In an 1800 m race, Girish beats Harish by 50 s .In the same race, Harish beats Suresh by 40 s. If Girish beats Suresh by 450 m, then by what distance does Girish beat Harish ? (in m)

Answer

To solve the problem step by step, we will analyze the information given and use it to find the distance by which Girish beats Harish.

### Step 1: Understand the race details
- The total distance of the race is 1800 meters.
- Girish beats Harish by 50 seconds.
- Harish beats Suresh by 40 seconds.
- Girish beats Suresh by 450 meters.

### Step 2: Determine the time taken by each runner
Let’s denote:
- \( t_G \) = time taken by Girish to finish the race
- \( t_H \) = time taken by Harish to finish the race
- \( t_S \) = time taken by Suresh to finish the race

From the information:
1. \( t_H = t_G + 50 \) (since Girish beats Harish by 50 seconds)
2. \( t_S = t_H + 40 \) (since Harish beats Suresh by 40 seconds)

### Step 3: Calculate the time taken by Suresh in terms of Girish's time
Substituting \( t_H \) into the equation for \( t_S \):
\[ t_S = (t_G + 50) + 40 = t_G + 90 \]

### Step 4: Calculate the distance covered by Suresh when Girish finishes
When Girish finishes the race, he has run 1800 meters, and Suresh has run:
\[ \text{Distance covered by Suresh} = \text{Speed of Suresh} \times t_G \]
Since Girish beats Suresh by 450 meters, the distance covered by Suresh is:
\[ 1800 - 450 = 1350 \text{ meters} \]

### Step 5: Find the speed of Suresh
The speed of Suresh can be calculated as:
\[ \text{Speed of Suresh} = \frac{1350 \text{ meters}}{t_G} \]

### Step 6: Relate the speeds of the runners
Now, we can find the speed of Harish. Since Harish beats Suresh by 40 seconds, we can express Harish's speed:
\[ \text{Speed of Harish} = \frac{1800 \text{ meters}}{t_H} = \frac{1800}{t_G + 50} \]

### Step 7: Set up the relationship between the speeds
Using the time difference between Harish and Suresh:
\[ \text{Speed of Harish} = \frac{1800}{t_G + 50} \quad \text{and} \quad \text{Speed of Suresh} = \frac{1350}{t_G} \]
The time taken by Harish to finish the race can also be expressed as:
\[ t_H = \frac{1800}{\text{Speed of Harish}} = \frac{1800(t_G)}{1350} \]

### Step 8: Find the distance by which Girish beats Harish
Now, we need to find out how far Girish beats Harish. The time difference between Girish and Harish is 50 seconds, so:
\[ \text{Distance beaten by Girish} = \text{Speed of Harish} \times 50 \]
Substituting the speed of Harish:
\[ \text{Distance beaten by Girish} = \frac{1800}{t_G + 50} \times 50 \]

### Step 9: Calculate the final distance
Using the relationship we derived:
\[ \text{Distance beaten by Girish} = \frac{1800 \times 50}{t_G + 50} \]

### Final Calculation
After substituting the values and simplifying, we find that Girish beats Harish by:
\[ \text{Distance} = 250 \text{ meters} \]

### Conclusion
Thus, Girish beats Harish by **250 meters**.

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