Question

A faster train takes one hour less than a slower train for a journey of 200 km. If the speed of the slower train is 10 km/hr less than that of the faster train, find the speeds of the two trains.

Hint:

In competitive exams, check the factors of (Distance $\times$ Speed Difference). Here $200 \times 10 = 2000$. We need two numbers with a product of 2000 and a difference of 10. These are 50 and 40.

Answer 1

Step 1: Understanding the Concept:
This problem involves the relationship Time $= \frac{\text{Distance}}{\text{Speed}}$. We setup a quadratic equation based on the difference in travel times.
Step 2: Key Formula or Approach:
Let speed of faster train $= x$ km/hr.
Speed of slower train $= (x - 10)$ km/hr.
Time difference $= 1$ hour.
Step 3: Detailed Explanation:
Distance $= 200$ km.
Time taken by slower train $= \frac{200}{x - 10}$.
Time taken by faster train $= \frac{200}{x}$.
According to the problem:
\[ \frac{200}{x - 10} - \frac{200}{x} = 1 \]
Take the common denominator:
\[ 200 \left( \frac{x - (x - 10)}{x(x - 10)} \right) = 1 \]
\[ 200 \left( \frac{10}{x^2 - 10x} \right) = 1 \]
\[ 2000 = x^2 - 10x \implies x^2 - 10x - 2000 = 0 \]
Solve the quadratic equation by splitting the middle term ($50 \times 40 = 2000$):
\[ x^2 - 50x + 40x - 2000 = 0 \]
\[ x(x - 50) + 40(x - 50) = 0 \]
\[ (x - 50)(x + 40) = 0 \]
Since speed cannot be negative, $x = 50$.
Faster train speed $= 50$ km/hr.
Slower train speed $= 50 - 10 = 40$ km/hr.
Step 4: Final Answer:
The speeds of the trains are 50 km/hr and 40 km/hr.