One tap fills a pool. The other one empties it. It's a word problem.In a pool there are two taps, one for filling and one for emptying. Once, when the pool was empty they opened the filling tap for 4 hours. Afterwards, they opened by mistake the emptying tap and after 2 hours the pool was filled 80% from its volume. When the pool was completely filled it turned out that the filling tap filled like water amount of 1 1 2 pools and the emptying tap emptied water amount of 1 2 pool.So what I did is power / rate time (hours) Total Filling tap until 80% 1 x 6 1 x × 6 = 6 x Emptying tap until 80% 1 y 2 1 y × 2 = 2 y Filling tap until the end 1 x 1.5 x 1.5 Emptying tap until the end 1 y 0.5 y 0.5 I can write that: 6 x − 2 y = 0.8And here I'm stuck. I don't know how to use the information of the last two rows of my chart. If I would know the time it took to fill 100% the pool then I could sum up the two last rows from the middle column. Or the time it took to fill up the remaining 20%.Sorry if my translation of the question wasn't so good.

Question

One tap fills a pool. The other one empties it. It&#039;s a word problem.In a pool there are two taps, one for filling and one for emptying. Once, when the pool was empty they opened the filling tap for 4 hours. Afterwards, they opened by mistake the emptying tap and after 2 hours the pool was filled 80% from its volume. When the pool was completely filled it turned out that the filling tap filled like water amount of   1      1    2   pools and the emptying tap emptied water amount of       1    2   pool.So what I did is                                      power / rate                          time (hours)                          Total                                      Filling tap until 80%                                      1            x                                    6                                      1            x                    ×          6          =                      6            x                                                Emptying tap until 80%                                      1            y                                    2                                      1            y                    ×          2          =                      2            y                                                Filling tap until the end                                      1            x                                                1.5            x                                    1.5                                      Emptying tap until the end                                      1            y                                                0.5            y                                    0.5                    I can write that:      6    x    −      2    y    =  0.8And here I&#039;m stuck. I don&#039;t know how to use the information of the last two rows of my chart. If I would know the time it took to fill 100% the pool then I could sum up the two last rows from the middle column. Or the time it took to fill up the remaining 20%.Sorry if my translation of the question wasn&#039;t so good.

Rebekah Zimmerman · Accepted Answer

Let&#039;s say the time between when the pool was 80% filled and when the pool was completely filled is   z hours. The filling tap had been running for   6  +  z hours and the emptying tap had been running for   2  +  z hours. Thus, we have the two following equations:            6      +      z        x    =  1.5  →  z  =  1.5  x  −  6            2      +      z        y    =  0.5  →  z  =  0.5  y  −  2Set both equations equal to each other using the Transitive Property of Equality:  1.5  x  −  6  =  0.5  y  −  2  →  1.5  x  =  0.5  y  +  4  →  3  x  =  y  +  8  →  x  =            y      +      8        3  Substitute this into your first equation:      6                  y        +        8            3        −      2    y    =  0.8  →      18          y      +      8        −      2    y    =  0.8Multiply everything by   5  y  (  y  +  8  )            90      y      (      y      +      8      )              y      +      8        −            10      y      (      y      +      8      )        y    =  4  y  (  y  +  8  )  →  90  y  −  10  (  y  +  8  )  =  4      y    2    +  32  y  →  80  y  −  80  =  4      y    2    +  32  yPut everything on one side of the equation and then divide by   4:  4      y    2    −  48  y  +  80  =  0  →      y    2    −  12  y  +  20  =  0This quadratic can be factored into   (  y  −  10  )  (  y  −  2  )  (  y  −  10  )  (  y  −  2  )  =  0  →  y  −  10  =  0     OR     y  −  2  =  0  →  y  =  10     OR     y  =  2Thus,   y  =  10 or   y  =  2. However, if   y  =  2, then, when we plug back into our   z  =  0.5  y  −  2 formula, we get   z  =  0, which doesn&#039;t make any sense since   z is the amount of time between the pool being 80% filled and the pool being completely filled. Therefore,   y  =  10Finally, substitute this into   x  =            y      +      8        3    x  =            10      +      8        3    =      18    3    =  6Therefore, the filling tap takes 6 hours to fill a pool while the emptying tap takes 10 hours to fill a pool.(Also, if you&#039;re curious, you can plug   y  =  10 back into the   z  =  0.5  y  −  2 formula or plug   x  =  6 back into the   x  =  1.5  x  −  6 formula to find that there were 3 hours between the pool being 80% filled and the pool being completely filled, meaning this whole process took   4  +  2  +  3  =  9 hours overall.)

One tap fills a pool. The other one empties it. It's a word problem. In a pool there are two taps,

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