A fast train takes 3 hours less than a slow train for a journey of 600 km. If the speed of the slow train is 10 km/hr less than that of the fast train, find the speeds of the two trains.

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Points to Remember:

  • This is a mathematical word problem involving speed, time, and distance.
  • The core concept is the relationship between speed, time, and distance (Distance = Speed × Time).
  • We need to set up and solve a system of equations to find the speeds of the two trains.

Introduction:

This question is a classic example of a problem solvable using simultaneous equations. The fundamental relationship governing the motion of objects is the equation: Distance = Speed × Time. We are given that the distance of the journey is 600 km, and the difference in travel time between a fast and a slow train is 3 hours. We will use this information, along with the relationship between the speeds of the two trains, to determine their individual speeds.

Body:

1. Defining Variables and Setting up Equations:

Let’s define:

  • x = speed of the fast train (in km/hr)
  • y = speed of the slow train (in km/hr)

We know that:

  • y = x – 10 (The speed of the slow train is 10 km/hr less than the fast train)
  • Time taken by fast train = 600/x hours
  • Time taken by slow train = 600/y hours

The problem states that the fast train takes 3 hours less than the slow train. Therefore:

600/x = 600/y – 3

2. Solving the Equations:

We now have a system of two equations with two variables:

  1. y = x – 10
  2. 600/x = 600/y – 3

We can substitute equation (1) into equation (2):

600/x = 600/(x – 10) – 3

Now, we solve for x:

600(x – 10) = 600x – 3x(x – 10)
600x – 6000 = 600x – 3x² + 30x
3x² – 30x – 6000 = 0
x² – 10x – 2000 = 0

This is a quadratic equation. We can solve it using the quadratic formula or factoring. Factoring gives us:

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(x – 50)(x + 40) = 0

This gives us two possible solutions for x: x = 50 or x = -40. Since speed cannot be negative, we discard x = -40.

Therefore, x = 50 km/hr (speed of the fast train).

Substituting this value back into equation (1):

y = x – 10 = 50 – 10 = 40 km/hr (speed of the slow train).

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3. Verification:

Let’s verify our solution:

  • Time taken by fast train = 600 km / 50 km/hr = 12 hours
  • Time taken by slow train = 600 km / 40 km/hr = 15 hours

The difference is 15 – 12 = 3 hours, which matches the problem statement.

Conclusion:

The speed of the fast train is 50 km/hr, and the speed of the slow train is 40 km/hr. This solution was obtained by formulating a system of equations based on the relationship between speed, time, and distance, and then solving these equations simultaneously. The solution was verified to ensure its accuracy. This problem highlights the practical application of mathematical concepts in everyday scenarios. Further analysis could involve exploring the impact of varying speeds on fuel efficiency or travel costs for the trains. A holistic approach to transportation planning should consider factors beyond just speed, including safety, environmental impact, and passenger comfort.

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