Rate/Work problem

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Sun Jul 29, 2012 10:25 am

Hello,

This is from MGMAT Strategy Guide 3, P. 43:

Twelve identical machines, running continuosly at the same constant rate, take 8 days to complete a shipment. How many additional machines, each running at the same constant rate, would be needed to reduce the time required to complete a shipment by 2 days?

Ans: 4 additional machines

My solution was as follows:

12 machines - 8 days
1 machine - x days

So, 12x = 8 => x = 8/12 = 3/4

1 machine - 3/4 days
x machines - 6 days

=> 6 = (3/4)x
=> x = 8 machines

However, this is in-correct since 8 < 12

Can you please assist here? Thanks for your help.

Best Regards,
Sri

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Source: Beat The GMAT — Problem Solving | Sun Jul 29, 2012 10:25 am

AbhiJ: Master | Next Rank: 500 Posts; Posts: 381; Joined: Wed May 19, 2010 10:15 pmFollowed by: 2 members; Thanked: 41 times

by AbhiJ » Sun Jul 29, 2012 11:38 am

12 machines = 8 days
1 machine = 8 *12 = 96 days

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ksc1940: Senior | Next Rank: 100 Posts; Posts: 68; Joined: Wed Oct 19, 2011 2:27 pmFollowed by: 2 members; Thanked: 2 times

by ksc1940 » Sun Jul 29, 2012 12:17 pm

The answer i'm getting is 4 machines.

So we have 12 machines working together to get a job done in 8 days. Question is asking how many additional machines are needed to do the job in 6 days.

W=rt, and when w=1 (referring to the whole job), time and rate are reciprocals of each other. When t=8, the rate of all 12 machines combined is 1/8. Divide that by 12, and you get a rate of 1/96 for each individual machine. The combined rate of all 12 machines with the rate of the new machines will give us 1/6, the total rate. The reciprocal of that would give us the time we're looking for, 6 days. Hence, 1/8+x=1/6, x=1/24. Keep in mind that the rate of EACH machine is 1/96, and x here is the combined rate of the new machines. Since 1/24 is 4 times larger than 1/96, this means that 4 new machines need to be added to do the job in 6 days.

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eagleeye: Legendary Member; Posts: 520; Joined: Sat Apr 28, 2012 9:12 pmFollowed by: 49 members; Thanked: 339 times; GMAT Score:770

by eagleeye » Sun Jul 29, 2012 1:01 pm

gmattesttaker2 wrote:Hello,

This is from MGMAT Strategy Guide 3, P. 43:

Twelve identical machines, running continuosly at the same constant rate, take 8 days to complete a shipment. How many additional machines, each running at the same constant rate, would be needed to reduce the time required to complete a shipment by 2 days?

Ans: 4 additional machines

My solution was as follows:

12 machines - 8 days
1 machine - x days

So, 12x = 8 => x = 8/12 = 3/4

1 machine - 3/4 days
x machines - 6 days

=> 6 = (3/4)x
=> x = 8 machines

However, this is in-correct since 8 < 12

Can you please assist here? Thanks for your help.

Best Regards,
Sri

This is how I typically to these questions using logic.

To do the task in 8 days, we need = 12 machines

To do the task in 1 day, we need = 12*8 machines (Less days means that we need more machines)

Reduction by 2 days means, now we have 6 days to do the job.
To do the task in 6 days, we need = 12*8/6 machines (More days (from 1 to 6) means we need fewer machines).

Hence additional machines = New number - old number = 12*8/6 - 12 = 16-12 = 4 additional machines.

Here's another way of doing it:

Let number of additional machines required be x. So new total number = 12+x.

Amount = Number of machines*Time taken.

Since Amount of work remains constant:

New number* New time = Old number * Old time
(12+x)*6 = 12*8
=> 12+x = 12*8/6 = 2*8 = 16
=> x = 16-12 = 4

Cheers!

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everything's eventual: Master | Next Rank: 500 Posts; Posts: 101; Joined: Sun Jun 03, 2012 10:10 pmFollowed by: 1 member; Thanked: 10 times

by everything's eventual » Wed Aug 22, 2012 1:42 am

Since 12 machines are working at a constant rate, we can do it this way :

12 machines do one work in 8 days.

Therefore one machine does one work in 96 days.

1- 1/96

x - 1/6

Cross multiply and solve for x.

x= 16

No. of additional machines = 16 - 12 = 4

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Wed Aug 22, 2012 9:02 am

gmattesttaker2 wrote:Hello,

This is from MGMAT Strategy Guide 3, P. 43:

Twelve identical machines, running continuosly at the same constant rate, take 8 days to complete a shipment. How many additional machines, each running at the same constant rate, would be needed to reduce the time required to complete a shipment by 2 days?

The shipment is to be completed in 6 days instead of 8.
6/8 = 3/4.
Time and rate are RECIPROCALS: to complete the shipment in 3/4 the time, the work must be produced at 4/3 the rate.
Thus, the number of machines must increase by 1/3:
(1/3)12 = 4.

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cypherskull: Senior | Next Rank: 100 Posts; Posts: 94; Joined: Sat Mar 31, 2012 3:39 am; Location: Calcutta; Thanked: 8 times

by cypherskull » Wed Aug 22, 2012 12:02 pm

GMATGuruNY wrote:
gmattesttaker2 wrote: Thus, the number of machines must increase by 1/3:
(1/3)12 = 4.
Hi Mitch...I couldn't follow the quoted portion of ur solution. Could you please explain?

Regards,
Sunit

________________________________

Kill all my demons..And my angels might die too!

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Wed Aug 22, 2012 12:10 pm

cypherskull wrote:
GMATGuruNY wrote: Thus, the number of machines must increase by 1/3:
(1/3)12 = 4.
Hi Mitch...I couldn't follow the quoted portion of ur solution. Could you please explain?

To complete the work in 3/4 the time, the work must be produced at 4/3 the rate.
Thus, the rate of the 12 machines must increase by 1/3.
The machines cannot work faster; to increase the rate, we must increase the number of machines.
For the rate to increase by 1/3, the number of machines must increase by 1/3.
Since (1/3)12 = 4, the number of machines must increase by 4.

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gmat6087: Senior | Next Rank: 100 Posts; Posts: 53; Joined: Sat Oct 29, 2011 3:49 amFollowed by: 1 member; Thanked: 2 times

by gmat6087 » Fri Oct 05, 2012 5:20 am

Another easier approach:

since total work done is constant:

m1*d1=m2*d2 (m1,m2 number of men/machines. d1,d2 number of days taken.)

Substitute the values

12*8=m2*6 (work to be reduced by 2 days i.e 8-2=6)

=>m2=16
So 16-12=4 more machines required

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