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2020.1 INTRODUCTION(i) Any contest of speed in running, driving, ridin---gor-rowing is called a Race(ii) The point from where a race begins is called the Starting Point(iii) The point where the race finishes is called the Finishing Point or Winning Post(iv) The person who first reaches the finishing point is called the Winner

20.2 lWO PARTICIPANTSIn a contest with 2 partictlants, one is the winner and the other is the loser.(i) The winner can giv fallow the loser a start of t seconds or x metres, i.e.

start distance = x ~ tres and start time = t seconds.(ii) The winner can be_L the loser by t seconds or x metres, i.e. beat distance = x metres and

beat time = t seco tis ~ Loser can run x metres in t seconds.Let A beats BWinner's (A) di(a) A beats B by x metre

+---L= Length of race --~X--------------Y,""*---x --~X-------+-------Y

Loser's (B) distance = L - x --L-x ----+ZA and B start together at XWhen A finishes at Y, Breaches Z .

(b) A gives B a start of x metreLoser's (B) distance = L - x X-------r-----------y-x --~Z-L-x----+

(c) A beats B by t secondsA starts at X, but B starts at Z i.e. x metresahead of A at the same moment.

X YA and B starts together at XA finishes at Y, but t secondsbefore B finishes,x--------------Y

Winner's (A) time= Loser's (B) time - t

(d) A gives B a start of t seconds. A starts t seconds after B starts at X

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0-2 Quantitative Aptitude for Competitive ExaminationsFrom the above figures, following formulae have been derived for a race (contest) of two participants,(a) Winner's distance = Length of race(b) Loser's distance = Winner's distance - (beat distance + start distance)(c) Winner's time = Loser's time - (beat time + start time)Winner's time Loser's time beat time + start time(d) Loser's distance Winner's distance beat distance + start distance(e) H a race ends in a dead lock, i.e. both reach the winning post together

then beat time = 0 and beat distance = 0

0.3 THREE PARTICIPANTSA, B, and C participate in a race. The length of the race is L metres.

Assuming, A as the winner, i.e. A gets the Ist position, in the race,if A beats B by x metres and~--L-------_O-----""*""-----A o

A beats C by y metres~----L --------;~-------~---AB x

he value of x and y will decide the lInd position in the race.r < y, then B will beat C, i.e. B gets the IInd position.r > y, then C will beat B, i.e C gets the llnd positionow. the following relation is used for three participants in a race of same length

L - xu) X z3 = L (xu - XI~ )ength of race = Lt beats llnd by a distance = xl2st beats IIIrd by a distance = xl3nd beats IIIrd by a distance = x23

if the length of race (L) changes in each case, i.e.t beats llnd by xl2 metres in a race of LI metrest beats IIIrd by xl3 metres in a race of ~ metresnd beats IIIrd by x23 metres in a race of L2 metres.

X12 ' x13 , and x23 are to be converted on a desired length of race, say, L metres.xb xl 2 xL. =t:

x12 ' xLl 3 =-L3X 23x2 3 = y- xL

2hen using the formula (L - x 1 2 ) xl3 =L (x1 3 - x12 )e unknowns can be found out.

C y

(a)

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Races 20-3Solved Example.

E-l In one kilometer race, A beats B by 36 metres or 9 seconds. Find A 's time over the course.S-1 Here A is the winner and B is the loser.Using the formula, 20.2

Winner's time beat time + start time-

E-2

beat distance + start distance9+0 . /

1,000-36 = 36+0 \A's time = ;6 x 964 s = 241 s.

