In a kilometre race, A, B and C are three participants. A can give B a start of 50 m and C a start of 69 m. The start which B can allow C, is

### Concept & Start Allowances When a runner "gives a start" to another, it means the first runner covers the full race distance while the second runner covers the total distance minus the start margin in the exact same time. ### Step-by-Step Solution - **Given:** Race = $1000$ m. - When $A$ runs $1000$ m, $B$ runs $1000 - 50 = 950$ m. - When $A$ runs $1000$ m, $C$ runs $1000 - 69 = 931$ m. - This establishes the simultaneous positions of $B$ and $C$: when $B$ covers $950$ m, $C$ covers $931$ m. - We need to find the start $B$ can give $C$ in a full $1000$ m race. - When $B$ runs $1000$ m, distance covered by $C$ is: $$ \text{Distance}_C = \frac{931}{950} \times 1000 = \frac{931}{95} \times 100 = 9.8 \times 100 = 980 \text{ m} $$ - Thus, $B$ can allow $C$ a start of $1000 - 980 = 20$ m. ### Exam Strategy & Shortcut Compare $B$ and $C$ directly. $C$ lags behind $B$ by $19$ m ($950 - 931$) in a $950$ m stretch. To find the lag over $1000$ m, scale it proportionally: $\frac{19}{950} \times 1000 = \frac{1}{50} \times 1000 = 20$ m. ### Common Pitfall A very common error is to simply subtract the two starts ($69 - 50 = 19$ m). This is incorrect because $19$ m is the gap when $B$ is at $950$ m, not at the finish line ($1000$ m). ### Final Answer Therefore, the correct answer is **20 m**.