Moody takes 30 seconds to finish riding an escalator if he walks on it at his normal speed in the same direction. He takes 20 seconds to finish riding the escalator if he walks at twice his normal speed in the same direction. If Moody decides to stand still on the escalator, then the time, in seconds, needed to finish riding the escalator is

Question

Accepted Answer

1. **Define Variables:**  
   - Let Moody's walking speed be $$ x $$ steps/sec.  
   - Let the escalator's speed be $$ y $$ steps/sec.

2. **Scenario 1: Normal Speed (30 seconds):**  
   - Total steps covered = $$ (x + y) 	imes 30 $$.

3. **Scenario 2: Twice Normal Speed (20 seconds):**  
   - Total steps covered = $$ (2x + y) 	imes 20 $$.

4. **Equating Total Steps:**  
   Since the total number of steps on the escalator remains constant:  
   $$
   (x + y) 	imes 30 = (2x + y) 	imes 20.
   $$  
   Simplify:  
   $$
   30x + 30y = 40x + 20y.
   $$  
   Rearrange terms:  
   $$
   10y = 10x \implies x = y.
   $$

5. **Total Steps on Escalator:**  
   Substitute $$ x = y $$ into either equation to find total steps:  
   $$
   	ext{Total Steps} = (x + y) 	imes 30 = (2y) 	imes 30 = 60y.
   $$

6. **Time Taken When Standing Still:**  
   If Moody stands still, only the escalator moves at a speed of $$ y $$ steps/sec:  
   $$
   	ext{Time Taken} = \frac{	ext{Total Steps}}{	ext{Escalator Speed}} = \frac{60y}{y} = 60 	ext{ seconds}.
   $$

Thus, if Moody decides to stand still, it will take **60 seconds**.