Question 299160
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a plane flew 500 km from Atlanta to Louisville. when returning to Atlanta the flight took 1/2 hour less time. 
if the rate to Louisville was 50km/hr faster than the rate returning, find the two rates in km/hr.
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After reading this problem and its solution in the post by @mananth, I have two notices.


Firstly, the problem's formulation in the post is absurdist and self-contradictory.
Indeed, it says that the flight to Atlanta took 1/2 hour less time
and the rate to Louisville was 50 km/h faster than the rate returning.


So, the problem's creator was so hurry that missed circumstances.


I will edit the problem as it is shown below, to make sense from nonsense:


<pre>
      A plane flew 500 km from Atlanta to Louisville. When returning to Atlanta the flight took 1/2 hour {{{highlight(highlight(more))}}} time. 
      If the rate to Louisville was 50km/h faster than the rate returning, find the two rates in km/hr.
</pre>

Secondly, in the post by @mananth, both the solution and the answer are incorrect due to arithmetic errors.


      I came to bring a correct solution to this my edited formulation.



<pre>
Let the rate to Louisville be x km/h.

Time of this flight is  {{{500/x}}}  hours.


The rate to Atlanta (returning) will be (x-50) km/h.

The time of this returning flight is  {{{500/(x-50)}}} hours.


Time equation is

    {{{500/(x-50)}}} - {{{500/x}}} = {{{1/2}}}  of an hour.   <<<---===  It says literally what the problem says


Simplify and find x

    2*500*x - 2*500*(x-50) = x*(x-50),

    1000x - 1000x + 50000 = x^2 - 50x,

    x^2 - 50x + 50000 = 0,

    (x-250)*(x+200) = 0,

    x = 250  or  x = -200.


We choose the positive root and reject the negative one.


<U>ANSWER</U>.  The rate from Atlanta to Louisville was 250 km/h.  The rate of the returning trip from Louisville to Atlanta was 250-50 = 200 km/h.


<U>CHECK</U>.  The tome flying from Atlanta to Louisville was  {{{500/250}}} = 2 hours.

        The time flying from Louisville to Atlanta (returning) was  {{{500/(250-50)}}} = 2.5 hours, in 0.5 hours longer.

        Precisely correct.
</pre>

The problem edited and converted from self-contradictory to self-consistent 
and then solved in a right way.