Discussion
It takes eight hours for a 600 km journey, if 120 km is done by train and the rest by car. It takes 20 minutes more, if 200 km is done by train and the rest by car. The ratio of the speed of the train to that of the cars is:
Answer: C
Write \( x = \tfrac{1}{v_{\text{train}}} \) and \( y = \tfrac{1}{v_{\text{car}}} \) to make both conditions linear.
\[ 120x + 480y = 8, \qquad 200x + 400y = 8\tfrac{1}{3} = \frac{25}{3} \]
Dividing the first by 8 and the second by \( \tfrac{25}{3} \):
\[ 15x + 60y = 1, \qquad 24x + 48y = 1 \]
Equating and clearing:
\[ 24(1 - 60y) = 15(1 - 48y) \quad\Longrightarrow\quad 9 = 720y \quad\Longrightarrow\quad y = \frac{1}{80} \]
\[ x = \frac{1 - 60/80}{15} = \frac{1}{60} \]
So the train does 60 kmph and the car 80 kmph:
\[ 60 : 80 = 3 : 4 \]
The reciprocal substitution is what turns two awkward equations into a pair of linear ones.
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