In a running race, when one runner allows another runner to stay ahead at the start of the race, then it is termed as startup and when runners reach the finishing line at the same time, then it is termed as dead heat. In a race of 4 km distance, Anu wins by 600m over Binu. Binu can give a startup of 200m to Caira in a 4 km race. By how much distance should Caira get startup so that the race between Anu and Caira ends in dead heat in the same race of 4 km?
To solve this problem, let's first understand the distances involved: - Anu wins by 600 meters over Binu in a 4 km (4000 meters) race. This means when Anu finishes 4000 meters, Binu has run 3400 meters. - Binu can give a 200-meter startup to Caira in a 4 km race. This means when Binu finishes 4000 meters, Caira has run 3800 meters. Now we need to find the startup distance that Caira should get so that she finishes at the same time as Anu. 1. Calculate the speeds of Anu, Binu, and Caira relative to each other. - Anu's speed relative to Binu: Anu runs 4000 meters while Binu runs 3400 meters. \[ \text{Relative speed of Anu to Binu} = \frac{4000}{3400} = \frac{20}{17} \] - Binu's speed relative to Caira: Binu runs 4000 meters while Caira runs 3800 meters. \[ \text{Relative speed of Binu to Caira} = \frac{4000}{3800} = \frac{20}{19} \] - Anu's speed relative to Caira: \[ \text{Relative speed of Anu to Caira} = \frac{20}{17} \times \frac{20}{19} = \frac{400}{323} \] 2. Use the relative speed to find the required startup distance. - When Anu runs 4000 meters, we need to find the distance Caira should run for them to finish simultaneously. - Let the required startup distance for Caira be \( x \) meters. 3. Set up the equation to solve for \( x \): \[\text{Anu's distance} = 4000 \text{ meters}, \quad \text{Caira's distance} = 4000 - x \text{ meters} \] \[ \frac{4000}{4000 - x} = \frac{400}{323} \] \[4000 \times 323 = 400 \times (4000 - x)\] \[ 1292000 = 1600000 - 400x \] \[ 400x = 1600000 - 1292000\] \[400x = 308000 \] \[ x = \frac{308000}{400} = 770\]Therefore, Caira should get a startup of 770 meters. Answer: C (770m)
