#### IMPORTANT TERMS

(i)

**Race:**A competition in running, riding, swimming, cycling etc.
(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

**Ans 9.**

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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