Train A takes 1 hour more than train B to travel a distance of 720 km. Due to engine trouble speed of train B falls by a third, so it takes 3 hours more than Train A to complete the same journey ? What is the speed of Train A (in km/hr) ?

Options

A. 80
B. 90
C. 60
D. 70

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information given.

### 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 1 hour more than Train B to travel 720 km. The time taken by Train A can be expressed as:
\[ \text{Time taken by Train A} = \frac{720}{A} \]
The time taken by Train B is:
\[ \text{Time taken by Train B} = \frac{720}{B} \]
From the information given, we can set up the first equation:
\[ \frac{720}{A} = \frac{720}{B} + 1 \tag{1} \]

### Step 3: Set Up the Second Equation
Due to engine trouble, the speed of Train B falls by a third, so the new speed of Train B becomes:
\[ \text{New speed of Train B} = \frac{2}{3}B \]
The time taken by Train B at this reduced speed is:
\[ \text{Time taken by Train B (new)} = \frac{720}{\frac{2}{3}B} = \frac{720 \cdot 3}{2B} = \frac{1080}{B} \]
According to the problem, this new time taken by Train B is 3 hours more than the time taken by Train A:
\[ \frac{1080}{B} = \frac{720}{A} + 3 \tag{2} \]

### Step 4: Solve the First Equation
From equation (1):
\[ \frac{720}{A} - \frac{720}{B} = 1 \]
Multiplying through by \( AB \):
\[ 720B - 720A = AB \]
Rearranging gives:
\[ AB = 720(B - A) \tag{3} \]

### Step 5: Solve the Second Equation
From equation (2):
\[ \frac{1080}{B} - \frac{720}{A} = 3 \]
Multiplying through by \( AB \):
\[ 1080A - 720B = 3AB \]
Rearranging gives:
\[ 3AB - 1080A + 720B = 0 \tag{4} \]

### Step 6: Substitute Equation (3) into Equation (4)
Substituting \( AB = 720(B - A) \) into equation (4):
\[ 3(720(B - A)) - 1080A + 720B = 0 \]
This simplifies to:
\[ 2160B - 2160A - 1080A + 720B = 0 \]
Combining like terms:
\[ 2880B - 3240A = 0 \]
Dividing through by 60:
\[ 48B = 54A \]
Thus:
\[ B = \frac{54}{48}A = \frac{9}{8}A \tag{5} \]

### Step 7: Substitute Back to Find Speeds
Substituting equation (5) back into equation (1):
\[ \frac{720}{A} = \frac{720}{\frac{9}{8}A} + 1 \]
This simplifies to:
\[ \frac{720}{A} = \frac{720 \cdot 8}{9A} + 1 \]
Cross-multiplying gives:
\[ 720 \cdot 9 = 720 \cdot 8 + 9A \]
Thus:
\[ 6480 = 5760 + 9A \]
Solving for \( A \):
\[ 9A = 6480 - 5760 = 720 \]
\[ A = \frac{720}{9} = 80 \text{ km/hr} \]

### Final Answer
The speed of Train A is \( 80 \) km/hr.