To solve this problem, we need to determine the total number of steps in the escalator when it is turned off.

Step 1: Variables and relationships

Let:

• $N$ be the total number of steps on the escalator when it is turned off.
• $v$ be the speed of the escalator in steps per second.
• $s_M$ and $s_B$ be the speeds of Mike and Bryan in steps per second, respectively.

Given:

1. Mike takes 30 steps while the escalator moves $30 + N - 30s_M$.
2. Bryan takes 27 steps while the equatorm will applyTo solve the problem systematically, we establish the relationships between the variables:

Definitions and Setup

• Let $N$ be the total number of steps on the escalator when it is stationary.
• The escalator moves at a constant rate of $v$ steps per second.
• Mike's step rate is $5x$ steps per second.
• Bryan's step rate is $4x$ steps per second.

When the escalator is moving:

• Mike takes 30 steps to reach the top.
• Bryan takes 27 steps to reach the top.

The total number of steps covered by Mike or Bryan is the sum of:

1. Steps they take themselves.
2. Steps the escalator moves during their travel.

Step 1: Express the total distance

The time Mike and Bryan take to reach the top can be calculated from their respective step counts and rates:

• Mike's travel time: $t_M = \frac{30}{5x} = \frac{6}{x}$.
• Bryan's travel time: $t_B = \frac{27}{4x}$.

The escalator moves at $v$ steps per second, so the distance moved by the escalator during this time is:

• For Mike: $v \cdot t_M = v \cdot \frac{6}{x}$.
• For Bryan: $v \cdot t_B = v \cdot \frac{27}{4x}$.

The total distance covered is equal to the stationary escalator length $N$:

• For Mike: $30 + v \cdot \frac{6}{x} = N$.
• For Bryan: $27 + v \cdot \frac{27}{4x} = N$.

Step 2: Solve for $v$ and $N$

From the two equations:

1. $N = 30 + v \cdot \frac{6}{x}$,
2. $N = 27 + v \cdot \frac{27}{4x}$.

Equating the two expressions for $N$: $30 + v \cdot \frac{6}{x} = 27 + v \cdot \frac{27}{4x}.$

Simplify: $3 = v \left( \frac{27}{4x} - \frac{6}{x} \right).$

Combine terms inside the parentheses: $\frac{27}{4x} - \frac{6}{x} = \frac{27 - 24}{4x} = \frac{3}{4x}.$

Substitute back: $3 = v \cdot \frac{3}{4x}.$

Solve for $v$: $v = 4x.$

Step 3: Substitute $v$ back to find $N$

Using $v = 4x$ in Mike’s equation for $N$: $N = 30 + \frac{6}{x} \cdot 4x.$

Simplify: $N = 30 + 24 = 54.$

Step 4: Verify for Bryan

For Bryan: $N = 27 + \frac{27}{4x} \cdot 4x.$

Simplify: $N = 27 + 27 = 54.$

Final Answer

When the escalator is stationary, the total number of steps is: $\boxed{54 \text{ steps.}}$

Additional Notes

• Mike and Bryan would both need to climb 54 steps to reach the top without the escalator moving.

Let me know if you want a detailed breakdown or further explanation!

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Related Questions:

1. How would the solution change if Mike and Bryan walked at the same speed?
2. What is the relationship between the step counts and escalator speed in general?
3. How can we use relative rates to solve similar problems?
4. If the escalator moved faster, how would their step counts change?
5. How could this problem be modeled graphically?

Tip: Breaking a problem into small steps and validating results helps avoid calculation errors!