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# A certain dairy packing plant has two machines, P and Q

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Joined: 29 Jul 2013
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A certain dairy packing plant has two machines, P and Q  [#permalink]

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01 Aug 2013, 01:45
1
13
00:00

Difficulty:

65% (hard)

Question Stats:

70% (03:00) correct 30% (03:10) wrong based on 269 sessions

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A certain dairy packing plant has two machines, P and Q, that process milk at constant rates of 30 gallons per minute and 45 gallons per minute, respectively. A day's run of milk can be processed by machine P operating alone in 6 hours, by machine Q operating alone in 4 hours, or by both machines operating simultaneously in 2.4 hours. If a day's run of milk is processed using machine Q alone for half the time and both machines together for half the time, how many hours does it take to complete the run?

A. 1.5
B. 3.0
C. 3.75
D. 4.2
E. 5.0

I stumbled upon this one yesterday and it kind of confused me, though I have pretty much understood the logic behind work problems. Ideas? Thanks in advance!

OA B
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Re: A certain dairy packing plant has two machines, P and Q  [#permalink]

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01 Aug 2013, 01:52
4
1
$$rate*time=work$$
$$45=$$rate Q/min
$$30=$$rate P/min
$$30+45=$$rate Q+P/min

Let $$t$$ be half of the time necessary, expressed in minutes. The total work of a day is $$6 h*60 min*30r/min$$.

$$45*t + t(30+45)=6*60*30$$ or $$120t=6*60*30$$, $$t=90$$ minutes. $$t$$ is half of the time , so the total is $$180$$ minutes or $$3$$ hours.

Hope it's clear
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##### General Discussion
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Re: A certain dairy packing plant has two machines, P and Q  [#permalink]

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01 Aug 2013, 02:12
2
1
DrFeelgood wrote:
A certain dairy packing plant has two machines, P and Q, that process milk at constant rates of 30 gallons per minute and 45 gallons per minute, respectively. A day's run of milk can be processed by machine P operating alone in 6 hours, by machine Q operating alone in 4 hours, or by both machines operating simultaneously in 2.4 hours. If a day's run of milk is processed using machine Q alone for half the time and both machines together for half the time, how many hours does it take to complete the run?

A. 1.5
B. 3.0
C. 3.75
D. 4.2
E. 5.0

I stumbled upon this one yesterday and it kind of confused me, though I have pretty much understood the logic behind work problems. Ideas? Thanks in advance!

OA B

machine p ==>30 gallon per minute =30*60 gallon per hour
machine q ==>45 gallon per minute =45*60 gallon per hour
machine p+q ==>75 gallon per minute =75*60 gallon per hour

for a days run milk required = run machine p for 6 hours = $$6*30*60$$ gallon

If a day's run of milk is processed using machine Q alone for half the time and both machines together for half the time, how many hours does it take to complete the run?
lets total time taken = $$2t$$
t time with machine q and t time with machine p+q

therefore
milk required for a day = $$45*60*t + 75*60*t$$
now we know milk required for a day = $$6*30*60$$ gallon
therefore
$$6*30*60$$ gallon = $$45*60*t + 75*60*t$$
cancel out all $$60$$ from both sides
$$6*30 = 45*t+75*t solving t=1.5 hence 2t=3$$

hence B
HOPE IT HELPS
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Re: A certain dairy packing plant has two machines, P and Q  [#permalink]

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01 Aug 2013, 02:16
1
Both your answers are very helpful, indeed! Thanks a lot! I somehow managed to confuse myself with the "half the time" aspect of the exercise. I started thinking in terms of 24 hours etc etc yet it was that simple (signing it with a "t"). A simple reminder that I should keep things simple and to the point! The only logical thinking that I did was to eliminate answers A, D and E since the time needed to do the run, has to be longer than 2.4 hours and less than 4.
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Re: A certain dairy packing plant has two machines, P and Q  [#permalink]

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01 Aug 2013, 02:20
DrFeelgood wrote:
Both your answers are very helpful, indeed! Thanks a lot! I somehow managed to confuse myself with the "half the time" aspect of the exercise. I started thinking in terms of 24 hours etc etc yet it was that simple (signing it with a "t"). A simple reminder that I should keep things simple and to the point! The only logical thinking that I did was to eliminate answers A, D and E since the time needed to do the run, has to be longer than 2.4 hours and less than 4.

