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: Understand the Relationship Between Speed, Distance and Time
We use the basic formula: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] Both trains travel the same distance (200 km), but their speeds are different. Because the slower train takes 1 hour more, we form an equation using the difference in their travel times.

Step 2: Assume the Variables Clearly
Let the speed of the faster train be \( x \) km/hr.
Then the speed of the slower train will be \( x - 10 \) km/hr.

Distance \( = 200 \) km.

Time taken by faster train: \[ \frac{200}{x} \] Time taken by slower train: \[ \frac{200}{x - 10} \] According to the question: \[ \frac{200}{x - 10} - \frac{200}{x} = 1 \]
Step 3: Solve the Equation Step by Step
Take common denominator: \[ \frac{200x - 200(x - 10)}{x(x - 10)} = 1 \] Simplify numerator: \[ 200x - 200x + 2000 \] \[ \frac{2000}{x(x - 10)} = 1 \] Multiply both sides by \( x(x - 10) \): \[ 2000 = x(x - 10) \] \[ 2000 = x^2 - 10x \] Bring all terms to one side: \[ x^2 - 10x - 2000 = 0 \]
Step 4: Solve the Quadratic Equation
We factorise. We need two numbers: • Product = −2000 • Sum = −10 These numbers are −50 and 40. \[ x^2 - 50x + 40x - 2000 = 0 \] \[ x(x - 50) + 40(x - 50) = 0 \] \[ (x - 50)(x + 40) = 0 \]
Step 5: Find the Valid Solution
\[ x - 50 = 0 \Rightarrow x = 50 \] \[ x + 40 = 0 \Rightarrow x = -40 \] Speed cannot be negative, so we reject −40. \[ x = 50 \]
Step 6: Find Both Speeds
Faster train speed: \[ 50 \text{ km/hr} \] Slower train speed: \[ 50 - 10 = 40 \text{ km/hr} \]
Step 7: Final Answer
\[ \boxed{\text{Faster train speed } = 50 \text{ km/hr}} \] \[ \boxed{\text{Slower train speed } = 40 \text{ km/hr}} \]
Verification (Very Important for Exams)
Time taken by faster train: \[ \frac{200}{50} = 4 \text{ hours} \] Time taken by slower train: \[ \frac{200}{40} = 5 \text{ hours} \] Difference: \[ 5 - 4 = 1 \text{ hour} \quad ✔ \] Hence, the solution is correct.