Races a n d Games

Introduction 1. Race: A contest of speed is called a race. 2. Racecourse: The ground or path on which contests are

arranged is called a racecourse. 3. Starting Point: The point from where a race begins is

called the starting point. 4. Finishing Point: The point where the race finishes is

called finishing point or winning post. 5. Winner: The person who first reaches the finishing point

is called the Winner. 6. 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.

Now, suppose A and B are two participants in a race. If, before the start of the race, A is at the starting point and B is ahead of A by 25 metres, then A is said to give B a start of 25 metres. To cover a race of 100 metres in this case, A will cover a distance of 100 metres and B will cover 100 - 25 = 75 metres only. Note: In the above case, we may say that "A has given a lead of 25 metres to B."

7. Games: If we say that it is a game of 100, then the person among the participants who scores 100 points first is the winner. If, when A scores 100 while B scores only 80 points, then we say that "A can give 20 points to B" or, "A can give B 20 points" in a game of 100.

Rule 1 Involving Two Participants In a contest with two participants, one is the winner and the other is the loser. a) The winner can give or allow the loser a start of t seconds or x metres, i.e.

start distance = x metres and start time = t seconds. b) The winner can beat the loser by t seconds or x metres, i.e.

beat distance = x metres and beat time = t seconds Now, consider the following cases,

I. A beats B < L = Length of race •

Winner's (A) distance = L p , Q

A beats B by 'x' metres

Loser's (B) distance =<L-x) m p ! L-JC

A and B start together at P When A finishes at Q, B reaches R

II. A gives B a start of x metres X"' •(

Loser's (B) distance = ( L - x) m P * * i (

R* L-x *

A starts at P, but B starts at R at the same time.

III. A beats B by t seconds

A and B starts together at P Winner's (A) time = Loser's (B) time - 1

A finishes at Q but t seconds before B finishes

IV. A gives B a start of t seconds

A starts t seconds after B starts at P From the above figures, we have the following formulae for a race of two participants. (i) Winner's distance = Length of race (ii) Loser's distance = Winner's distance - (beat distance + start distance) (iii) Winner's time =• Loser's time - (beat time + start time)

(iv) Winner's time Loser's time Loser's distance Winner's distance

beat time + start time

beat distance + start distance (v) I f a race ends in a dead lock, i.e. both reach the winning

post together then beat time = 0 and beat distance = 0

Illustrative Example Ex.: In one kilometre race, A beats B by 36 metres or 9

seconds. Find A's time over the course. Soln: Here A is the winner and B is the loser.

Races and Games 507

C B

182 169

182

1 6 < { ^ U82

350 325

.-. C beats B by 350 - 325 = 25 m. Quicker Method: Applying the above theorem, we have

the required answer = 31-18

1,200-18

13

x350

x350 = 25 metres. 182

Exercise L In a 100 metres race, A beats B by 10 metres and C by 13

metres. In a race of 180 metres, B will beat C by: a) 5.4 metres b) 4.5 metres c) 5 metres d) 6 metres

2 In a km race A beats B by 100 metres and C by 200 metres, by how much can B beat C in a race of 1350 metres? a) 150 metres b) 160 metres c) 140 metres d) 13 5 metres

3. In a 100 metres race A can beat B by 10 metres, and B can beat C by 10 metres. By how much can A beat C in the same race? / a) 10m b)12m c)19m d) Can't be determined

4. A can beat B by 25 metres in a — km race, and B can

beat C by 20 metres in a — km race. By how much can A beat C in a km race? a) 130 m b) 126 m c) 136 m d) Data inadequate

5. In a race of600 m, A can beat B by 60 m and in a race of 500 m, B can beat C by 50 m. By how many metres will A beat C in a race of400 m?

1 a) 70 m b) 76 m c) 77 - m d) None of these

6. In a race of600 m, A can beat B by 50 m and in a race of 500 m, B can beat C by 60 m. By how many metres will A beat C in a race of400 m?

a) 76 m

Answers

b) 7 6 j m c)77m d) 77y m

( 200-100 } b - * 2 a ; H i n H T o ^ ^ J x l 3 5 ° = , 5 0 m e t r e s -3. c; Hint: We can apply the given rule, in this problem also.

Now, applying the given rule we have

/ xV-10 ,xl00 =io

U o o - i o J w

or,x-10 = 9 .-. x=19m. 4. c; Hint: A can beat B by (25 x 4 =) 100 metres in a km race

B can beat C by (20 x 2 =) 40 metres in a km race. Now, applying the given rule, we have ( x-100 U000-100 or,*- 100 = 36

xlOOO =40

x= 100 + 36 = 136 metres.

60 5. b; Hint: A can beat B by -r^r* 400 = 40 m in 400 m race

600

B can beat C by ^ * 400 = 40 m in 400 m race.

