It takes one minute for a passenger standing on an escalator to reach the top. If the escalator does not move it takes him `3` minute to walk up. How long will it take for the passenger to arrive at the top if he walks up the moving escalator ?

Options

A. 30 sec
B. 45 sec
C. 40 sec
D. 35 sec

Explanation

To solve the problem, we need to determine how long it will take for a passenger to reach the top of the escalator when he walks up the moving escalator. We can break down the solution into the following steps:

### Step 1: Define Variables
- Let the length of the escalator be \( L \).
- Let the speed of the escalator be \( e \) (in meters per second).
- Let the speed of the passenger walking be \( p \) (in meters per second).

### Step 2: Determine Speeds
1. The time taken by the passenger standing on the escalator to reach the top is 1 minute (60 seconds). Therefore, we can express the speed of the escalator:
\[ e = \frac{L}{60} \]
2. The time taken by the passenger to walk up the escalator when it is not moving is 3 minutes (180 seconds). Thus, we can express the speed of the passenger:
\[ p = \frac{L}{180} \]

### Step 3: Calculate the Combined Speed
When the passenger walks up the moving escalator, the effective speed of the passenger with respect to the ground (or crown) is the sum of his walking speed and the escalator speed:
\[ \text{Effective speed} = p + e = \frac{L}{180} + \frac{L}{60} \]

### Step 4: Find a Common Denominator
To add the two fractions, we need a common denominator. The least common multiple of 180 and 60 is 180. Thus, we rewrite the second term:
\[ e = \frac{L}{60} = \frac{3L}{180} \]
Now we can add:
\[ p + e = \frac{L}{180} + \frac{3L}{180} = \frac{4L}{180} = \frac{L}{45} \]

### Step 5: Calculate Time Taken to Reach the Top
The time taken to reach the top when walking on the moving escalator can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{L}{\frac{L}{45}} = 45 \text{ seconds} \]

### Conclusion
Therefore, the time it will take for the passenger to arrive at the top if he walks up the moving escalator is **45 seconds**.

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