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Machines X and Y produced identical bottles at different constant rate

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Machines X and Y produced identical bottles at different constant rate [#permalink] New post 08 Nov 2011, 19:21
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Machines X and Y produced identical bottles at different constant rates. Machine X, operating alone for 4 hours, filled part of a production lot; then Machine Y, operating alone for 3 hours, filled the rest of this lot. How many hours would it have taken Machine X operating alone to fill the entire production lot?

(1) Machine X produced 30 bottles per minute.
(2) Machine X produced twice as many bottles in 4 hours as Machine Y produced in 3 hours.


[Reveal] Spoiler:
My work:

Bottles = (rateX)4 + (rateY)3 *Did not forget to convert minutes to hours


1) RateX = 30 b/min = 1800 b/hr
Xbottles = 1800(4) = 7200 bottles

Insuff

2) Bottles Y = 1/2 Bottles X

I went with putting them together, getting the total number of bottles produced and backing out the rate for the answer. WRONG.

I have the answer, I just don't understand it.


OPEN DISCUSSION OF THIS QUESTION IS HERE: machines-x-and-v-produced-identical-bottles-at-different-104208.html
[Reveal] Spoiler: OA

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Re: Machines X and Y produced identical bottles at different constant rate [#permalink] New post 08 Nov 2011, 20:39
I recommend you to approach the question as mentioned below -

As question is asking to find time taken by Machine X operating alone to fill the entire production lot and we've know that production lot (a) partily filled by X in 4 hrs (b) Rest filled by Y in 3 hrs.
As we dont have any information about Part(b), we need to find or establish a relationship between X and Y so that we can calculate the whole period in terms of X. Fortunately, The relation is clearly given in (2) Machine X produced twice as many bottles in 4 hours as Machine Y produced in 3 hours.

Hence, B is the answer. No calculation required.

Hope it helps!
Cheers!
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Re: Machines X and Y produced identical bottles at different constant rate [#permalink] New post 09 Nov 2011, 12:00
Thanks. Your explanation is so much clearer.

Good luck with your GMAT!
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Re: Machines X and Y produced identical bottles at different constant rate [#permalink] New post 30 Nov 2011, 23:38
(1) x filled 30 x 4 = 120 bottles, but we do not know the what portion is 120. insufficient.

(2) y = 3b
x = 6b
total 9b
so 2 sufficient.
Ans. B
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Re: Machines X and Y produced identical bottles at different constant rate [#permalink] New post 12 Nov 2014, 13:30
X Y total
Time 4 3 7
rate x y
bottles 4x 3y 4x+3y

from statement 1 - x= 30b/min = 1800b/hr, nothing is known of y here, so insuff
from statement 2 - 4x = 3y * 2, x = 3y/2.
total bottles = 4*3y/2 + 3y = 9y bottles to be manufactured by x alone in how much time is the question.
let the time be T.
so, 3y/2*T = 9y, T = 6 hours.
sufficient.
Answer [b]
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Re: Machines X and Y produced identical bottles at different constant rate [#permalink] New post 12 Nov 2014, 17:52
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galaxyblue wrote:
Machines X and Y produced identical bottles at different constant rates. Machine X, operating alone for 4 hours, filled part of a production lot; then Machine Y, operating alone for 3 hours, filled the rest of this lot. How many hours would it have taken Machine X operating alone to fill the entire production lot?

(1) Machine X produced 30 bottles per minute.
(2) Machine X produced twice as many bottles in 4 hours as Machine Y produced in 3 hours.


[Reveal] Spoiler:
My work:

Bottles = (rateX)4 + (rateY)3 *Did not forget to convert minutes to hours


1) RateX = 30 b/min = 1800 b/hr
Xbottles = 1800(4) = 7200 bottles

Insuff

2) Bottles Y = 1/2 Bottles X

I went with putting them together, getting the total number of bottles produced and backing out the rate for the answer. WRONG.

I have the answer, I just don't understand it.


