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Four large and 2 small pumps can fill a swimming pool in 2 hours. Two large and 6 small pumps can also fill the same swimming pool in 2 hours. How long does it take 8 large and 8 small pumps to fill 50% of the swimming pool. (NOTE: all the large pump have same power and all the small pumps have the same power).

 Jan 28, 2019
 #1
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0

24 minutes

 Jan 28, 2019
 #2
avatar+18754 
+2

x=small pump rate    y = large pump rate

4y + 2x  = Swimming pool (in 2 hours)    = 1 pool/2hr

 

AND:

 

2 y + 6x  = swimming pool  (in 2 hours) = 1 pool/2hr

 

From the first equation    2x = 1/2 - 4y

                                          x = 1/4-2y     substitute this into the second equation

 

2y + 6 ((1/4-2y) = 1/2   Solve for y

2y + 6/4 -12y = 1/2

-10y = -1

     y = 1/10 pool/ hr       Sub this into 1st(or 2nd) equation:

                             4(1/10) + 2x = 1/2          solve for x = 1/20 pool/hr

 

Now    :    for the final part of the question.....

                 ( 8(y) + 8(x)) t = 1/2 pool

                 ( 8(1/10) + 8(1/20) ) t = 1/2

                    (16/20 + 8/20) t = 1/2

                      (    24/20 ) t = 1/2

                         t = 1/2 * 20/24 = 20/48 hr    =    20/48*60 = 25 mins     

 Jan 28, 2019
 #3
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+2

4L + 2S =2L + 6S, solve for L
L = 2S - This means the capacity of 1 large pump = 2 small pumps.
4L + 2S = Equivalent to 5 Large pumps in capacity.
8 Large pumps + 8 Small pumps = Equivalent in Capacity to 8+4 = 12 Large pumps.
12 large pumps / 5 Large pumps = 2 2/5 times - filling capacity of 8 Large and 8 small pumps.
Therefore it takes: 120 minutes / (12/5) =50 - minutes to fill the pool.
50/2 =25 minutes to fill 50% of the pool.

 Jan 28, 2019
edited by Guest  Jan 28, 2019
 #4
avatar+102372 
+1

Let (1/x) be the amount of the pool filled  by a large pump in 1 minute

Let (1/y) be the amount of the pool filled by a small pump in one minute

 

So we have this system

 

4(1/x)(120) + 2(1/y)(120) =  1   ⇒  480/x + 240/ y = 1     (1)

 

2(1/x)(120) + 6(1/y)(120) = 1   ⇒   240/ x + 720/y = 1    ⇒  -480/x - 1440/ y = -2   (2)

 

Add (1) and (2)

 

-1200/ y = - 1    ⇒  y = 1200

 

And   

480/x + 240/1200 = 1

480/x + 1/5 = 1

480/x = 4/5

x/480 = 5/4

x = 480 (5/4) = 600

 

 

So.....the large pump fills 1/600 of the pool in a minute and the small pump fills 1/1200 pool every minute

So....in each minute they filll  1/400 of the pool working together

So.....8 times this = 8/400 of the pool  = 1/50 of the pool

 

So.....  (1/50) * minutes = 1/2

 

Minutes = 50/2 =   25 minutes

 

 

cool cool  cool

 Jan 29, 2019

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