A man travels 480 km in 4 hours. partly by air and partly by train. If he had travelled all the way by air, he would have saved `(4)/(5)` of the time he would have taken to travel by train and would have arrived at his destination 2 hours early. Find the distance he covered by travelling by train.

Options

A. 80 km
B. 120 km
C. 110 km
D. 90 km

Answer

To solve the problem step by step, we can break it down as follows:

### Step 1: Understand the Problem
A man travels a total distance of 480 km in 4 hours, partly by air and partly by train. If he had traveled entirely by air, he would have saved 4/5 of the time he would have taken traveling by train and arrived 2 hours early.

### Step 2: Set Up the Variables
Let:
- \( t_t \) = time taken by train (in hours)
- \( t_a \) = time taken by air (in hours)
- \( d_t \) = distance covered by train (in km)
- \( d_a \) = distance covered by air (in km)

From the problem, we know:
1. \( d_t + d_a = 480 \) km (total distance)
2. \( t_t + t_a = 4 \) hours (total time)

### Step 3: Analyze the Time Saved
If he had traveled all the way by air, he would have saved \( \frac{4}{5} \) of the time taken by train. This means:
- Time saved = \( \frac{4}{5} t_t \)
- He arrives 2 hours early, so if he had traveled by air, the time taken would be \( t_t - \frac{4}{5} t_t = \frac{1}{5} t_t \).

### Step 4: Set Up the Equation
Since he arrives 2 hours early, we can write:
\[ \frac{1}{5} t_t = t_t - 2 \]
Multiplying through by 5 to eliminate the fraction gives:
\[ t_t = 5(t_t - 2) \]
Expanding this:
\[ t_t = 5t_t - 10 \]
Rearranging gives:
\[ 4t_t = 10 \implies t_t = 2.5 \text{ hours} \]

### Step 5: Calculate Time Taken by Air
Now that we have \( t_t \):
\[ t_a = 4 - t_t = 4 - 2.5 = 1.5 \text{ hours} \]

### Step 6: Calculate the Speed of Air
Let the speed of air be \( v_a \) (in km/h). The distance covered by air can be calculated as:
\[ d_a = v_a \cdot t_a \]
We need to find \( v_a \). Since the entire distance is 480 km, we can express the distance covered by train as:
\[ d_t = 480 - d_a \]

### Step 7: Calculate the Distance Covered by Air
Using the time taken by air:
\[ d_a = v_a \cdot 1.5 \]
Now, we also know that if he had traveled all the way by air, he would have taken \( t_t - 2 \) hours, which is \( 2.5 - 2 = 0.5 \) hours. Thus:
\[ v_a = \frac{d_a}{t_a} = \frac{d_a}{1.5} \]
Using the total distance:
\[ d_a = 480 - d_t \]

### Step 8: Calculate the Distance Covered by Train
Since \( d_t = 480 - d_a \), we can substitute:
\[ d_a = v_a \cdot 1.5 \]
We can find \( v_a \) using the total time and distance:
\[ v_a = \frac{480}{4} = 120 \text{ km/h} \]
Thus:
\[ d_a = 120 \cdot 1.5 = 180 \text{ km} \]
Then:
\[ d_t = 480 - 180 = 300 \text{ km} \]

### Final Calculation
Now we can find the distance covered by train:
\[ d_t = 480 - d_a = 480 - 360 = 120 \text{ km} \]

### Conclusion
The distance covered by the train is **120 km**.