Question

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

Answer 1

Moody walks on an escalator. When he walks at his normal speed, it takes 30 seconds to ride the escalator. When he walks at twice his normal speed, it takes 20 seconds. If he stands still, how much time will it take to finish riding the escalator?

Step 1: Let Speeds Be

• Moody’s normal walking speed = \( W \) units/sec
• Escalator’s speed = \( E \) units/sec

Effective speed while walking normally: \( W + E \) 
Effective speed while walking at double speed: \( 2W + E \)

Step 2: Total Distance of Escalator = 1 unit (assumed)

Now form two equations based on time = distance / speed:

• Equation 1: \( (W + E) \times 30 = 1 \Rightarrow 30W + 30E = 1 \)
• Equation 2: \( (2W + E) \times 20 = 1 \Rightarrow 40W + 20E = 1 \)

Step 3: Solve the Equations

From Equation 1:

\[ 30W + 30E = 1 \Rightarrow E = \frac{1 - 30W}{30} \]

Substitute into Equation 2:

\[ (2W + \frac{1 - 30W}{30}) \times 20 = 1 \]

Simplify:

\[ 2W + \frac{1 - 30W}{30} = \frac{1}{20} \]

Multiply all terms by 30:

\[ 60W + 1 - 30W = 1.5 \Rightarrow 30W + 1 = 1.5 \Rightarrow 30W = 0.5 \Rightarrow W = \frac{1}{60} \]

Step 4: Find Escalator Speed

\[ E = \frac{1 - 30 \cdot \frac{1}{60}}{30} = \frac{1 - 0.5}{30} = \frac{0.5}{30} = \frac{1}{60} \]

Step 5: Time if Moody Stands Still

If Moody stands still, his speed is just \( E = \frac{1}{60} \).
So time = distance / speed = \( \frac{1}{\frac{1}{60}} = 60 \) seconds

Final Answer:

\[ \boxed{60 \text{ seconds}} \]

Answer 2

Moody walks on an escalator. When walking at his normal speed, he takes 30 seconds to finish the ride. When walking at twice his normal speed, it takes 20 seconds. If Moody stands still, how long will it take him to complete the ride?

Step 1: Let Speeds Be

• Moody’s walking speed = \( x \) steps/sec
• Escalator speed = \( y \) steps/sec

Total number of steps of the escalator = Distance = constant.

Step 2: Total Steps Covered in Each Case

When Moody walks at normal speed: \[ \text{Total steps} = 30(x + y) \] When Moody walks at double speed: \[ \text{Total steps} = 20(2x + y) \]

Step 3: Equating Both Distances

Since both expressions represent the same total distance: \[ 30(x + y) = 20(2x + y) \]

Step 4: Expand Both Sides

\[ 30x + 30y = 40x + 20y \Rightarrow 30x - 40x + 30y - 20y = 0 \Rightarrow -10x + 10y = 0 \Rightarrow x = y \]

Step 5: Find Time When Moody Stands Still

If Moody stands still, his speed is 0. So the only motion comes from the escalator: \[ \text{Speed} = y, \quad \text{Distance} = 30(x + y) = 30(2y) = 60y \]

Time = \( \frac{\text{Distance}}{\text{Speed}} = \frac{60y}{y} = 60 \) seconds

Final Answer:

\[ \boxed{60 \text{ seconds}} \]