Important Facts and Formulae: 

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

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 called starting point.

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

Dead Heat Race:
If all the persons contesting a race reach the goal exactly 
at the same time,then the race is called 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 satrtint point and B is ahead
of A by 12 metres. Then we say that "A gives B a start 12 metres.

->To cover a race of 100metres in this case,A will have to cover 
100m while B will have to cover 88m=(100-12) 

->In a100m race 'A can give B 12m' or 'A can give B a start of 
12m' or 'A beats B by 12m'means that while A runs 100m B runs 88m.

Games:
A game of 100m,means that the person among the contestants who 
scores 100 points 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'.
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Problems:

1) In a 1 km race,A beats B by 28 m or 7sec.Find A's time over the
course? 
Sol: B covers 28 m in 7sec so,B's time over the course 
         = 7/28 *1000 =250 sec
    A's time over the course =250-7 =243 sec = 4min ,3 sec. 

2) A runs 1 3/4 times as fast as B.If A gives B a start of 84 m,how
far must be winning post be so that A and B might reach it at the
same time ?

Sol: Ratio of rates of A:B =7/4 :1 =7 :4 In a game of
7 m A gains 3m over B. 3m are gained by A in a race of 7m 84 m are
gained by A in a race of 7*84/3 =196 m Winning post must be 196m 
away from the starting point. 

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

Sol: clearly A beats B by
10 sec. Distance covered by B in 10 sec =1000/200 *10 =50 m A beats
B by 50 metres. 

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

Sol: 8000 m -------60*60 sec 100m ------- 60*60*1000/8000 =45 sec.
Time taken by A to cover 100m =45 sec. 
B covers 100-4 m =96 min 45 sec =60 sec 
B's speed =96 *60*60/60*1000 =5.76 km/hr 

5) A and B take part in 100m race .A runs at 5km per hour. A gives
B a start of 8 m and still beats him by 8 sec. What is the 
speed of B?

Sol : A'speed = 5km/hr =5*5/18 =25/18 m/s Time taken by A to
cover 100m =100*18/25 =7.2 sec 
Time taken by A to cover 92m = 72+8 =80 sec 
B's speed =92*18/80*5 = 4.14 kmph. 

6) A runs 1 2/3 times as fast as B.If A gives B a start of 80 m,
how far must the winning post be so that A and B might reach if 
at the same time? 

Sol: Ratio of the speed of A and B =5/3 :1 Thus
in a race of 5m ,A gains 2m over B 2m are gained by in a race of 
5m 80 m will be gained by A in a race of 5/2* 80 =200 m 

7) A,B and C are three contestans in a kmrace .If A can give B a 
start of 40 m and A can give C a start of 64 m,how amny metres 
start can b give C? 

Sol: A covers 100m,B covers (1000-40) =960 m C covers 1000-64 m 
or 936m when B covers 960 m,C covers 936 m when B covers 1000 m,
C covers 936*1000/960 m =975 m B can give C a start of 1000-975
 or 25 m. 

8) In a 100m race,A covers the distance in 36 sec and B in 
45 sec.In this race A beats B by?

Sol: Distance covered by B in 9secs =100*9/45 =20 m A beats
B by 20m 

9) In a 200 m race A beats B by 35m or 7 sec.What is the A's time
over the course? 

Sol: B runs 35 m in 7sec. B covers 200m in =7*200/35 = 40 sec.
B's time over the course =40 sec 
A's time over the course =40-7 =33 sec.

10) In a 300 m race A beats B by 22.5 m or 6 sec.What is the
B's time over the course? 

Sol: B runs 22.5 m in 6sec.
 B runs 300m in =6*300*2/45 =80 sec.
 B's time over the course =80 sec.
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11) A can run 22.5 m while B runs 25 m.In a kilometre race B 
beats A by? 

Sol: B runs 25 m ,A runs 45/2 m B runs 1000 A runs 
= 1000*45/2*25 =900m B beats A by 100m

12)In a 500 m race, the ratio of the speeds of two contestants 
A and B is 3:4.A has a start of 140 m.Then ,A win by B? 

Sol: The speeds of A and B =3:4 To reach the winning post A will
have to cover a distance of 500-140 m i.e 360 m while A covers 3m,
b covers 4m A covers 360m B covers 4/3*360=480m Thus when A 
reaches the winning post, b covers 480m and therefore remains 
20m behind. A wins by 20m.

13) In a 100m race,A can beat B by 25 m and B can beat c by 4m.
In the same race A can beat C by/ 

Sol: If A:B =100 :75 B :C=100 :96 then A :C =A/B*B/C=100/75* 100/96
               = 100/72 A beats C by 100 -72 m=28 m

14) In a 100 race,A can give B 10 mand C 28 m.In the same race B
can give C?

Sol: A:B =100 :90 A :C=100 :72 B:C =B/A *A/C =90/100*100/72
                                             =90/72 
When B runs 90 C runs 72 when B runs 100 C runs
            =72*100/90 B beats C by 20m

15) In a 100m race ,A beats B by 10 m and C by 13 m.In the race 
of 180m. B will beat c by? 

Sol :A : B =100 :90 A/B =100/90 A/C =100/87 B/C =B/A *A/C =90 /87 
When B runs 90m C runs 87 When B runs 180 m then C runs 
=87 *180/90 B beats C by 180-174= 6m

16) In a race of 200 m, A can beat B by 31 m and C by 18m.In arace 
of 350 m, C will beat B by?

Sol : In a race of 200 m A :B =200:169 A :C =200 :182 C/B =C/A*A/B 
=182/200*200/169 When C runs 182 m B runs 169 
when C runs 350 m B runs =350*169/182 =325m

17) In agame of 100 points A can give B 20 points and C 28 points 
then B can give C?

Sol: In a game of 100 points.A :B =100 :80 A :C =100 :72 B/C=B/A*A/C 
=80/100*100/72 = 80/72 when B runs 80m C runs 72 when B runs
 100m C runs =100*72/80 =90 B can give C 10 points in agame of 100.

18)At a game of billiards,A can give B 15 points in 60 and A can 
give C 20 points in60.How amny points can B give C in a game 
of 90? 

Sol: A:B =60:45 A:C =60:40 B/C =B/A*A/C =45/60*60/40 =90/80
 B can give C 10 points in agame of 90.

19) in agame of 80 points,A can give B 5 points and C 15 points.
Then how many points B can give C in agame of 60? 

Sol: A :B =80 :75 A :C =80:65 B/C=B/A*A/C = 75/80*80/65 
                                          =15/13 B:C =60:52


    

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