Raman covers 1 km in 8 min while Suman covers the same distance in 10 min. By what distance does Raman beat suman ?

Options

A. (A)150 m
B. (B)65 m
C. (C)190 m
D. (D)200 m

Answer

To solve the problem of how much distance Raman beats Suman by, we will follow these steps:

### Step 1: Calculate the speed of Raman and Suman
- **Raman's speed**: He covers 1 km in 8 minutes.
- Convert 8 minutes to hours:
\[ \text{Time in hours} = \frac{8}{60} = \frac{2}{15} \text{ hours} \]
- Speed = Distance / Time
\[ \text{Speed of Raman} = \frac{1 \text{ km}}{\frac{2}{15} \text{ hours}} = 1 \times \frac{15}{2} = \frac{15}{2} \text{ km/h} = 7.5 \text{ km/h} \]

- **Suman's speed**: She covers 1 km in 10 minutes.
- Convert 10 minutes to hours:
\[ \text{Time in hours} = \frac{10}{60} = \frac{1}{6} \text{ hours} \]
- Speed = Distance / Time
\[ \text{Speed of Suman} = \frac{1 \text{ km}}{\frac{1}{6} \text{ hours}} = 1 \times 6 = 6 \text{ km/h} \]

### Step 2: Determine the time taken by both to finish the race
- **Time taken by Raman**:
\[ \text{Time} = \frac{1 \text{ km}}{7.5 \text{ km/h}} = \frac{1}{7.5} \text{ hours} = \frac{2}{15} \text{ hours} \]

- **Time taken by Suman**:
\[ \text{Time} = \frac{1 \text{ km}}{6 \text{ km/h}} = \frac{1}{6} \text{ hours} \]

### Step 3: Calculate the time difference
- **Time difference**:
\[ \text{Time taken by Suman} - \text{Time taken by Raman} = \frac{1}{6} - \frac{2}{15} \]
- To subtract these fractions, find a common denominator (which is 30):
\[ \frac{1}{6} = \frac{5}{30}, \quad \frac{2}{15} = \frac{4}{30} \]
\[ \text{Time difference} = \frac{5}{30} - \frac{4}{30} = \frac{1}{30} \text{ hours} \]

### Step 4: Calculate the distance Suman covers in that time
- **Distance covered by Suman in the time difference**:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
- Suman's speed is 6 km/h, and the time difference is \(\frac{1}{30}\) hours:
\[ \text{Distance} = 6 \text{ km/h} \times \frac{1}{30} \text{ hours} = \frac{6}{30} = \frac{1}{5} \text{ km} \]

### Step 5: Convert the distance to meters
- Since 1 km = 1000 meters:
\[ \text{Distance in meters} = \frac{1}{5} \text{ km} \times 1000 \text{ meters/km} = 200 \text{ meters} \]

### Final Answer:
Raman beats Suman by **200 meters**.

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