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

Case 1: Total distance of the race = 150m.
R gets a 5m head start, so R only needs to run:

Distance covered by R=150−5=145m
R beats S by 35m, which means when R reaches the finish line (145m covered), S is 35m behind the finish line.

Distance covered by S=150−35=115m
Since time taken by both is the same, the ratio of their speeds is equal to the ratio of distances covered:

Speed of S
Speed of R
​
=
115
145
​
=
23
29
​

Case 2: A new race of 100m. S gets a 25m head start.
Distance S needs to run = 100−25=75m.
Let's find out how much distance R covers in the time S takes to complete the remaining 75m:

Distance covered by R=75×
Speed of S
Speed of R
​
=75×
23
29
​
=
23
2175
​
≈94.565m
Since the total track is 100m and R has only covered 94.565m when S completes the race, S wins the race.
The margin of victory for S is:

Margin=100−94.565=5.435m≈5.44m
Thus, S wins by 5.44m.