A drunked person is 10 step away from a manhole. Suddenly he begins to walk towards the manhole. Suddenly he begins to walk towards the manhole. He walks two steps forward and one step beckward in one second. Find out
(i) The time after which the person will fall into the manhole?

Answer

To solve the problem, we need to determine how long it takes for the drunkard to reach the manhole, given that he is initially 10 steps away and walks 2 steps forward and 1 step backward every second.

### Step-by-Step Solution:

1. **Understanding the Movement**:
- The drunkard walks 2 steps forward and then takes 1 step backward in 1 second.
- This means that in 1 second, the net movement of the drunkard is:
\[ \text{Net steps per second} = 2 \text{ (forward)} - 1 \text{ (backward)} = 1 \text{ step forward} \]

2. **Calculating the Total Steps to the Manhole**:
- The drunkard starts 10 steps away from the manhole.
- Since he moves 1 step forward every second, we can calculate how many seconds it will take to cover the 10 steps:
\[ \text{Total time (in seconds)} = \text{Total steps} = 10 \text{ steps} \]

3. **Time Calculation**:
- It will take him 10 seconds to reach the 10th step (the manhole).
- However, we need to consider the last movement. In the 10th second, he will move from the 9th step to the 11th step (which is beyond the manhole). So we need to calculate the exact time when he reaches the manhole.

4. **Movement in the 10th Second**:
- In the 10th second, he will take 2 steps forward from the 9th step and then take 1 step backward.
- Thus, he will reach the 10th step (the manhole) before he takes the backward step.

5. **Final Calculation**:
- He reaches the manhole during the 10th second. Since he moves 2 steps forward in that second, he will reach the manhole at:
\[ \text{Time to reach manhole} = 10 \text{ seconds} - \text{time to take 1 backward step} \]
- The backward step does not matter since he has already reached the manhole.

6. **Conclusion**:
- Therefore, the total time taken for the drunkard to fall into the manhole is:
\[ \text{Total time} = 10 \text{ seconds} \]

### Final Answer:
The time after which the person will fall into the manhole is **10 seconds**.

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