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

Question

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?

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!

Show Spoiler

OA B

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

Zarrolou · Accepted Answer

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?

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!

Show Spoiler

OA B

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

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

blueseas · Answer

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

DrFeelgood · Answer

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.

blueseas · Answer

Thats the great way to eliminate 3 options.

SVaidyaraman · Answer

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

nhanlevent · Answer

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.

TGC · Answer

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

EMPOWERgmatRichC · Answer

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.

Let's TEST Answer B....

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.

Final Answer:  B

GMAT assassins aren't born, they're made,
Rich

gracie · Answer

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

malevg · Answer

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.

shisingh · Answer

I had one doubt: Q operating alone in 4 hours will be 45*4*60 which is not equal to P alone = 30*6*60 but both should be ideally equal as day's run of milk is constant

Bunuel · Answer

45 * 4 * 60 = 30 * 6 * 60 = 10800

HarshavardhanR · Answer

Hey!

45 x 4 x 60 = 180 x 60
30 x 6 x 60 = 180 x 60.

shubhamagarwal44 · Answer

Usually we shouldn't add individual rates in Work-Rate Problems right? Why is that not the case here? Why are we adding 30+45 for total work done.

Bunuel · Answer

We add the rates because both machines are working on the same milk run at the same time. Their combined rate is 30 + 45 = 75 gallons per minute, which means together they process 75 gallons every minute.

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

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