In a 150-m race, if R gets a 5-metre head start, then he would beat S by 35 metres. If S gets a 25-metre head start in a 100-metre race, then _______ would win the race by ________ metres.

Given:

In a 150-metre race, R gets a 5-metre head start and beats S by 35 metres.

Formula used:

Let the speed of R be r and the speed of S be s.

Time taken by R to run 145 metres = Time taken by S to run 115 metres

Time = Distance / Speed

Calculation:

145 / r = 115 / s

⇒ r / s = 145 / 115 = 29 / 23

In a 100-metre race, S gets a 25-metre head start:

So, R runs 100 metres and S runs 75 metres.

Let the time taken by R be t_R and time taken by S be t_S.

t_R = 100 / r

t_S = 75 / s

Using the speed ratio r/s = 29/23:

t_S = 75 / s = 75 / (23r/29)

⇒ t_S = 75 × (29 / 23r)

⇒ t_S = (75 × 29) / (23 × r)

⇒ t_S = (2175) / (23r)

Comparing t_R and t_S:

t_R = 100 / r

t_S = 2175 / (23r)

⇒ 100 / r > 2175 / (23r)

⇒ 100 > 2175 / 23

⇒ 100 > 94.57

Therefore, S finishes the 75 meters first. To find by how much S wins:

Distance covered by R when S finishes:

t_S = 75 / s

Distance covered by R in this time:

D = r × t_S = r × (75 / s)

Using r/s = 29/23:

D = r × (75 / (23r/29)) = 75 × (29 / 23)

D = 75 × (29 / 23) = 75 × 1.26 = 94.56

Therefore, R covers 94.56 meters when S finishes 75 meters.

Remaining distance for R:

100 - 94.56 = 5.44 meters

∴ The correct answer is S, winning by 5.44 meters.