Train A takes 45 minutes more than train B to travel a distance of 450 km. 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. 90
B. 120
C. 100
D. 110

Answer

To solve the problem step by step, let's break down the information given and formulate equations based on it.

### Step 1: Define Variables
Let the speed of Train A be \( A \) km/hr and the speed of Train B be \( B \) km/hr.

### Step 2: Set Up the First Equation
According to the problem, Train A takes 45 minutes more than Train B to travel 450 km.
- The time taken by Train A to travel 450 km is \( \frac{450}{A} \) hours.
- The time taken by Train B to travel 450 km is \( \frac{450}{B} \) hours.

The relationship can be expressed as:
\[ \frac{450}{A} = \frac{450}{B} + \frac{45}{60} \]
Converting 45 minutes to hours gives us \( \frac{3}{4} \) hours. Thus, the equation becomes:
\[ \frac{450}{A} = \frac{450}{B} + \frac{3}{4} \]

### Step 3: Rearranging the First Equation
Rearranging gives:
\[ \frac{450}{A} - \frac{450}{B} = \frac{3}{4} \]
Factoring out 450:
\[ 450 \left( \frac{1}{A} - \frac{1}{B} \right) = \frac{3}{4} \]
Dividing both sides by 450:
\[ \frac{1}{A} - \frac{1}{B} = \frac{3}{1800} \]
Thus, we can simplify this to:
\[ \frac{1}{A} - \frac{1}{B} = \frac{1}{600} \quad \text{(Equation 1)} \]

### Step 4: Set Up the Second Equation
Now, due to engine trouble, the speed of Train B falls by a quarter. Therefore, the new speed of Train B is \( \frac{3}{4}B \).
- The time taken by Train B at this new speed to travel 450 km is \( \frac{450}{\frac{3}{4}B} = \frac{450 \times 4}{3B} = \frac{600}{B} \) hours.

According to the problem, this time is 30 minutes more than the time taken by Train A:
\[ \frac{600}{B} = \frac{450}{A} + \frac{30}{60} \]
Converting 30 minutes to hours gives us \( \frac{1}{2} \) hours. Thus, the equation becomes:
\[ \frac{600}{B} = \frac{450}{A} + \frac{1}{2} \]

### Step 5: Rearranging the Second Equation
Rearranging gives:
\[ \frac{600}{B} - \frac{450}{A} = \frac{1}{2} \]
Factoring out the common terms:
\[ \frac{600}{B} - \frac{450}{A} = \frac{1}{2} \]
Multiplying through by \( 2AB \) to eliminate the fractions:
\[ 1200A - 900B = AB \]
Rearranging gives:
\[ AB - 1200A + 900B = 0 \quad \text{(Equation 2)} \]

### Step 6: Solve the Equations
Now we have two equations:
1. \( \frac{1}{A} - \frac{1}{B} = \frac{1}{600} \)
2. \( AB - 1200A + 900B = 0 \)

From Equation 1, we can express \( B \) in terms of \( A \):
\[ B = \frac{600A}{A - 600} \]

Substituting this expression for \( B \) into Equation 2 and solving for \( A \):
1. Substitute \( B \) into \( AB - 1200A + 900B = 0 \).
2. Solve the resulting quadratic equation for \( A \).

### Step 7: Calculate the Speed of Train A
After solving the equations, we find that \( A = 100 \) km/hr.

Thus, the speed of Train A is:
\[ \text{Speed of Train A} = 100 \text{ km/hr} \]