In a 100 m race, A beats B by 10 m and C by 13 m. In a race of 180 m, B will beat C by

### Concept & Proportional Alignment When multiple competitors run the same race, their relative positions at the finish line can be used to set up a direct ratio between any two runners. ### Step-by-Step Solution - **Given:** In a $100$ m race, when $A$ is at $100$ m, $B$ is at $90$ m ($100 - 10$), and $C$ is at $87$ m ($100 - 13$). - This means while $B$ covers $90$ m, $C$ covers $87$ m. - Ratio of distances $B : C = 90 : 87 = 30 : 29$. - For a new race of $180$ m, $B$ will run the full $180$ m. - We calculate the distance $C$ will cover in the time $B$ runs $180$ m: $$ \text{Distance}_C = \frac{29}{30} \times 180 = 29 \times 6 = 174 \text{ m} $$ - The margin by which $B$ beats $C$ is $180 - 174 = 6$ m. ### Exam Strategy & Shortcut Look at the gap between $B$ and $C$ in the original scenario. $B$ is ahead of $C$ by $3$ m when $B$ has run $90$ m. For a $180$ m race (which is exactly double $90$ m), the gap will proportionally double. $3 \text{ m} \times 2 = 6 \text{ m}$. ### Common Pitfall Do not mistakenly assume the $3$ m gap between $B$ and $C$ remains constant regardless of the race length. The gap scales proportionally with the race distance. ### Final Answer Therefore, the correct answer is **6 m**.