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

Correct Answer: 60

We are given that Moody walks on an escalator, and the time taken depends on his walking speed and the speed of the escalator. Let’s define the following variables:

• \( W \) = Moody’s normal walking speed (in units per second)
• \( E \) = Speed of the escalator (in units per second)

Step 1: Set up the equations based on the given information

- When Moody walks at his normal speed, his effective speed is the sum of his walking speed and the escalator's speed, i.e. \( W + E \). He completes the escalator ride in 30 seconds, so: \[ (W + E) \times 30 = 1 \] - When Moody walks at twice his normal speed, his effective speed is \( 2W + E \), and he completes the ride in 20 seconds: \[ (2W + E) \times 20 = 1 \]

Step 2: Solve the equations

From the first equation: \[ 30(W + E) = 1 \] We can express \( E \) in terms of \( W \): \[ 30W + 30E = 1 \] \[ E = \frac{1 - 30W}{30} \] Now, substitute this expression for \( E \) into the second equation: \[ (2W + \frac{1 - 30W}{30}) \times 20 = 1 \] Simplify: \[ 2W + \frac{1 - 30W}{30} = \frac{1}{20} \] Multiply through by 30 to eliminate the fraction: \[ 60W + 1 - 30W = \frac{30}{20} \] \[ 30W + 1 = 1.5 \] \[ 30W = 0.5 \] \[ W = \frac{0.5}{30} = \frac{1}{60} \]

Step 3: Find \( E \)

Substitute \( W = \frac{1}{60} \) into the expression for \( E \): \[ E = \frac{1 - 30 \times \left( \frac{1}{60} \right)}{30} \] \[ E = \frac{1 - 0.5}{30} = \frac{0.5}{30} = \frac{1}{60} \]

Step 4: Time taken to ride the escalator without walking

If Moody stands still, his effective speed is just the speed of the escalator, \( E \). The time needed to finish riding the escalator will be: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{1}{E} = \frac{1}{\frac{1}{60}} = 60 \text{ seconds} \]

Final Answer:

If Moody stands still on the escalator, it will take him 60 seconds to finish riding it.

Conclusion:

The correct answer is \( \boxed{60} \) seconds.

Answer 2

Let Moody travel at a speed of \( x \) steps per second, and the escalator at \( y \) steps per second. We know:

Step 1: When Moody travels at normal speed

Moody will complete his trip on the escalator in 30 seconds. Therefore, the total number of steps taken is: \[ 30(x + y) \text{ steps}. \]

Step 2: When Moody's speed doubles

If Moody's speed doubles, the time it takes him is reduced to 20 seconds. The total number of steps is now: \[ 20(2x + y) \text{ steps}. \]

Step 3: Solving the relationship between the speeds

By equating the total number of steps in both cases: \[ 30(x + y) = 20(2x + y) \] Simplifying the equation: \[ 30x + 30y = 40x + 20y \] \[ 30x - 40x = 20y - 30y \] \[ -10x = -10y \] \[ x = y \]

Step 4: Determining the time if Moody stands still

When \( x = y \), the total number of steps taken is: \[ 30(x + y) = 60y \text{ steps}. \] The time taken if Moody stands still (with only the escalator moving) is: \[ \frac{60y}{y} = 60 \text{ seconds}. \]

Final Answer:

If Moody stands still, it will take him 60 seconds to finish riding the escalator.

Conclusion:

The time taken to finish riding the escalator when Moody stands still is \( \boxed{60} \) seconds.