RACES AND GAMES OF SKILL

Race: A contest of speed over a specified distance is called race.

Eg: Running, Driving, Riding, Sailing, Rowing, etc… are the areas where the Race is concerned.

Race Course: The ground or path on which the races are arranged is called a race course.

Note:

1. The point from where a race begins is called the starting point.
2. The point where the race finishes is called the winning post or finishing point or goal.
3. The person who first reaches the goal is the winner.

Dead-heat Race: If all the contestants of a race reach the goal exactly at the same time, then the race is called a dead-heat race.

Important Note:

If P and Q are two contestants in a race, then the following statements and their corresponding meanings are very useful in solving the problems.

Statements	Mathematical Interpretation
P beats Q by t seconds	P finishes the race t seconds before Q finishes.
P gives Q a start of t seconds	P starts t seconds after Q starts from the same point.
P gives Q a start of x metres	While P starts at the starting point, Q starts x metres ahead from the starting point at the same time.
Game of 100	A game in which the participant scoring 100 points first is the winner.
In a game of 100, “P can give Q 20 points”	While P scores 100 points, Q scores only 100 – 20 = 80 points.

Formulae:

Two participants

1. If P is n times as fast as Q and P gives Q a start of x metres, then the length of the race course, so that both P and Q reach the winning post at the same time must be,

metres

1. P can run x m race in t₁ seconds and Q in t₂ seconds, then P beats Q by a distance

metres    [Here t₂ > t₁]

Three participants:

Suppose A, B, and C participate in a race. The length of the race is L meters. If A is the winner , i.e A gets the first position, in the race.

If A beats B by x meters              and                 A beats C by y meters

◄══════ L ══════════►                                                      ◄══════ L ════════════►

O———–­­­–○—————A                                 O———-­­­–○——————-A

B<————->                                                    C<—————->

X mts                                                                           y mts

The value of x and y decide will decide the II position in the race.

If x < y then B will beat C, i.e B gets the II position.

If x > y then C will beat B, i.e C gets the II position.

The following rule is used for three participants in a race of same length.

(L – x₁₂ ) x₂₃ = L (x₁₃ – x₁₂ )