Let A will beat C in a race of400 m by x m. Now, applying the given rule we have

x-40 x400 =40

6.d

400-40 J or,x-40 = 36 .•. x=40+36=76m. Note: Try to solve this type of question by Rule - 2 also.

Rule 8 Theorem: A can give B x metres and Cy metres (y >x)ina R metres race, while B can give C't' seconds over the course. Then the time taken to run R metres by (i) A is given by

(R-xXR-y) . ( y - x ) .

seconds and (Hi) C is given by t

Illustrative Example

^ seconds,(ii) B is given by 1

fR-x^

R - y y - x

U - x J seconds.

\.

I d ; Hint: Here y 2 >y\, hence formula will change as r \ yi-y\

A can give B 20 m and C 25 m in a 100 m race, while B can give C one second over the course. How long does each take to run 400 m?

Soln: Detail Method: A:B:C = I00:80:75

xx.

. required answer = ( 13-10 > 3

xl80 = —xl80 100-10 j 90

= 6 metres.

, / l Q O ^ B:C = 80:75 = 8 0 [ W J : 7 5 | — J =

100 375

.Cruns 100 375 25 m in 1 second.

508

C runs 100 m in — * 100 = 16 seconds.

Now, B runs 100 m in 16 -1 = 15 seconds. And A runs 100 m in the same time as B runs 80 m

ie., J O O x ^ ~ ^ seconds.

Quicker Method: Applying the above theorem, we have

(i) time taken by A = 1 100

(l00-20Xl00-25) (25-20)

(ii) time taken by B = 1

80x75 500

100-25 25-20

= 12 sec.

= — = \5 sec. 5

(iii) time taken by G = 1 (100-20 "l 80 -

= — = 16 ^ 25-20 J 5

sec.

Exercise 1. A can give B 40 metres and C 82 metres in a 880 metres

race while B can give C 9 seconds over the course. Find the time C takes to run 880 metres. a)lmin b) 180 min c)3min d) 60 sec

2 A can give B 10 metres and C 20 metres in a 100 metres race. B can give C 1 second over the course of 100 metres. How long does each take to run 100 metres? a) 7.2 sec, 8 sec, 9 sec b) 6.2 sec, 8 sec, 10 sec c) 7.2 sec, 9 sec, 10 sec d) Data inadequate

3. A can give B 40 metres and C 80 metres in a 400 metres race. B can give C 4 seconds over the course of 400 metres. How long does A take to run 400 metres? a) 28 sec b) 28.2 sec c) 28.8 sec d) 29 sec

Answers

1. c; Hint: Required answer = 9

2. a; Hint: Time taken by A

880-40 82-40

(100-10)(100-20) 20-10

Time taken by B :

Time taken by C =

1 100

72 10

= 180 sec=3 min.

«7.2 sec

' l 0 0 - 2 0 \ _ 8 0 ~ 10 ^ 20-10

100-10 90 10

= 8 sec

= 9 sec

3. c; Hint: Time taken by A

^ 20-10 (400-40)(100-80)

80-40 4

400 = 28.8 sec

PRACTICE BOOK ON QUICKER MATHS

Rule 9 Theorem: In a game of 'x'points, A can give B x, points

and C x2 points ( x 2 > x , ) then B can give

X T — X i

x - x 1 J points.

Illustrative Example Ex.: In a game of 100 points, A can give B 25 points and C

31 points, then how many points can B give C? Soln: Detail Method: A : B: C = 100:75 :69

7> 100 75x

B:C = ^ = 75-69 — 100 92

100

69x-75

.-. B can give C 8 points. Quicker Method: Applying the above theorem, we have

( 31 — 25 N

the required answer = ^ ^ J Q O - 2 5

Exercise 1. In a game

100x6 „ . = ——— = 8 points.

5.

— e of 100 points, A can given B 20 points and C 28 points. Then, B can give C: a) 8 points b) 10 points c) 14 points d) 40 points In a game of250 points A can give B 50 points and C 70 points. How many can B give G? a) 20 points b) 25 points c) 30 points d) None of these A can give B 20 points, A can give C 32 points and B can give C 15 points. How many points make the game? a) 1000 b)100 c)500 d)250 A can give B 20 points in 100 and B can give C 20 points in 100. How many in 100 can A give C? a) 26 b)36 c)46 d)30 A can give B 25 points, A can give C 40 points and B can m\ / A r . 7 n " ">» t< - Urt, ./ «-» '•". ' n n i n t c m',1L'r> thf* ( r a m p 9 give C 20 points a)200 b)150 c)100 d)120 A can give B 15 points, A can give C 22 points and B can give C 10 points. How many points make the game?

b)60 c)80 d)90

toints, A can give C 40 points and B c How many points make the game? 50 C ) 100 d) 120

a) 50

Answers l .b ••

3b; Hint: * 32-20

2.b

= 15 V x-20 j or, 12x = 15x - 300 or, -3x=-300 • x= 100 points

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