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

Question

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 journey, what was its first average speed?

Accepted Answer

Step 1: Let first average speed of the train be $x$ km/hr.
$\frac{54}{x} + \frac{63}{x + 6} = 3$
Step 2: $\Rightarrow 54(x + 6) + 63x = 3x(x + 6)$
$\Rightarrow 54x + 324 + 63x = 3x^2 + 18x$
$\Rightarrow 3x^2 - 99x - 324 = 0$
$\Rightarrow x^2 - 33x - 108 = 0$
Step 3: $\Rightarrow (x - 36)(x + 3) = 0$
$x = 36$, $x = -3$ (rejected)
First average speed was $36$ km/hr.

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