Thats the great way to eliminate 3 options.
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Re: A certain dairy packing plant has two machines, P and Q  [#permalink]

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01 Aug 2013, 03:03
1. A day's run of milk is processed by p and q together and also by q alone.
2. Let the total time taken be x.
3. x = time taken by p and q together/ 2 + time taken by q alone /2
4. i.e., 1/x = (1/4 + 1/6)/2 + (1/4)/2
5. x= 3
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Re: A certain dairy packing plant has two machines, P and Q  [#permalink]

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06 Aug 2013, 01:45
2
x = total time

1/x = 1/2*1/4 (Q alone) + 1/2*1/2.4 (P & Q simultaneously) = 1/8 + 1/2*24/10 = 1/8 + 5/24 = 8/24

--> x = 24/8 = 3 --> B.
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Re: A certain dairy packing plant has two machines, P and Q  [#permalink]

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25 Sep 2013, 22:40
1
Rate (P) = 1/2
Rate (Q) = 3/4

P+Q (2.4)= work done = 3

Let 't' be the total time for the scenario then

(3/4) t/2 + t/2 (3/4 + 1/2) = 3

t=3
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Re: A certain dairy packing plant has two machines, P and Q  [#permalink]

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23 Dec 2014, 11:06
1
1
Hi All,

This question can be solved by TESTing THE ANSWERS.

We're given rates for Machine P (30 gallons/minute) and Machine Q (45 gallons/minute) and we're told how long it would take each machine to finish a job (6 hours and 4 hours respectively). We can use this information to figure out the TOTAL JOB....

Total Job = (30 gallons/min)(60 mins)(6 hours) = 10,800 Gallons

We're then given a specific set of circumstances to calculate: Machine Q works by itself for a period of time, then the two machine work together for the SAME amount of time that Machine Q worked by itself. We're told that this situation will complete the entire job. We're asked how long it would take to complete the job under these circumstances.

Logically, since Machine Q can do the entire job on its own in 4 hours, with the added help of Machine P, the amount of time MUST be LESS than 4 hours.

If the Total job was 3 hours, then Machine Q would work for 1.5 hours by itself, then Machines P and Q would work for another 1.5 hours together. We can now calculate how many Gallons are processed in this situation...

Machine Q = (45 gallons/min)(90 minutes) = 4,050 gallons
Machines P+Q = (45+30 gallons/min)(90 minutes) = 6,750 gallons
Total = 4,050 + 6,750 = 10,800 gallons.
This IS the exact TOTAL we are supposed to end up with, so the total time MUST be 3 hours.

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A certain dairy packing plant has two machines, P and Q  [#permalink]

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Updated on: 30 Jul 2017, 14:50
DrFeelgood wrote:
A certain dairy packing plant has two machines, P and Q, that process milk at constant rates of 30 gallons per minute and 45 gallons per minute, respectively. A day's run of milk can be processed by machine P operating alone in 6 hours, by machine Q operating alone in 4 hours, or by both machines operating simultaneously in 2.4 hours. If a day's run of milk is processed using machine Q alone for half the time and both machines together for half the time, how many hours does it take to complete the run?

A. 1.5
B. 3.0
C. 3.75
D. 4.2
E. 5.0

let w=portion of work done by Q alone
1-w=portion of work done by P and Q
w/(1/4)=(1-w)/(1/4+1/6)
w=3/8 of run completed by Q alone
(3/8)/(1/4)=3/2 hours for Q alone
(3/2)*2=3 hours total to complete run
B

Originally posted by gracie on 29 Jul 2017, 22:38.
Last edited by gracie on 30 Jul 2017, 14:50, edited 1 time in total.
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Re: A certain dairy packing plant has two machines, P and Q  [#permalink]

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30 Jul 2017, 01:55
DrFeelgood wrote:
A certain dairy packing plant has two machines, P and Q, that process milk at constant rates of 30 gallons per minute and 45 gallons per minute, respectively. A day's run of milk can be processed by machine P operating alone in 6 hours, by machine Q operating alone in 4 hours, or by both machines operating simultaneously in 2.4 hours. If a day's run of milk is processed using machine Q alone for half the time and both machines together for half the time, how many hours does it take to complete the run?

A. 1.5
B. 3.0
C. 3.75
D. 4.2
E. 5.0

I stumbled upon this one yesterday and it kind of confused me, though I have pretty much understood the logic behind work problems. Ideas? Thanks in advance!

OA B

Done in 30 seconds: Q alone does the work in 4 hours, 2 machines together in 2,4 hours. Therefore, eliminate A, D and E. Taking into account that Q operated for half the time and both machines together for half the time - the answer should be somewhere close to the average of these times. C is very close to the time that needed for Q if operating alone. B is the answer.
Re: A certain dairy packing plant has two machines, P and Q   [#permalink] 30 Jul 2017, 01:55
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