Ravi, who lives in the contryside, caught a train for home earlier than usual yesterday. His wife normally drives to the station to meet him. But yesterday he set out on foot from the station to meet his wife on the way. He reached home 12 minutes earlier then he would have done had he waited at the station for his wife. The car travels at a uniform speed, which is 5 times Ravi's speed on foot. Ravi reached home at exactly 6 O'clock. At what time would he have reached home if his wife, forewarned of his plan, had met him at the station?

Options

A. `5:48`
B. `5:24`
C. `5:00`
D. `5:36`

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Define Variables
Let Ravi's speed on foot be \( x \) km/min. Therefore, his wife's speed in the car is \( 5x \) km/min.

### Step 2: Determine the Time Saved
Ravi reached home 12 minutes earlier than he would have if he waited for his wife at the station. This means that the time it took for Ravi to walk home and the time it would have taken for his wife to drive to the station and then home is 12 minutes longer than the time it took for Ravi to walk home.

### Step 3: Calculate the Distance
Let the distance from the station to their home be \( d \) km. The time taken by Ravi to walk home is \( \frac{d}{x} \) minutes. The time taken by his wife to reach the station and then home is \( \frac{d}{5x} + \frac{d}{5x} = \frac{2d}{5x} \) minutes.

### Step 4: Set Up the Equation
According to the information given:
\[ \frac{d}{x} + 12 = \frac{2d}{5x} \]

### Step 5: Solve the Equation
Multiply through by \( 5x \) to eliminate the denominators:
\[ 5d + 60x = 2d \]
Rearranging gives:
\[ 5d - 2d = -60x \]
\[ 3d = -60x \]
\[ d = 20x \]

### Step 6: Determine the Time Taken by Ravi's Wife
Now, we can find out how long it would take for Ravi's wife to reach home if she had known about his plan. The time taken by his wife to reach home from the station would be:
\[ \text{Time} = \frac{d}{5x} = \frac{20x}{5x} = 4 \text{ minutes} \]

### Step 7: Calculate Total Time if She Met Him at the Station
If Ravi had waited for his wife at the station, she would have taken 4 minutes to reach him and then they would have taken the same time to reach home, which is 4 minutes. Therefore, the total time taken would be:
\[ 4 \text{ minutes (to reach Ravi)} + 4 \text{ minutes (to reach home)} = 8 \text{ minutes} \]

### Step 8: Calculate the Time Ravi Would Have Reached Home
Since Ravi reached home at 6:00 PM, if he had waited for his wife, he would have arrived home at:
\[ 6:00 PM + 8 \text{ minutes} = 6:08 PM \]

### Final Answer
If Ravi's wife had met him at the station, he would have reached home at **6:08 PM**.

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