A person, "A" starts descending from the first floor of a building to the ground floor on a descending escalator, while another person, B simultaneously starts ascending from the ground floor of the building to the first floor, using the same escalator. lf the speed of B is twice that of A, and A and B take 30 steps and 120 steps to reach their respective destinations, then find the possible number of steps that are visible on the escalator when it is stationary.

This problem involves two people, A and B, using an escalator where A is descending and B is ascending simultaneously. The speeds of A and B are given relative to each other, and we need to determine the number of visible steps on the escalator when it is stationary.

### Step-by-step solution:

#### Step 1: Define the variables
Let:
- \( N \) be the total number of steps on the escalator when it is stationary.
- \( v \) be the speed of the escalator in terms of steps per unit time.
- \( s_A \) be the speed at which person A descends (in steps per unit time).
- \( s_B \) be the speed at which person B ascends (in steps per unit time).
- \( t_A \) be the time taken by A to reach the ground floor.
- \( t_B \) be the time taken by B to reach the first floor.

From the problem statement:
- The speed of B is twice the speed of A: \( s_B = 2s_A \).
- A takes 30 steps, and B takes 120 steps.

#### Step 2: Express the relationship between time, steps, and speed

For person A (who is descending):
- In time \( t_A \), A takes 30 steps and the escalator moves downward by \( vt_A \) steps.
- The total number of steps \( N \) is covered by A's 30 steps plus the steps moved by the escalator:
  \[
  N = 30 + vt_A
  \]
  
For person B (who is ascending):
- In time \( t_B \), B takes 120 steps and the escalator moves downward by \( vt_B \) steps.
- Since B is moving against the escalator, the total number of steps \( N \) is covered by B's 120 steps minus the steps moved by the escalator:
  \[
  N = 120 - vt_B
  \]

#### Step 3: Relate the times for A and B
Since the speeds of A and B are given relative to each other:
\[
t_B = \frac{t_A}{2}
\]
because B is moving twice as fast as A, they reach their destinations in half the time.

#### Step 4: Solve the system of equations

From the two expressions for \( N \):
\[
N = 30 + vt_A
\]
and
\[
N = 120 - vt_B.
\]
Substitute \( t_B = \frac{t_A}{2} \) into the second equation:
\[
N = 120 - v\left(\frac{t_A}{2}\right).
\]
Now we have two equations:
1. \( N = 30 + vt_A \)
2. \( N = 120 - \frac{vt_A}{2} \)

Equating these two:
\[
30 + vt_A = 120 - \frac{vt_A}{2}.
\]

#### Step 5: Simplify and solve for \( vt_A \)
Multiply through by 2 to eliminate the fraction:
\[
60 + 2vt_A = 240 - vt_A.
\]
Combine like terms:
\[
3vt_A = 180.
\]
Solve for \( vt_A \):
\[
vt_A = 60.
\]

#### Step 6: Find \( N \)
Substitute \( vt_A = 60 \) into the first equation for \( N \):
\[
N = 30 + 60 = 90.
\]

Thus, the total number of visible steps on the escalator when stationary is \( \boxed{90} \).

### Conclusion:
The possible number of steps visible on the escalator when it is stationary is 90.

---

Do you need more details on any of the steps? Here are some additional questions you might explore:

1. What happens if A and B move at the same speed instead of different speeds?
2. How would the time taken by A and B change if the escalator's speed was doubled?
3. What if the escalator were stationary—how many steps would A and B take then?
4. Can we generalize the relationship between the number of steps and the speeds for different ratios between A and B's speeds?
5. How would the result change if A's speed were thrice that of B instead of half?

### Tip:
Whenever solving such problems, focus on clearly defining relationships between speed, time, and distance/steps to simplify the process of setting up equations.

A person, A, starts descending from the first floor of a building to the ground floor on a descending escalator, while another person, B, simultaneously starts ascending from the ground floor of the building to the first floor, using the same escalator. If the speed of B is twice that of A, and A and B take 30 steps and 120 steps to reach their respective destinations, then find the possible number of steps that are visible on the escalator when it is stationary.

Solve the relative motion problem where two people, A and B, are using an escalator in opposite directions. A takes 30 steps, and B takes 120 steps. Learn how to calculate the possible number of visible steps on the escalator when stationary using a system of simultaneous equations. Suitable for Grades 9-12.

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