Races and Games

A race is a contest in which competitors cover a fixed distance in the shortest possible time — running, riding, driving, sailing, and rowing are all common forms. The path the competitors follow is the racecourse, the point where the race begins is the starting point, and the point where it ends is the finishing point. Because a race is fundamentally about covering distance in time, every problem in this topic reuses the time–speed–distance relationships you've already studied, just applied to two or more competitors at once.

Several fixed phrases describe how competitors are compared:

A gives B a start of $x$ metres means B begins the race already $x$ metres ahead of the point where A starts. So to complete a race of $y$ metres, A must cover the full $y$ metres while B only needs to cover $(y - x)$ metres.

A beats B by $x$ metres means that at the moment A reaches the finish, B is still $x$ metres short of it — so in the same time, A has covered $y$ metres while B has covered only $(y - x)$ metres.

A gives B a start of $t$ minutes means B is allowed to begin $t$ minutes before A does, with both reaching the finish at the same instant. …