A person decided to walk on an escalator which is moving at constant rate (speed). When he moves at the rate 1 step/sec. then he reaches top in 20 steps. Next day he goes 2 steps/sec. and reaches top in 32 steps. If speed of escalator is n steps/sec. Find the value of n

Answer

To solve the problem, we need to find the speed of the escalator (n) based on the information given about the person's walking speed and the number of steps taken on two different days.

### Step-by-Step Solution:

1. **Determine the time taken on Day 1:**
- The person walks at a speed of 1 step/sec and takes 20 steps.
- Time taken (t1) on Day 1 = Number of steps / Walking speed = 20 steps / 1 step/sec = 20 seconds.

2. **Determine the time taken on Day 2:**
- The person walks at a speed of 2 steps/sec and takes 32 steps.
- Time taken (t2) on Day 2 = Number of steps / Walking speed = 32 steps / 2 steps/sec = 16 seconds.

3. **Calculate the effective speed of the person on both days:**
- On Day 1, the effective speed with respect to the escalator is (n + 1) steps/sec (where n is the speed of the escalator).
- On Day 2, the effective speed with respect to the escalator is (n + 2) steps/sec.

4. **Calculate the total distance covered (length of the escalator):**
- The distance covered on Day 1 = Effective speed × Time = (n + 1) × 20.
- The distance covered on Day 2 = Effective speed × Time = (n + 2) × 16.

5. **Set the distances equal to each other:**
- Since the length of the escalator is the same on both days, we can set up the equation:
\[ (n + 1) \times 20 = (n + 2) \times 16 \]

6. **Expand and simplify the equation:**
- Expanding both sides gives:
\[ 20n + 20 = 16n + 32 \]
- Rearranging the equation:
\[ 20n - 16n = 32 - 20 \]
\[ 4n = 12 \]

7. **Solve for n:**
- Dividing both sides by 4:
\[ n = 3 \text{ steps/sec} \]

### Final Answer:
The speed of the escalator (n) is **3 steps/sec**.