A thief seeing a policeman from a distance of 200 metres, starts running with a speed of 8 km/hr. The policeman gives chase immediately with a speed of 9 km/hr and the thief is caught. What is the distance run by the thief?

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the speeds and initial distance
- The thief runs at a speed of 8 km/hr.
- The policeman runs at a speed of 9 km/hr.
- The initial distance between the thief and the policeman is 200 meters.

### Step 2: Calculate the relative speed
- Since both the thief and the policeman are running in the same direction, we find the relative speed by subtracting the speed of the thief from the speed of the policeman.

\[ \text{Relative Speed} = \text{Speed of Policeman} - \text{Speed of Thief} = 9 \text{ km/hr} - 8 \text{ km/hr} = 1 \text{ km/hr} \]

### Step 3: Convert the relative speed to meters per second
- To convert km/hr to m/s, we use the conversion factor \( \frac{5}{18} \).

\[ \text{Relative Speed in m/s} = 1 \text{ km/hr} \times \frac{5}{18} = \frac{5}{18} \text{ m/s} \]

### Step 4: Calculate the time taken for the policeman to catch the thief
- The time taken can be calculated using the formula:

\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

Here, the distance is 200 meters and the speed is \( \frac{5}{18} \) m/s.

\[ \text{Time} = \frac{200 \text{ meters}}{\frac{5}{18} \text{ m/s}} = 200 \times \frac{18}{5} = 720 \text{ seconds} \]

### Step 5: Calculate the distance run by the thief
- Now that we have the time, we can calculate the distance run by the thief using the formula:

\[ \text{Distance} = \text{Speed} \times \text{Time} \]

The speed of the thief is 8 km/hr, which we need to convert to m/s:

\[ 8 \text{ km/hr} \times \frac{5}{18} = \frac{40}{18} \text{ m/s} = \frac{20}{9} \text{ m/s} \]

Now, we can calculate the distance:

\[ \text{Distance run by the thief} = \left(\frac{20}{9} \text{ m/s}\right) \times 720 \text{ seconds} = 1600 \text{ meters} \]

### Final Answer
The distance run by the thief is **1600 meters**.

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