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Can I apply this formula differently?

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by yumi2012 » Fri Aug 09, 2013 7:40 am
Working alone, printers x, y, and z can do a certain printing job, consisting of a large number of pages, 12, 15, and 18 hours, respectively. What is the ratio of the time it takes printer x to do the job, working at its rate, to time it takes printers y and z to do the job, working together at their individual rates?

(A) 4/11
(B) 1/2
(C) 15/22
(D) 22/15
(E) 11/4

--------------------------------------------

From one of solution replies under this daily problem postion, I found the following formula.

"There's another formula you can use when it's exactly 2 workers:
Comb time = (a*b)/(a+b) "

Can I use this formula for 3 workers like (a*b*c)/(a+b+c)?

Thanks,
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by GMATGuruNY » Fri Aug 09, 2013 8:26 am
yumi2012 wrote: "There's another formula you can use when it's exactly 2 workers:
Comb time = (a*b)/(a+b) "

Can I use this formula for 3 workers like (a*b*c)/(a+b+c)?

Thanks,
The expression in red is incorrect.

RATE and TIME are RECIPROCALS.
If the 3 times are a, b and c, then the 3 rates are 1/a, 1/b and 1/c.
When the 3 workers work together, their combined rate = 1/a + 1/b + 1/c = bc/abc + ac/abc + ab/abc = (bc + ac + ab)/abc.
Thus, the time when they work together = the reciprocal of their combined rate = abc/(ab + ac + bc).

Many rate problems can be solved by plugging in a value for the job.
Working alone, printers x, y, and z can do a certain printing job, consisting of a large number of pages, 12, 15, and 18 hours, respectively. What is the ratio of the time it takes printer x to do the job, working at its rate, to time it takes printers y and z to do the job, working together at their individual rates?

(A) 4/11
(B) 1/2
(C) 15/22
(D) 22/15
(E) 11/4
Let the job = the LCM of 12, 15, and 18 = 180 units.

Rate for x = w/t = 180/12 = 15 units per hour.
Rate for y = w/t = 180/15 = 12 units per hour.
Rate for z = w/t = 180/18 = 10 units per hour.

Combined rate for y and z = 12+10 = 22 units per hour.
Time for y and z together = w/t = 180/22 = 90/11 hours.

(time for x)/(time for y and z) = 12 / (90/11) = (12*11) / 90 = 22/15.

The correct answer is D.
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by [email protected] » Fri Aug 09, 2013 6:31 pm
Hi yumi2012,

It's worth noting that this question does NOT ask for what happens when the three machines work together, it asks you to compare one machine by ITSELF vs. the two others working TOGETHER.

So we need to compare printer x to printers y and z

We can use the Work Formula: (ab)/(a+b) for the 2 machines working together.

Printer x = 12 hours
Printers y and z = (15x18)/(15+18) = 270/33 = 90/11

The ratio ends up as 12 : 90/11

This is the same as 12/(90/11) = (12x11)/90 = 132/90 = 22/15

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by Brent@GMATPrepNow » Sat Aug 10, 2013 12:17 pm
yumi2012 wrote:Working alone, printers x, y, and z can do a certain printing job, consisting of a large number of pages, 12, 15, and 18 hours, respectively. What is the ratio of the time it takes printer x to do the job, working at its rate, to time it takes printers y and z to do the job, working together at their individual rates?

(A) 4/11
(B) 1/2
(C) 15/22
(D) 22/15
(E) 11/4
For work questions, there are two useful rules:
Rule #1: If a person can complete an entire job in k hours, then in one hour, the person can complete 1/k of the job
Example: If it takes Sue 5 hours to complete a job, then in one hour, she can complete 1/5 of the job. In other words, her work rate is 1/5 of the job per hour

Rule #2: If a person completes a/b of the job in one hour, then it will take b/a hours to complete the entire job
Example: If Sam can complete 1/8 of the job in one hour, then it will take him 8/1 hours to complete the job.
Likewise, if Joe can complete 2/3 of the job in one hour, then it will take him 3/2 hours to complete the job.

Let's use these rules to solve the question. . . .

Printer Y takes 15 hours to complete a job. So, by rule #1, printer Y's rate is 1/15 of the job per hour
Printer Z takes 18 hours to complete a job. So, by rule #1, printer Z's rate is 1/18 of the job per hour
So, their combined rate per hour = 1/15 + 1/18
= 6/90 + 5/90
= 11/90
So, working together, printers Y and Z can complete the 11/90 of the job in one hour.
When we apply rule #2, we can conclude that, working together, printers Y and Z will complete the entire job in 90/11 hours.


What is the ratio of the time it takes printer X to do the job, working at its rate, to time it takes printers y and z to do the job?
So, (time for X to complete)/ (time for Y & Z to complete) = 12/(90/11)
= (12)(11/90)
= [spoiler]22/15[/spoiler]
= D

Cheers,
Brent
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