There are several important things you should know to solve work problems:

1. Time, rate and job in work problems are in the same relationship as time, speed (rate) and distance in rate problems.

time*speed=distance <--> time*rate=job \ done. For example when we are told that a man can do a certain job in 3 hours we can write: 3*rate=1 --> rate=\frac{1}{3} job/hour. Or when we are told that 2 printers need 5 hours to complete a certain job then 5*(2*rate)=1 --> so rate of 1 printer is rate=\frac{1}{10} job/hour. Another example: if we are told that 2 printers need 3 hours to print 12 pages then 3*(2*rate)=12 --> so rate of 1 printer is rate=2 pages per hour;

So, time to complete one job = reciprocal of rate. For example if 6 hours (time) are needed to complete one job --> 1/6 of the job will be done in 1 hour (rate).

2. We can sum the rates.

If we are told that A can complete one job in 2 hours and B can complete the same job in 3 hours, then A's rate is rate_a=\frac{job}{time}=\frac{1}{2} job/hour and B's rate is rate_b=\frac{job}{time}=\frac{1}{3} job/hour. Combined rate of A and B working simultaneously would be rate_{a+b}=rate_a+rate_b=\frac{1}{2}+\frac{1}{3}=\frac{5}{6} job/hour, which means that they will complete \frac{5}{6} job in one hour working together.

3. For multiple entities: \frac{1}{t_1}+\frac{1}{t_2}+\frac{1}{t_3}+...+\frac{1}{t_n}=\frac{1}{T}, where T is time needed for these entities to complete a given job working simultaneously.

For example if:
Time needed for A to complete the job is A hours;
Time needed for B to complete the job is B hours;
Time needed for C to complete the job is C hours;
...
Time needed for N to complete the job is N hours;

Then: \frac{1}{A}+\frac{1}{B}+\frac{1}{C}+...+\frac{1}{N}=\frac{1}{T}, where T is the time needed for A, B, C, ..., and N to complete the job working simultaneously.

For two and three entities (workers, pumps, ...):

General formula for calculating the time needed for two workers A and B working simultaneously to complete one job:

Given that t_1 and t_2 are the respective individual times needed for A and B workers (pumps, ...) to complete the job, then time needed for A and B working simultaneously to complete the job equals to T_{(A&B)}=\frac{t_1*t_2}{t_1+t_2} hours, which is reciprocal of the sum of their respective rates (\frac{1}{t_1}+\frac{1}{t_2}=\frac{1}{T}).

General formula for calculating the time needed for three A, B and C workers working simultaneously to complete one job:

T_{(A&B&C)}=\frac{t_1*t_2*t_3}{t_1*t_2+t_1*t_3+t_2*t_3} hours.

BACK TO THE ORIGINAL QUESTION:
Machines X and Y produced identical bottles at different constant rates. Machine X, operating alone for 4 hours, filled part of a production lot; then Machine Y, operating alone for 3 hours, filled the rest of this lot. How many hours would it have taken Machine X operating alone to fill the entire production lot?

You can solve this question as Karishma proposed in her post above or algebraically:

Let the rate of X be x bottle/hour and the rate of Y y bottle/hour.
Given: 4x+3y=job. Question: t_x=\frac{job}{rate}=\frac{job}{x}=?

(1) Machine X produced 30 bottles per minute --> x=30*60=1800 bottle/hour, insufficient as we don't know how many bottles is in 1 lot (job).

(2) Machine X produced twice as many bottles in 4 hours as Machine Y produced in 3 hours --> 4x=2*3y, so 3y=2x --> 4x+3y=4x+2x=6x=job --> t_x=\frac{job}{rate}=\frac{job}{x}=\frac{6x}{x}=6 hours. Sufficient.

Answer: B.

OPEN DISCUSSION OF THIS QUESTION IS HERE: machines-x-and-v-produced-identical-bottles-at-different-104208.html
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Re: Machines X and Y produced identical bottles at different constant rate   [#permalink] 12 Nov 2014, 17:52
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