In a 1000 m race, Swaroop beats Laxman by either 100 m or 20 s and Laxman beats Sashidhar by 25 s. By what distance can Swaroop beat Sashidhar in the same race ?

Answer

To solve the problem step by step, we can break it down as follows:

### Step 1: Understand the Race Conditions
Swaroop beats Laxman by either 100 meters or 20 seconds in a 1000-meter race. This means that when Swaroop finishes the race, Laxman has either run 900 meters (if we consider the distance) or taken 20 seconds longer than Swaroop to finish the race.

### Step 2: Calculate Laxman's Speed
Since Swaroop beats Laxman by 100 meters, we can find Laxman's speed.

Let’s assume Swaroop's speed is \( v_s \) and Laxman's speed is \( v_l \).

From the information given:
- When Swaroop finishes 1000 meters, Laxman has run 900 meters.
- The time taken by Swaroop to finish the race is \( t_s = \frac{1000}{v_s} \).
- The time taken by Laxman to run 900 meters is \( t_l = \frac{900}{v_l} \).

Since Swaroop beats Laxman by 20 seconds:
\[ t_l = t_s + 20 \]
\[ \frac{900}{v_l} = \frac{1000}{v_s} + 20 \]

### Step 3: Express Laxman's Speed in Terms of Swaroop's Speed
From the equation above, we can express Laxman's speed:
\[ \frac{900}{v_l} = \frac{1000}{v_s} + 20 \]
Rearranging gives:
\[ \frac{900}{v_l} - \frac{1000}{v_s} = 20 \]

### Step 4: Calculate the Speed Ratio
From the distance and time relationship, we can derive the ratio of speeds:
- Swaroop's speed can be calculated as \( v_s = \frac{1000}{t_s} \).
- Laxman's speed can be calculated as \( v_l = \frac{900}{t_l} \).

### Step 5: Calculate the Time Difference
Since Laxman beats Sashidhar by 25 seconds, we can set up another equation:
Let Sashidhar's speed be \( v_sh \).
When Laxman finishes the race, Sashidhar has not finished yet, so:
\[ t_{sh} = t_l + 25 \]
Where \( t_{sh} \) is the time taken by Sashidhar to run 1000 meters.

### Step 6: Determine the Distance Swaroop Beats Sashidhar
Now we need to find out how far Swaroop beats Sashidhar.

Using the time difference:
- If Laxman takes \( t_l \) seconds to finish, then Sashidhar takes \( t_l + 25 \) seconds.
- The distance Sashidhar runs in \( t_l + 25 \) seconds can be calculated using his speed:
\[ \text{Distance}_{sh} = v_{sh} \times (t_l + 25) \]

### Step 7: Calculate the Distance
Using the relationship between speed and time, we can find the distance that Swaroop beats Sashidhar:
- Since we know Laxman runs 1000 meters in \( t_l \) seconds, we can find \( v_l \) and relate it to Sashidhar's speed.
- The distance Swaroop beats Sashidhar can be calculated from the time difference and speed.

### Final Calculation
Assuming Laxman's speed is \( v_l \) and Sashidhar's speed is slower, we can conclude:
- If Swaroop beats Laxman by 100 meters, and Laxman beats Sashidhar by 25 seconds, we can find the total distance Swaroop beats Sashidhar by adding the distances.

### Conclusion
Swaroop beats Sashidhar by 200 meters in the same race.

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