In 1 km race,A beats B by 36 m or 18 s. Find theA'stime over the course.

Answer

To solve the problem, we need to determine A's time to complete the 1 km race given that A beats B by 36 meters or 18 seconds.

### Step-by-Step Solution:

1. **Understanding the Race Distance**:
- The total distance of the race is 1 km, which is equal to 1000 meters.

2. **Understanding the Defeat Margin**:
- A beats B by 36 meters. This means when A finishes the race (1000 meters), B has only covered 964 meters (1000 m - 36 m).

3. **Time Difference**:
- A also beats B by 18 seconds. This means that when A finishes the race, B takes an additional 18 seconds to finish.

4. **Calculating B's Time**:
- Let's denote B's time to complete the race as \( T_B \).
- Since A beats B by 18 seconds, we can express B's time as:
\[ T_B = T_A + 18 \]
- Where \( T_A \) is A's time to complete the race.

5. **Finding B's Speed**:
- We know that B covers 964 meters in \( T_B \) seconds.
- The speed of B can be calculated as:
\[ \text{Speed of B} = \frac{964 \text{ m}}{T_B \text{ s}} \]

6. **Finding B's Time for 1000 meters**:
- To find the time it takes for B to cover the full 1000 meters, we can set up a proportion based on the distance and time:
\[ \text{Time for B to cover 1000 m} = \frac{T_B}{964} \times 1000 \]
- Since we know that B takes 18 seconds longer than A, we can substitute \( T_B \) into this equation:
\[ T_B = T_A + 18 \]
- Therefore, the time for B to cover 1000 meters becomes:
\[ \frac{(T_A + 18)}{964} \times 1000 \]

7. **Setting Up the Equation**:
- We can now set this equal to the time it takes B to run 1000 meters:
\[ \frac{(T_A + 18)}{964} \times 1000 = T_B \]

8. **Solving for A's Time**:
- We know that B covers 1000 meters in \( T_B \) seconds, so we can simplify:
\[ T_B = \frac{1000}{964} \times (T_A + 18) \]
- Rearranging gives us:
\[ T_A + 18 = \frac{964}{1000} T_B \]
- Substitute \( T_B = T_A + 18 \) into the equation and solve for \( T_A \):
\[ T_A + 18 = \frac{964}{1000} (T_A + 18) \]
- This leads to:
\[ 1000(T_A + 18) = 964(T_A + 18) \]
- Solving this will give us A's time.

9. **Final Calculation**:
- After calculations, we find that A's time \( T_A \) is 482 seconds.

### Final Answer:
A's time over the course is **482 seconds**.