Question

Anand travelled 300 km by train and 200 km by taxi. It took him 5 hours and 30 minutes. However, if he travels 260 km by train and 240 km by taxi, he takes 6 minutes more. The speed of the train is:

• A. 100 km/h
• B. 120 km/h
• C. 80 km/h
• D. 110 km/h

Hint:

Form two equations in 1/speed for train and taxi from the two travel scenarios, then eliminate one variable.

Answer 1

The Correct Option is A

Step 1: Set up equations using time = distance / speed.
Let the speed of the train be \(x\) km/h and the speed of the taxi be \(y\) km/h. In the first case, total time = 5 h 30 min = \(5.5\) h: \[ \frac{300}{x} + \frac{200}{y} = 5.5 \] In the second case, total time is 6 minutes more, that is 5 h 36 min = \(5.6\) h: \[ \frac{260}{x} + \frac{240}{y} = 5.6 \]

Step 2: Eliminate \(y\) between the two equations.
Multiply the first equation by \(1.2\) so the taxi term matches the second equation's coefficient: \[ \frac{360}{x} + \frac{240}{y} = 6.6 \] Subtracting the second equation from this: \[ \frac{360}{x} - \frac{260}{x} = 6.6 - 5.6 \implies \frac{100}{x} = 1 \implies x = 100 \]

Step 3: Verify with the taxi speed.
From the first equation: \(\frac{300}{100} + \frac{200}{y} = 5.5 \implies 3 + \frac{200}{y} = 5.5 \implies y = 80\). Checking the second case: \(\frac{260}{100} + \frac{240}{80} = 2.6 + 3 = 5.6\), which matches.

Final Answer:
The speed of the train is 100 km/h. \[ \boxed{x = 100 \text{ km/h}} \]