = 9 machines\)

so, 3 more machines.

This is how I did it

6 machines in 1hr do 1/12 job
in 1 hr, 1 machine do 1/(12*6) job
In 8 days 1 machine do 8/(12*6)=1/9

Thus there is a need of 9 machine.

But I forgot to minus and get the difference so still get this question wrong.

\(Rate of machine = \frac{1}{m}\)

Draw the equation for 6 machines take 12 days:
\(\frac{6}{m}=\frac{1}{12}\)
\(m=72 days\)

Let's look for the number of machines to work for 8 days:
\(\frac{N}{72}=\frac{1}{8}\)
\(N=\frac{72}{8}=9 machines\)

Answer: 9 - 6 machines = 3 machines more

B

6 x 12 = (6+x) x 8

9 = 6+x

x=3

You can solve this problem without any equations. Just to calculate the rate of 1 machine.

The rate of 1 machine is \(\frac{1}{6*12}=\frac{1}{72}\) part per day.

So, 1 machine in 8 days can do \(\frac{8}{72}=\frac{1}{9}\).

So, to finish the job in 8 days we need 9 machines. Therefore 3 additional machines are required.

The answer is B.

Somehow, I found below method to be working for me -
6 Machines in 12 Days do 1 Work i.e. in notation form
6M * 12D = 1W,
so 1M * 1D = [1/(6*12)] W
How many Machines in 8 Days will complete 1 Work? i.e.
xM * 8D = [(x*8)/(6*12)] W = 1W

Solving for x in [(x*8)/(6*12)] = 1, we get x = 9 i.e. 9 Machines in 8 Days will do 1 Work. So 9-6=3 Machines more are required.

Ans - B

Here, 6 × 1/X = 1/12 or, X=72 . 2nd case, n × 1/72 = 1/8 or, n = 9 . additional = 9-6 = 3

6 Machines do \(\frac{1}{12}\) work in 1 day

We require \(\frac{1}{8}\) work to be done in 1 day

Multiplication factor\(= \frac{\frac{1}{8}}{\frac{1}{12}} = \frac{1}{8} * 12 = \frac{3}{2}\)

Additional machines required \(= 6 * \frac{3}{2} - 6 = 9 - 6 = 3\)

Answer = B

Six Machines:combined rate=6x
Time=12
Let Job be J

6x*12=J

Let M be the number of machines required to complete job in 8 days
Then
Mx*8=J

6x*12=Mx*8

M=9

Additional=9-6=3

Is this approach correct??

Hi All,

We're told that 6 machines, working at the same constant rate, can finish a job in 12 days. We're asked for the number of EXTRA machines will you need to finish the job in 8 days?

This type of question is easiest to solve if you convert the info into "work units" Here, we start with 6 machines working for 12 days. This equals (6)(12) = 72 machine-days of work to finish the job.

If we wanted to finish the job in 8 days, it would still take 72 machine-days of work, so we'd need 72/8 = 9 machines.

We already have 6 machines, so we need 3 MORE machines to get the job done in 8 days.

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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We are given that six machines, each working at the same constant rate, together can complete a certain job in 12 days. Since rate = work/time and if we consider work as 1, the rate of the six machines is 1/12.

We need to determine how many additional machines, each working at the same constant rate, will be needed to complete the same job in 8 days. In other words we need to determine how many additional machines are needed to work at a rate of 1/8. Since each machine works at the same constant rate, we can use a proportion to first determine the number of machines needed to work at the rate of 1/8.

Our proportion is as follows: 6 machines is to a rate of 1/12 as x machines is to a rate of 1/8.

6/(1/12) = x/(1/8)

72 = 8x

x = 9

So we need 9 – 6 = 3 more machines.

Answer: B
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Solution:

We know M1D1 = M2D2

=> M1 = 6 , D1 = 12 ,D2 = 8

Using the values, D2 = 6*12/8

= 72/8

=9 machines

=>Additionally = 9-6

= 3 machines (option b)

Devmitra Sen (GMAT SME)


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Bunuel andothers with more common sense.
I understand the mechanism of how the answer is derived. Cant believe I am asking this, but please bear with me slowness.

If we are reducing the time to complete the job by 33%, since 12 days to 8 days, is it wrong to proportionally increase the number of machines by that amount. Although the time and work are inversely proportional, we could not assume the machine's function and time is going to increase/decrease by the same amount. or originally I was solving by 6machines/12 days=8 days/x machines and solve for x for the incremental requirement.