IMPORTANT TERMS
(i) Race: A competition in running, riding, swimming, cycling etc.
(ii) Race Course: The plane on which competitions of races are contested.
(iii) Starting Point: The point from which the race starts .
(iv) Winning Point (Post) : The last point at which race is completed.
(v) Dead Heat: A contest of race in which all the competitors set the same time and no one is winner.
(vi) A Game of Hundred: A game in which the players agree that whoever first score hundred points is the winner.
(vii) while A scores 100 points, B scores only 75 points, then we say that 'In a game of 100 A can give B 25 points'.
(viii) In a race of 100 metres A beats B by 20 metres, means while A runs 100 metres B runs 80 metres.
Ques 1.
A can run 1 km in 5 minutes 48 seconds and B in 6 minutes. How many metres start can A give B in 1 km race so that the race may end in a dead heat?
Ans 1. A can give B (6 minutes - 5 minutes 48 seconds) or 12 seconds start. Now we must find the distance B can run in 12 seconds.
The distance run by B in 6 minutes = 1000 m
∴ The distance run by B in 12 seconds = 1000 x 12 / 6 x 60 = 100/3 = 33 1/3 m
∴ A can give B 33 1/3 metres' Start.Ques 2.
A and B run a km race A wins by 1 minute. A and C run a km race and A wins by 375 metres. B and C run a km race and B wins By 30 seconds .Find the time that C take to run a km race.
Ans 2. A beats B by 1 minute (60 seconds) and B beats C by 30 Seconds
∴ A beats C by ( 60 + 30) or 90 seconds
But A beats C by 375 m
∴ But A runs C by 375 m.
∴ C runs 1000 m in 90/375 x 1000 = 240 seconds = 4 minutes.
Ques 3.
In a race of 800 m, A can beat B by 74 m and in a race of 600 m, B beat C by 50 m. By how many metres will A beat C in a race of 500 m ?
Ans 3. While B runs 726 m, A runs 800 m. While, C runs 550 m, B runs 600 m.
. . While B runs 726m, C runs 550/600 x 726 = 665.5 m.
In a race of 800 m, A gives ( 800- 665.5) i.e. 134.5 m start to C, so in a race of 500 m A
gives 134.5/800x 500 m start to C i.e. 84.06 m start to C.
Ques 4.
In a game of billiards , A can give B 10 points in 60 and he can give C 15 in 60 .How many points can B give C in a game of 90 ?
Ans 4. A scores 60 while B scores 50, and C scores 45.
. . B scores 90 while C scores 45/50 x 90 = 81
Hence, in a game of 90 , B can give C ( 90-81) or 9 points .
Ques 5.
A can gives B 40 points, A can give C 64 points, and B ca give C 30 points .How many points do make the game ?
Ans 5. Let x points make the game, so that , according to the question,
ifA has x points,then B has ( x-40) points.
B has x points , then C has ( x- 30) Points.
C has (x -64) points , then A has x points
. . By chain rule
x x ( x-64) = ( x- 40) ( x- 30)x
=> x2 - 64x = x2 - 70x + 1200
=> x = 2000
Hence, 200 points make the game
Ques 6.
Two men A and B walk round a circle 1500 metres in circumference. A walks at the rate of 140 metres, and B at the rate 80 metres per minute .if they start from the same point, and walk in the same direction, when will they be together again ?
Ans 6. A and B will be together again for the first time , when A has gained one complete round on B.
Now A gains ( 140- 80) or 60 metres on B in 1 minute
. . A gains 1500 metres in 1500/60 or 25 minutes.
Ques 7.
Two men A and B run a 10 km race on a course 400 m round .If their rates be 5 : 4 , how often does the winner pass the other?
Ans 7. Since the rates are 5 :4 , a runs 5 rounds while B runs 4 rounds
. . A passes B each time that a has runs 5 rounds i.6. 5 x 2/6 or 3 kms
. . ( 400m = 2/5 km)
Now 2 kms is contained 5 times in 10 kms
. . A passes B 5 times.
Ques 8.
Three girls Seema , Kiran and Nitu walk round a circle 3 km in cricumference at the rates of 200, 150 and 125 metres per minute respectively. if they all start together and walk in the same direction,when will they first be together again?
Ans 8. Seema is quickest girl one complete round on Neetu, the slowest girl, in 3000/200-125 or 40 minutes .
Seema gains one complete round on Kiran , the next slowest girl in 3000/200-150 or 40 minutes.
Thus, Seema and Nitu are together after every 40 minutes , and Seema and Kiran are together after 60 minutes.
'Hence, Seema , Kiran and Nitu will be together in the time which is LCM of 40 and 60. The LCM of 40 and 60 is 120 minutes or 2 hours.
Ques 9.
A and B start from the same point and travel in the same direction round a circular track 3 km in circumference ,If their speed be 200 and 150 metres a minute, when will they first together again at the starting point ?
(a) 40 min
(b) 55 min
(c) 1 hour
(d) 45 min
L.C.m of 15 min and 20 min = 60 min = 1 hr.
Hence, they will meet first together at the starting point after 1 hour.
Ques 10.
Two men A and B run a 500 m race, A having 140 m start their speeds are 3 :4 . Then , A wins By:
(a) 10 m
(b) 20 m
(c) 40 m
(d) 60 m
Ans 10. Here, greatest have to cover a distance of ( 500-140) = 360 m to reach the winning point while B 500 m Now, A covers 3m while B covers 4 m
. . A covers 360 m while B covers 4/3 x 360 = 480 m
Hence, B will be ( 500- 480) = 20 m behind the winning post .
. . A wins by 20 min.
Ques 11.
in a race of 200 metres, A can give a start of 10 metres to A C give a start of 20 metres to B. The start that C can give to A, in the same race is
(a) 30m
(b) 29 m
(c) 27 m
(d) 25 m
Ans 11. B covers 200 m while A 190 m
B covers 180 m while A 190/200x180 = 171 m
Hence, C covers 200 m while A = 171 m
So, C can give to A ( 200 - 171) = 29 m Start.
Ques 12.
In a race of 300 metres A rates B by 15 metres or 5 seconds , Hoe much time A take to complete the race ?
(a) 105 sec.
(b) 100 sec.
(c) 95 sec.
(d) 90 sec.
Ans 12. From the question it is clear that speed of B = 15/5 = 3 m/s
the time taken by B to cover 300 m = 300 / 3 = 100 sec.
Hence, Time taken by A to complete the race = 100-5 = 95 sec.
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