A train travels a distance of $90$ km at a constant speed. Had the speed been $15$ km/h more, it would have taken $30$ minutes less for the journey. Find the original speed of the train.

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Step 1: Let the original speed be $x$ km/h. New speed $= (x + 15)$ km/h.
Step 2: $\frac{90}{x} - \frac{90}{x + 15} = \frac{1}{2}$
Step 3: $\Rightarrow x^2 + 15x - 2700 = 0$
Step 4: $\Rightarrow (x + 60)(x - 45) = 0 \Rightarrow x = 45$ (rejecting $x = -60$)
Step 5: The original speed of the train is $45$ km/h.

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