A policemen is chasing a thief. The distance, between them is 100 m and the ratio of their speeds is 5 : 4. What is the distance covered by the thief before being caught ?

Options

A. 80 m
B. 200 m
C. 400 m
D. 600 m

Answer

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

### Step 1: Understand the Given Information
- The distance between the policeman and the thief is 100 meters.
- The ratio of their speeds is 5:4 (policeman:thief).

### Step 2: Assign Variables for Speeds
Let the speed of the policeman be \(5x\) and the speed of the thief be \(4x\), where \(x\) is a common factor.

### Step 3: Set Up the Distance Equation
When the policeman catches the thief, the distance covered by the thief is \(d\) meters, and the distance covered by the policeman is \(d + 100\) meters (since he starts 100 meters behind).

### Step 4: Use the Formula for Time
Since both the policeman and the thief will take the same time to reach the point where the policeman catches the thief, we can set up the equation using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).

For the thief:
\[ \text{Time}_{\text{thief}} = \frac{d}{4x} \]

For the policeman:
\[ \text{Time}_{\text{policeman}} = \frac{d + 100}{5x} \]

### Step 5: Set the Times Equal
Since both times are equal:
\[ \frac{d}{4x} = \frac{d + 100}{5x} \]

### Step 6: Cross Multiply to Solve for \(d\)
Cross multiplying gives:
\[ 5d = 4(d + 100) \]

### Step 7: Expand and Rearrange the Equation
Expanding the right side:
\[ 5d = 4d + 400 \]
Now, subtract \(4d\) from both sides:
\[ 5d - 4d = 400 \]
This simplifies to:
\[ d = 400 \]

### Step 8: Conclusion
The distance covered by the thief before being caught is **400 meters**.

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