A policaman starts to chase a thief. When the thief goes 10 steps the policeman moves 8 steps. 5 steps of the policeman is equal to 7 steps of the thief. The ratio of the speed of the policement and the thief is

Options

A. `25:28`
B. `25:26`
C. `28:25`
D. `56:25`

Answer

To solve the problem, we need to determine the ratio of the speeds of the policeman and the thief based on the information provided.

### Step-by-Step Solution:

1. **Understanding the Steps**:
- When the thief takes 10 steps, the policeman takes 8 steps.
- We are given that 5 steps of the policeman are equal to 7 steps of the thief.

2. **Finding the Equivalent Step Lengths**:
- Let the length of one step of the policeman be \( P \) and the length of one step of the thief be \( T \).
- From the information given, we can express the relationship between the lengths of their steps:
\[ 5P = 7T \quad \Rightarrow \quad \frac{P}{T} = \frac{7}{5} \]

3. **Calculating the Distances Covered**:
- When the thief runs 10 steps, the distance covered by the thief is:
\[ \text{Distance by Thief} = 10T \]
- When the policeman runs 8 steps, the distance covered by the policeman is:
\[ \text{Distance by Policeman} = 8P \]

4. **Substituting the Step Lengths**:
- We can substitute \( P \) in terms of \( T \):
\[ P = \frac{7}{5}T \]
- Thus, the distance covered by the policeman becomes:
\[ \text{Distance by Policeman} = 8P = 8 \left(\frac{7}{5}T\right) = \frac{56}{5}T \]

5. **Calculating the Ratio of Speeds**:
- The speed of the policeman and the thief can be expressed as the distance covered per unit time. Since both are running for the same time, we can set up the ratio of their speeds:
\[ \text{Speed of Policeman} : \text{Speed of Thief} = \frac{56}{5}T : 10T \]
- Simplifying this ratio:
\[ = \frac{56}{5} : 10 = \frac{56}{5} : \frac{50}{5} = 56 : 50 \]

6. **Reducing the Ratio**:
- To simplify \( 56 : 50 \), we can divide both sides by 2:
\[ = 28 : 25 \]

### Final Answer:
The ratio of the speed of the policeman to the speed of the thief is \( 28 : 25 \).

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