Grade 12 Question

24 minutes

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     

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.

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