Question 1118151
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<pre>
Let x be the average speed of the slower train, in kilometers per hour.

Then the speed of the faster train is (x+25) km/h.


The slower train takes  {{{125/x}}} hours    to travel 125 kilometers.

The faster train takes  {{{150/(x+25)}}} hours  to travel 150 kilometers.


The condition says that the faster train takes 2 hours less:


{{{125/x}}} - {{{150/(x+25)}}} = 2   hours.     


It is yours "time" equation.     // As soon as I got this equation, I just know the answer: x = 25.  (I solved the equation in my head)
                                 // But I will pretend that I don't know the solution, and will solve it formally (i.e. "honestly")

To solve it, multiply both sides by x*(x+25). You will get


125*(x+25) - 150*x = 2x*(x+25),

125x + 125*25 - 150x = 2x^2 + 50x,

2x^2 + 75x - 125*25 = 0,

2x^2 - 50x + 125x - 125*25 = 0,   <<<---=== starting from this line, I work to factor left side

2x*(x-25) + 125(x-25) = 0,

(2x+125)*(x-25) = 0.


The only positive root is  x= 25,  and it is the only solution to the problem.


<U>Answer</U>.  The slower train average speed is 25 km/h.  The faster train 25+25 = 50 km/h.


<U>Check</U>.   {{{125/25}}} - {{{150/50}}} = 5 - 3 = 2.  ! The time equation is satisfied !
</pre>

Solved and checked.