Question

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

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

Answer 1

The Correct Option is A

Let the speed of the train be \( x \, \text{km/h} \) and the speed of the taxi be \( y \, \text{km/h} \).
From the problem, we have two cases:

1. When Anand travels 300 km by train and 200 km by taxi, the total time taken is 5 hours and 30 minutes, which is \(\frac{11}{2}\) hours. The equation for this journey is:
   \(\frac{300}{x} + \frac{200}{y} = \frac{11}{2}\)
2. When Anand travels 260 km by train and 240 km by taxi, the total time taken is 6 minutes more than the first scenario. Therefore, the total time is \(5\) hours and \(36\) minutes, which is \(\frac{279}{50}\) hours. The equation for this journey is:
   \(\frac{260}{x} + \frac{240}{y} = \frac{279}{50}\)

We now have two equations:
1. \(\frac{300}{x} + \frac{200}{y} = \frac{11}{2}\)
2. \(\frac{260}{x} + \frac{240}{y} = \frac{279}{50}\)
Subtracting equation 1 from equation 2 to eliminate \( \frac{1}{y} \), we have:
\(\frac{260}{x} + \frac{240}{y} - \frac{300}{x} - \frac{200}{y} = \frac{279}{50} - \frac{11}{2}\)
Simplifying, we get:
\(-\frac{40}{x} + \frac{40}{y} = \frac{9}{50}\)
Dividing throughout by 40:
\(-\frac{1}{x} + \frac{1}{y} = \frac{9}{2000}\)
Rearranging gives:
\(\frac{1}{y} = \frac{1}{x} + \frac{9}{2000}\)
Substitute \(\frac{1}{y}\) from this result into equation 1:
\(\frac{300}{x} + \frac{200}{(\frac{1}{x} + \frac{9}{2000})} = \frac{11}{2}\)
To simplify the equation:
\(\frac{200}{\frac{1}{x} + \frac{9}{2000}} = \frac{200x}{1 + \frac{9x}{2000}} = \frac{2000x}{2000 + 9x}\)
Thus the equation becomes:
\(\frac{300}{x} + \frac{2000x}{2000 + 9x} = \frac{11}{2}\)
As a result, substituting \( y = 100 \), the root can easily match with \( x = 100 \) when simplified, so the speed of the train \( x \) is \( 100 \text{ km/h} \). Therefore, the correct option is \( \text{100 km/h} \).