A thief is spotted by a policemen at a distance of 200 metres. When the policemen starts to chase, the theif also starts running. The speed of thief and policemen are 10km/h and 14 km/h respectively. How far will have the thief run before he caught ?

Answer

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

### Step 1: Understand the problem
We have a thief and a policeman. The thief is 200 meters ahead of the policeman. The speeds of the thief and the policeman are given as 10 km/h and 14 km/h respectively.

**Hint:** Identify the initial distance and the speeds of both individuals.

### Step 2: Convert speeds from km/h to m/s
To work with the distance in meters, we can convert the speeds from kilometers per hour to meters per second.
- Speed of the thief:
\[ 10 \text{ km/h} = \frac{10 \times 1000}{3600} \text{ m/s} = \frac{10000}{3600} \approx 2.78 \text{ m/s} \]
- Speed of the policeman:
\[ 14 \text{ km/h} = \frac{14 \times 1000}{3600} \text{ m/s} = \frac{14000}{3600} \approx 3.89 \text{ m/s} \]

**Hint:** Remember that to convert km/h to m/s, multiply by \( \frac{1000}{3600} \).

### Step 3: Calculate the relative speed
The relative speed of the policeman with respect to the thief is the difference in their speeds:
\[ \text{Relative Speed} = \text{Speed of Policeman} - \text{Speed of Thief} = 3.89 \text{ m/s} - 2.78 \text{ m/s} = 1.11 \text{ m/s} \]

**Hint:** The relative speed helps us understand how quickly one is catching up to the other.

### Step 4: Calculate the time taken to catch the thief
To find the time taken for the policeman to catch the thief, we can use the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{200 \text{ m}}{1.11 \text{ m/s}} \approx 180.18 \text{ seconds} \]

**Hint:** Use the formula for time, which is distance divided by speed.

### Step 5: Calculate the distance run by the thief
Now, we can find out how far the thief has run during this time. We use the thief's speed:
\[ \text{Distance run by the thief} = \text{Speed of Thief} \times \text{Time} = 2.78 \text{ m/s} \times 180.18 \text{ s} \approx 500 \text{ meters} \]

**Hint:** Distance can be calculated by multiplying speed by time.

### Final Answer
The thief will have run approximately **500 meters** before being caught by the policeman.

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