In a 1 km race, A beats B by 30 seconds and B beats C by 15 seconds. If A beats C by 180 m, Find the time taken by A to run 1 km.

Answer

To solve the problem step by step, we will analyze the information given and derive the time taken by A to run 1 km.

### Step 1: Define the time taken by A
Let the time taken by A to run 1 km be \( t \) seconds.

### Step 2: Determine the time taken by B
Since A beats B by 30 seconds, the time taken by B to run 1 km will be:
\[ t + 30 \text{ seconds} \]

### Step 3: Determine the time taken by C
B beats C by 15 seconds, so the time taken by C to run 1 km will be:
\[ (t + 30) + 15 = t + 45 \text{ seconds} \]

### Step 4: Analyze the distance difference between A and C
According to the problem, A beats C by 180 meters. This means when A finishes 1 km, C has only covered \( 1000 - 180 = 820 \) meters.

### Step 5: Calculate the time taken by C to cover 820 meters
Since C takes \( t + 45 \) seconds to cover 1 km (1000 meters), we can find the time taken by C to cover 820 meters using the proportion:
\[ \text{Time taken by C to cover 820 meters} = \frac{820}{1000} \times (t + 45) = 0.82(t + 45) \]

### Step 6: Set up the equation
Since A takes \( t \) seconds to cover 1000 meters, and C takes \( 0.82(t + 45) \) seconds to cover 820 meters, we can equate the two times:
\[ t = 0.82(t + 45) \]

### Step 7: Solve for \( t \)
Expanding the equation:
\[ t = 0.82t + 36.9 \]
Rearranging gives:
\[ t - 0.82t = 36.9 \]
\[ 0.18t = 36.9 \]
\[ t = \frac{36.9}{0.18} = 205 \text{ seconds} \]

### Conclusion
The time taken by A to run 1 km is \( 205 \) seconds.

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