26. RACES AND GAMES

IMPORTANT FACTS

Races: A contest of speed in running, riding, driving, sailing or rowing is called race

Course: The ground or path on which contests are made is called a race course.

Starting Point: The point from which a race begins is known as a starting point.

Winning Point or Goal: The point set to bound a race is called a winning paint or a goal.

Winner: The person who first reaches the winning point is called a winner.

 Dead Heat Race: If all the persons contesting a race reach the goal exactly at the same time, then the race is said to be a dead heat race.

Start: Suppose A and B are two contestants in a race. If before the start of the race, A is at the starting point and B is ahead of A by 12 metres, then we say that 'A gives  B, a start of 12 metres. '
To cover a race of 100 metres in this case, A will have to cover 100 metres while B  will
have to cover only (100 - 12) = 88 metres. i
In a 100 m race, 'A can give B 12 m' or 'A can give B a start of 12 m' or 'A beats  12 m' means that while A runs 100 m, B runs (100 - 12) = 88 m.

Games: 'A game of 100, means that the person among the contestants who scores 100m first is the winner.
If A scores 100 points while B scores only 80 points, then we say that 'A can give B 20 points.

 SOLVED EXAMPLES :

Ex. 1. In a km race, A beats B by 28 metres or 7 seconds. Find A's time over the course.

Sol.  Clearly, B covers 28 m in 7 seconds.
:. B's time over the course = (278 x 1000) sec = 250 seconds.
:. A's  time over the course = (250 - 7-) sec = 243 sec = 4 min. 3 sec.

Ex. 2. A  runs 1 ¾  times as fast as B. if A gives B a start of 84 m, bow far must
winning post be so that A and B might reach it at the same time?

Sol. Ratio of the rates of A and B =  7/4  : 1   = 7 : 4.
So, in a race of 7 m, A gains 3m over B.
:. 3 m are gained by A in a race of 7 m.
:. 84 m are gained by A in a race of (7/3 x 84) m = 196 m.
:. Winning post must be 196 m away from the starting point.


Ex. 3. A can run 1 km in 3 min. 10 sec. and B can cover the same distance in  3 min. 20 sec. By what distance can A beat B ?

Soln:Clearly, A beats B by 10 sec.
Distance covered by B in 10 sec. = (1000 x 10 )m = 50 m.
        200            
Therefore  A beats B by 50 metres.


Ex .4 . In a 100 m race, A runs at 8km  per hour. If A gives B a start of 4 m and still him by 15 seconds, what is the speed of B ?

Sol: Time taken by A to cover 100 m  =(60 X 60 / 8000) x 100 sec = 45 sec.
B covers (100 - 4) m  =  96 m   in  (45 + 15) sec = 60 sec.
B's speed  = (96 x 60 x 60  )km/hr = 5.76 km/hr.
                        60 x 1000


Ex. 5. A, Band C  are three contestants in a km race. If A can give B a start of 40 m
    and A can give C  a start of 64m  how many metre's  start can B give C ?

     Sol:   While A covers 1000 m, B covers (1000 - 40) m = 960 m and
     C covers (1000 - 64) m or 936 m.
 When B covers 960 m, C covers 936 m.

 Ex 6. In a game of 80 points; A can give B 5 points  and C 15  points. Then how many points  B can give C  in a game of 60 ?

Sol.    A: B = 80 : 75,   A : C = 80 : 65.
B/C  = ( B/ A *  A/C)  = (75 / 80 * 80 / 65)  = 15/13 = 60 /52 = 60: 5
Therfore ,In a game of 60, B can give C  8 points.

-- - - - - - - - - - - - - - - - - - - - - - - - - - -¬