In a race of 600 m, Anand beats Bharat by either 45 m or 9 s. Find the time taken by Anand and Bharat to finish the race

Answer

To solve the problem, we need to find the time taken by Anand and Bharat to finish a 600 m race, given that Anand beats Bharat by either 45 m or 9 seconds.

### Step 1: Understand the given information
- The total distance of the race is 600 m.
- Anand beats Bharat by 45 m. This means when Anand finishes the race, Bharat has run only 600 m - 45 m = 555 m.
- Anand also beats Bharat by 9 seconds. This means that when Anand finishes the race, Bharat takes 9 seconds more than Anand to finish.

### Step 2: Set up the equations
Let:
- \( t_A \) = time taken by Anand to finish the race (in seconds)
- \( t_B \) = time taken by Bharat to finish the race (in seconds)

From the information given:
1. When Anand finishes the race, Bharat has run 555 m. Therefore, we can express the speeds of Anand and Bharat as:
- Speed of Anand = \( \frac{600}{t_A} \)
- Speed of Bharat = \( \frac{555}{t_A} \)

2. From the second piece of information, we know that:
- \( t_B = t_A + 9 \)

### Step 3: Relate the speeds
Since speed is distance divided by time, we can set up the equation for Bharat's speed:
- Speed of Bharat = \( \frac{600}{t_B} \)

Substituting \( t_B \) from the previous step:
- Speed of Bharat = \( \frac{600}{t_A + 9} \)

### Step 4: Set the speeds equal
We can set the two expressions for Bharat's speed equal to each other:
\[ \frac{555}{t_A} = \frac{600}{t_A + 9} \]

### Step 5: Cross-multiply and solve for \( t_A \)
Cross-multiplying gives:
\[ 555(t_A + 9) = 600t_A \]
Expanding the left side:
\[ 555t_A + 4995 = 600t_A \]
Rearranging gives:
\[ 600t_A - 555t_A = 4995 \]
\[ 45t_A = 4995 \]
Dividing both sides by 45:
\[ t_A = \frac{4995}{45} = 111 \]

### Step 6: Find \( t_B \)
Now that we have \( t_A \), we can find \( t_B \):
\[ t_B = t_A + 9 = 111 + 9 = 120 \]

### Final Answer
- Time taken by Anand = 111 seconds
- Time taken by Bharat = 120 seconds