An engineer works at a factory out os town. A car is sent for him from the factory every day and arrives at the railway station at the same time as the train. One day the engineer arrived at the station one our before his usual time and without waiting for the car, started walking towards factory. On his way he met the car and reached his factory `10` minutes before the usual time. For how much time (in minutes) did the engineer walk before the met the car ? (The car moves with the same speed everyday.)

Options

A. 105 min
B. 55 min
C. 75 min
D. 45 min

Explanation

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

### Step 1: Define the Variables
Let:
- \( T \) = usual time taken by the engineer to reach the factory from the station (in minutes).
- \( t \) = time (in minutes) the engineer walked before meeting the car.
- The car travels at a constant speed and takes the same time every day to reach the station.

### Step 2: Understand the Scenario
- The engineer usually arrives at the station at time \( T \).
- On the day he arrived early, he reached the station at \( T - 60 \) minutes.
- He started walking towards the factory and met the car at some point on the way.
- He reached the factory 10 minutes earlier than usual, which means he arrived at \( T - 10 \) minutes.

### Step 3: Set Up the Equations
1. The engineer walked for \( t \) minutes and met the car.
2. The time taken by the car to reach the point where they met (let's call this point P) is the same as the time taken by the engineer to reach the factory after meeting the car.

From the information given:
- The total time for the engineer to reach the factory after meeting the car is \( T - 10 \).
- The time taken by the car to reach the station is \( T \), and it would take \( T - t \) to return to the point P after meeting the engineer.

### Step 4: Equate the Times
At the point of meeting (point P):
- The engineer has walked for \( t \) minutes, so he has \( T - 60 + t \) minutes total until he meets the car.
- The car, which started at the station, has traveled for \( t \) minutes to reach point P.

After meeting, the engineer takes \( T - 10 - (T - 60 + t) \) minutes to reach the factory:
- This simplifies to \( 50 - t \) minutes.

### Step 5: Formulate the Equation
Since the car takes the same time to return to the station after meeting the engineer:
- The time taken by the car to return to the station after meeting the engineer is equal to the time taken by the engineer to reach the factory after meeting the car.
- Therefore, we have:
\[ t + (50 - t) = T - t \]
This simplifies to:
\[ 50 = T - t \]

### Step 6: Solve for \( t \)
From the equation \( T - t = 50 \), we can express \( t \) as:
\[ t = T - 50 \]

### Step 7: Substitute Values
Since the engineer arrives 10 minutes earlier than usual, we know:
\[ T - 10 = T - 50 + t \]
This gives us:
\[ t = 55 \text{ minutes} \]

### Final Answer
The engineer walked for **55 minutes** before meeting the car.

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