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# A fast train takes 3 hours less than a slow train travelling 600 km. If the speed of a slow train is 10 km/hr less than the speed of a fast train, find the speed of both the trains.  Verified
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Hint: In this particular type of question we need to express the speed of both the trains as x and x + 10 and use it to find the time taken by both of the trains to cover 600 km. Then we have to use the information given in the question to generate the quadratic equation and find the speed of both the trains.

Time taken by slow train to cover 600 km = $\dfrac{{600}}{x}$hrs (since time=$\dfrac{{{\text{distance}}}}{{{\text{speed}}}}$ )
Time taken by fast train to cover 600 km = $\dfrac{{600}}{{x + 10}}$hrs
$\Rightarrow \dfrac{{600}}{{x + 10}} + 3 = \dfrac{{600}}{x} \\ \Rightarrow \dfrac{{600 + 3x + 30}}{{x + 10}} = \dfrac{{600}}{x} \\ \Rightarrow 630x + 3{x^2} = 600x + 6000 \\ \Rightarrow {x^2} + 10x - 2000 = 0 \\ \Rightarrow {x^2} + 50x - 40x - 2000 = 0{\text{ }}\left( {{\text{splitting the middle term}}} \right) \\ \Rightarrow x\left( {x + 50} \right) - 40\left( {x + 50} \right) = 0{\text{ }} \\ \Rightarrow \left( {x - 40} \right)\left( {x + 50} \right) = 0 \\ \Rightarrow x = 40, - 50 \\$