Train A takes 45 minutes more than train B to travel a distance of 450km. Due to engine trouble speed of train B falls by a quarter. So it takes 30 minutes more than Train A to complete the same journey. What is the speed of Train A (in km/hr)?

Options

A. A) 90
B. B) 120
C. C) 100
D. D) 110

Answer

To solve the problem step by step, let's break it down:

### Step 1: Define Variables
Let the speed of Train B be \( x \) km/hr. Then, the time taken by Train B to travel 450 km is given by:
\[ \text{Time taken by Train B} = \frac{450}{x} \text{ hours} \]
Since Train A takes 45 minutes more than Train B, the time taken by Train A is:
\[ \text{Time taken by Train A} = \frac{450}{x} + \frac{45}{60} = \frac{450}{x} + 0.75 \text{ hours} \]

### Step 2: Adjust for Train B's Speed Reduction
Due to engine trouble, the speed of Train B falls by a quarter. Therefore, the new speed of Train B is:
\[ \text{New speed of Train B} = x - \frac{x}{4} = \frac{3x}{4} \text{ km/hr} \]
The time taken by Train B with the reduced speed is:
\[ \text{New time taken by Train B} = \frac{450}{\frac{3x}{4}} = \frac{450 \times 4}{3x} = \frac{600}{x} \text{ hours} \]

### Step 3: Set Up the Equation for Time Difference
According to the problem, the new time taken by Train B is 30 minutes more than the time taken by Train A:
\[ \frac{600}{x} = \left(\frac{450}{x} + 0.75\right) + \frac{30}{60} \]
This simplifies to:
\[ \frac{600}{x} = \frac{450}{x} + 0.75 + 0.5 \]
\[ \frac{600}{x} = \frac{450}{x} + 1.25 \]

### Step 4: Solve for x
Rearranging the equation gives:
\[ \frac{600}{x} - \frac{450}{x} = 1.25 \]
\[ \frac{150}{x} = 1.25 \]
Multiplying both sides by \( x \):
\[ 150 = 1.25x \]
Dividing both sides by 1.25:
\[ x = \frac{150}{1.25} = 120 \text{ km/hr} \]

### Step 5: Calculate Speed of Train A
Now that we have the speed of Train B, we can find the speed of Train A. The time taken by Train B is:
\[ \text{Time taken by Train B} = \frac{450}{120} = 3.75 \text{ hours} \]
Thus, the time taken by Train A is:
\[ \text{Time taken by Train A} = 3.75 + 0.75 = 4.5 \text{ hours} \]
Finally, the speed of Train A is:
\[ \text{Speed of Train A} = \frac{450}{4.5} = 100 \text{ km/hr} \]

### Final Answer
The speed of Train A is **100 km/hr**.

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