question_answer

A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?

Answer:

Let the average speed of the train be \[x\text{ }km/hr\].

Then, new average speed of the tram \[=(x+6)\text{ }km/h\]

Time taken by train to cover \[54\text{ }km=\frac{54}{x}hrs\].

And time taken by train to cover \[63\text{ }km=\frac{63}{(x+6)}hrs.\]

According to the question,

\[\frac{54}{x}+\frac{63}{x+6}=3\]

\[\frac{54(x+6)+63x}{x(x+6)}=3\]

\[54x+324+63x=3x(x+6)\]

\[324+117x=3{{x}^{2}}+18x\]

\[3{{x}^{2}}-99x-324=0\]

\[{{x}^{2}}-33x-108=0\]

\[{{x}^{2}}-36x+3x-108=0\]

\[x(x-36)+3(x-36)=0\]

\[(x+3)(x-36)=0\]

\[x=-3\]or \[36\]

Since, speed cannot be negative

\[\therefore x=36\]

So, First speed of train\[=36\text{ }km/hr\]