A policeman saw a thief from a distance of 200m and started running towards him. If speed of policeman and thief are 12 km/h & 10 km/h respectively. Find the distance thief had run before caught.

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
The policeman sees a thief 200 meters away and starts running towards him. The speeds of the policeman and the thief are given as 12 km/h and 10 km/h, respectively. We need to find out how far the thief has run before being caught.

**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 distances in meters, we need to convert the speeds from kilometers per hour to meters per second. The conversion factor is:
1 km/h = (1000 m / 3600 s) = 5/18 m/s.

- Speed of policeman = 12 km/h = 12 * (5/18) = 10/3 m/s.
- Speed of thief = 10 km/h = 10 * (5/18) = 50/18 = 25/9 m/s.

**Hint:** Use the conversion factor to change the units for easier calculations.

### Step 3: Calculate the Relative Speed
Since both the policeman and the thief are moving in the same direction, the relative speed of the policeman with respect to the thief is:
Relative Speed = Speed of Policeman - Speed of Thief
= (10/3) m/s - (25/9) m/s.

To subtract these fractions, we need a common denominator:
- The common denominator of 3 and 9 is 9.
- Convert (10/3) to (30/9).

Now, calculate the relative speed:
Relative Speed = (30/9) - (25/9) = 5/9 m/s.

**Hint:** When dealing with speeds, remember to subtract the smaller speed from the larger speed when they are moving in the same direction.

### Step 4: Calculate the Time Taken to Catch the Thief
The time taken for the policeman to cover the initial distance of 200 meters at the relative speed of 5/9 m/s is given by:
Time = Distance / Relative Speed
= 200 m / (5/9 m/s) = 200 * (9/5) = 360 seconds.

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

### Step 5: Calculate the Distance Run by the Thief
Now that we know the time taken for the policeman to catch the thief, we can calculate how far the thief has run in that time. The distance run by the thief is given by:
Distance = Speed of Thief * Time
= (25/9 m/s) * 360 seconds.

Calculating this:
Distance = (25/9) * 360 = 1000 meters.

**Hint:** Remember that distance is calculated as speed multiplied by time.

### Conclusion
The distance the thief had run before being caught is 1000 meters, or 1 kilometer.

**Final Answer:** The thief ran 1000 meters before being caught by the policeman.