The problem states:

• It takes 48 seconds for a person to climb up a moving escalator.
• It takes 60 seconds to move down the same escalator when it is moving in the opposite direction.
• We are asked to find how long it would take for the person to move up the escalator if it is switched off.

Let's analyze this.

Solution

1. Let the escalator’s speed be $E$ (in terms of escalator length per second) and the person's walking speed be $P$ (in terms of escalator length per second).

2. When going up on the moving escalator, both the escalator and the person’s speeds combine, so the effective speed is $P + E$.

   • Time taken = 48 seconds.
   • Thus, the distance of the escalator (let’s call it $D$) can be expressed as: $D = 48 \times (P + E)$

3. When going down on the moving escalator, the escalator's speed opposes the person's speed, so the effective speed is $P - E$.

   • Time taken = 60 seconds.
   • Thus, the same distance $D$ can also be expressed as: $D = 60 \times (P - E)$

4. Setting up the equations: $48(P + E) = 60(P - E)$

5. Expanding and simplifying: $48P + 48E = 60P - 60E$ $108E = 12P$ $E = \frac{P}{9}$

6. Substitute $E = \frac{P}{9}$ back to find $D$: $D = 48(P + \frac{P}{9})$ $D = 48 \times \frac{10P}{9} = \frac{480P}{9}$

7. Calculating the time when the escalator is switched off:

   • When the escalator is off, only the person’s speed $P$ is effective. $\text{Time} = \frac{D}{P} = \frac{\frac{480P}{9}}{P} = \frac{480}{9} = 53.\overline{3} \text{ seconds}$

Final Answer:

It will take approximately $53.33$ seconds for the person to move up the escalator when it is switched off.

Would you like further details on any part of this solution, or do you have additional questions?

Related Questions

1. If the person increases their speed by 10%, how would the new time compare?
2. How would the solution change if the escalator length was known?
3. How long would it take to descend if the escalator was off?
4. What if the escalator speed was twice as fast?
5. Can you derive a general formula for cases like this?

Tip:

In problems with relative motion (like moving escalators or conveyor belts), remember to consider how speeds add or subtract based on direction.