A person travels a distance of 240 km, partly by train and the rest by bus.He takes 3 `(1)/(2)` hours if he travels 150 km by the train and the rest by bus.If he travels 140 km by bus and rest by train .He takes 3 `(2)/(3)` . What is the Speed of the tarin?

Options

A. A)72 km/h
B. B)75 km/h
C. C) 80 km/h
D. D) 70 km/h

Answer

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

### Step 1: Define Variables
Let the speed of the train be \( T \) km/h and the speed of the bus be \( B \) km/h.

### Step 2: Set Up the Equations
1. In the first scenario, the person travels 150 km by train and the remaining distance by bus (which is \( 240 - 150 = 90 \) km). The total time taken is \( 3 \frac{1}{2} \) hours, which is \( \frac{7}{2} \) hours.

The equation for this scenario is:
\[ \frac{150}{T} + \frac{90}{B} = \frac{7}{2} \]

2. In the second scenario, the person travels 140 km by bus and the remaining distance by train (which is \( 240 - 140 = 100 \) km). The total time taken is \( 3 \frac{2}{3} \) hours, which is \( \frac{11}{3} \) hours.

The equation for this scenario is:
\[ \frac{100}{T} + \frac{140}{B} = \frac{11}{3} \]

### Step 3: Solve the First Equation
From the first equation:
\[ \frac{150}{T} + \frac{90}{B} = \frac{7}{2} \]
Multiply through by \( 2TB \) to eliminate the denominators:
\[ 300B + 180T = 7TB \quad (1) \]

### Step 4: Solve the Second Equation
From the second equation:
\[ \frac{100}{T} + \frac{140}{B} = \frac{11}{3} \]
Multiply through by \( 3TB \) to eliminate the denominators:
\[ 300B + 420T = 11TB \quad (2) \]

### Step 5: Rearrange Both Equations
Rearranging equation (1):
\[ 7TB - 180T - 300B = 0 \quad (1) \]
Rearranging equation (2):
\[ 11TB - 420T - 300B = 0 \quad (2) \]

### Step 6: Eliminate One Variable
Now we can eliminate \( B \) by subtracting equation (1) from equation (2):
\[ (11TB - 420T - 300B) - (7TB - 180T - 300B) = 0 \]
This simplifies to:
\[ 4TB - 240T = 0 \]
Factoring out \( T \):
\[ T(4B - 240) = 0 \]
Since \( T \neq 0 \), we have:
\[ 4B - 240 = 0 \implies B = 60 \text{ km/h} \]

### Step 7: Substitute Back to Find \( T \)
Now substitute \( B = 60 \) into one of the original equations to find \( T \). Using equation (1):
\[ 300(60) + 180T = 7T(60) \]
This simplifies to:
\[ 180T = 420T - 18000 \]
Rearranging gives:
\[ 240T = 18000 \implies T = \frac{18000}{240} = 75 \text{ km/h} \]

### Final Answer
The speed of the train is \( \boxed{75} \) km/h.