A train has to travel the distance between Mumbai and Chennai, equal to 1000 km, at a constant speed. It travelled half of the way with the specified speed and stopped for 2 hours, to arrive at Chennai on time, it had to increase its speed by 40 km/h for the rest of the way. What is the ratio between the train’s speed in first half and that in second half?
Solution
Correct Answer: Option D
Let the regular speed of the train = x km/hr.
Now, time = distance/speed
∴ regular time taken by the train to travel the full distance = (1000/x) hrs
Case-1:
Distance travelled at regular speed = 1000/2 = 500km
∴ time taken to travel this distance = (500/x) hrs ---(1)
Waiting time = 1 hour
⇒ for the remaining 500 kms, the train increases its speed by 40 km/h (i.e. x + 40)
∴ time taken to travel last 500 km = 500/(x + 40) ---(2)
From the given data, we can clearly deduce that:
Regular time = (time taken to travel first 500 km) + (waiting time) + (time taken to travel last 500 km)
\(\begin{array}{l} \Rightarrow \frac{{1000}}{x} = \frac{{500}}{x} + 2 + \frac{{500}}{{x + 40}}\\ \Rightarrow \frac{{500}}{x} - \frac{{500}}{{x + 40}} = 2\\ \Rightarrow \frac{{40}}{{{x^2} + 40x}} = \frac{2}{{500}} \end{array}\)
⇒x2 + 40x – 10000 = 0
⇒ D = b2 – 4ac = 1600 - 4 × (-10000) = 41600
⇒ x = (-b ± √D)/2a
Omitting the negative value of speed,
x = 81.98 ≈ 82 km/h
∴ x + 40 = 121.98 ≈ 122 km/h
Required Ratio = 82 :122 = 41 : 61
Thus, required ratio is 41 : 61.
Practice More Questions on Our App!
Download our app for free and access thousands of MCQ questions with detailed solutions