Step 1: Understanding the Question:
The problem asks us to find the winning margin by which runner B will beat runner C in a new 180-meter race.
We are given data from a 100-meter race where A beats B by 10 meters.
In that exact same race, A beats C by 13 meters.
Step 2: Key Formula or Approach:
The phrase "A beats B by $X$ meters" implies that when A has completed the total race distance, B is exactly $X$ meters behind the finish line.
Because the runners are assumed to run at constant speeds, the ratio of the distances they cover in the same amount of time is exactly equal to the ratio of their respective speeds.
We will establish the ratio of the speed of B to the speed of C and use it to determine C's position when B finishes the 180m race.
Step 3: Detailed Explanation:
Let us consider the exact moment when runner A finishes the 100 m race.
At this instant, the distance covered by A is exactly 100 m.
Because A beats B by 10 m, the distance covered by B at this exact moment is $100 - 10 = 90$ m.
Similarly, because A beats C by 13 m, the distance covered by C at the same moment is $100 - 13 = 87$ m.
Since B and C have been running for the exact same amount of time, the ratio of their speeds is proportional to the distances they have covered.
Therefore, the ratio of the speed of B to the speed of C is given by:
\[ \frac{\text{Speed of B}}{\text{Speed of C}} = \frac{\text{Distance of B}}{\text{Distance of C}} = \frac{90}{87} \]
We can simplify this ratio by dividing both numbers by 3.
\[ \frac{\text{Speed of B}}{\text{Speed of C}} = \frac{30}{29} \]
Now, let us imagine a completely new race with a total distance of 180 meters.
When B finishes this new race, B will have covered exactly 180 m.
We need to find the distance covered by C in the same time it takes B to run 180 m.
Let the distance covered by C be denoted as $y$.
Using our established ratio, we can set up the proportion: \[ \frac{180}{y} = \frac{30}{29} \]
To solve for $y$, we can cross-multiply and isolate the variable.
\[ y = 180 \times \frac{29}{30} \]
Dividing 180 by 30 gives exactly 6.
\[ y = 6 \times 29 = 174 \text{ m} \]
This means that when B crosses the 180 m finish line, C is at the 174 m mark.
The distance by which B beats C is the remaining distance to the finish line.
Winning margin = $180 - 174 = 6$ meters.
Step 4: Final Answer:
In a race of 180 m, B will beat C by 6 m.