Question

Trains A and B start traveling at the same time towards each other with constant speeds from stations X and Y, respectively. Train A reaches station Y in 10 minutes while train B takes 9 minutes to reach station X after meeting train A. Then the total time taken, in minutes, by train B to travel from station Y to station X is

• A. 15
• B. 12
• C. 6
• D. 10

Answer 1

The Correct Option is A

1. Given:

• Train A reaches station Y in 10 minutes after meeting Train B
• Train B takes 9 minutes to reach station X after meeting Train A
• Let the meeting point be M

2. Let t be the time (in minutes) each train took to reach point M

From the diagram:

• Train A: X → M in \( t \) minutes, then M → Y in \( 10 - t \) minutes
• Train B: Y → M in \( t \) minutes, then M → X in 9 minutes

 

Since the distances XM and MY are same for both trains (they meet at the same point), the ratio of times should be inverse of speeds:

\[ \frac{t}{9} = \frac{10 - t}{t} \]

3. Solving the Equation

Cross-multiplying:

\[ t^2 = 9(10 - t) \] \[ t^2 = 90 - 9t \Rightarrow t^2 + 9t - 90 = 0 \]

Solving the quadratic:

\[ (t + 15)(t - 6) = 0 \Rightarrow t = -15 \text{ or } t = 6 \]

Since time cannot be negative, we take:

\[ t = 6 \]

4. Final Answer

Train B took:

\[ \text{From Y to M: } t = 6 \text{ minutes} \] \[ \text{From M to X: } 9 \text{ minutes} \] \[ \Rightarrow \text{Total time from Y to X = } 6 + 9 = \boxed{15} \text{ minutes} \]

Correct Answer: 15 minutes

Answer 2

1. Let the Speeds Be:

Let the speed of Train A be \( a \), and the speed of Train B be \( b \). \ Let \( D \) be the distance from station X to Y, and the trains meet at point \( x \) from station X.

Time taken by Train A to reach point \( x \) from X: \[ \frac{x}{a} \] 
Time taken by Train B to reach point \( x \) from Y: \[ \frac{D - x}{b} \]

Since both trains meet at the same time,

\[ \frac{x}{a} = \frac{D - x}{b} \quad \text{(1)} \]

2. Given Information

• Train A takes 10 minutes to reach station Y: \( \frac{D}{a} = 10 \)
• Train B takes 9 minutes to reach station X from meeting point: \( \frac{x}{b} = 9 \)

3. Substituting Into Equation

From \( \frac{D}{a} = 10 \Rightarrow a = \frac{D}{10} \)

From \( \frac{x}{b} = 9 \Rightarrow b = \frac{x}{9} \)

Substitute both into equation (1):

\[ \frac{x}{\frac{D}{10}} = \frac{D - x}{\frac{x}{9}} \Rightarrow \frac{10x}{D} = \frac{9(D - x)}{x} \]

Cross-multiplying:

\[ 10x^2 = 9D^2 - 9Dx \Rightarrow 10x^2 + 9Dx - 9D^2 = 0 \]

Solving the quadratic gives:

\[ x = \frac{3D}{5} \]

4. Time Taken by Train B

We already know:

\[ \frac{x}{b} = 9 \Rightarrow \frac{\frac{3D}{5}}{b} = 9 \Rightarrow \frac{3D}{5b} = 9 \Rightarrow \frac{D}{b} = 15 \]

So the total time taken by Train B to travel from station Y to X is: \[ \boxed{15 \text{ minutes}} \]

5. Final Answer

Correct Option: (A) 15 minutes