A' s time over the course is 241 s.A runs 1~ as fast as B. IfA gives B a start of 30 metres, how far must be the winning post, so thatthe race ends in a dead heat?Assuming L = distance of the winning post such that the race ends in a dead heat, i.e. both theparticipants A and B reach the winning post at the same time.. . time taken by A = time taken by BL L - 30 ( d )> -- - --- Since t = V.V - V=> 3 L= 120

Loser's distanceA's time

=>

S-2

length of race (distance) of winning post is 120 m.E-3 P runs a kilometer in 4 min. and Q in 4 min. 10 seconds. How many metres start can P give Q in

a kilometer race so that the race may end in a dead heat?S -3 P runs a kilometer in 4 min. (= 240 seconds)Q runs a kilometer in 4 min. 10 s (= 250 seconds)

. . P can beat Q by 10 secondsBut if P gives Q a start of 10 seconds or x metres so that the race may end in a dead heat, i.e,beat time = 0, beat distance = O.Then, using the formula 20.2

Loser's time beat time + start timeWinner's distance beat distance + start distance

250 0+10=> --- => x = 401,000 O+xHence if P gives Q a start of 40 metres in a race of one kilometer, the race will end in a dead heat.E-4 P can run a kilometer in 4 min. 50 seconds and Q in 5 min. By what distance can P beat Q?S -4 Here, P is the winner and Q is the loser. Using the formula,

Loser's time beat time + start timeWinner's distance beat distance + start distance

300 10+0 100=> beat distance =3metre> =1000 beat distance + 0P beats Q by 331 metres in a kilometer race.

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20-4 Quantttative Aptitude for Competitive ExaminationsE-S Ina race of 100 metres, A beats B by 4 metres and A beats C by 2 metres. By how many metres would

C beat B in a 100 metres race?s - s Here, the number of participants = 3

Length of race = L = 100 metresA becomes the winner (I st ) andC gets IInd position (Since 2 metres < 4 metres)B gets llIrd positionUsing the formula,(L - x12) X z 3 = L (x13 - x1 2 ) whereIst (A ) beats IInd (C ) by xl2 = 2 metresIst (A ) beats llIrd (B ) by xl3 = 4 metresllnd (C ) beats llIrd (B ) by X z 3 = ?Length of race L = 100 metres= > (100 - 2) x x23 = 100 (4 - 2) = > x23 = 2.04 metresHence C would beat B by 2.04 metres in a 100 metres race.

[Refer 20.3J

E-6 x, y, and z are the three contestants in a kilometer race. If x can give y a start of 50 metres and x canalso give z a start of 69 metres, how many metres start y can give z?

S-6 Here x becomes Ist, y becomes IInd and z becomes llIrd in the race (Since 50 < 69)Using the formula [Refer 20.3J(L - xl2 ) x23 = L (xl3 - xl2 ) where,Ist (X) gives Ilnd (Y ) a start of xl2 = 50 metresIst (X) gives llIrd (Z) a start of xl3 = 69 metresTInd (Y) gives llIrd (Z) a start of x23Length of race L = 1,000 metres= > (1,000 - 50) X z 3 = 1,000 (69 - 50)= > x23 = 20 metresHence Y gives Z a start of 20 metres

E-7 Ina 100 metres race, A runs at a speed of 2 metres/so If A gives B a start of 4 metres and still beatshim by 10 seconds, find the speed of B.

S - 7 Here A is the winner and B is the loser.Using the formula [20.2]Loser's time - Winner's time = beat time + start time= > B's time - A's time = 10 + 0

B's distance A's distance= > = 10B'sspeed A'sspeed100-4 100= > - - = 10 = > B's speed = 1.6 metre/sB's speed 2

Hence the speed of B is 1.6 metre/so

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Races 20-5-8 A can run 330 metres in 41 seconds and Bin 44 seconds. By how many seconds will B win if he has

30 metres start.S-8 Bruns 330 metres in 44 seconds

Bruns (330 - 30) metres in 3~ X 300si.e .. 40 s

But A runs 330 metres in 41 secondsSo, B wins by (41 - 40) seconds, i.e. 1s. - - -

E-9 A can give B 40 metres start and A can give C 50 metres start in a 200 metres race, while B can giveC two seconds over the course. How long does each take to run 200 metres?

S-9 Here, A comes Ist, B comes IInd and C comes IIIrd in the contest.Using the formula, [R efer 2 0.3 J

(L - x12) x23 = L(x13 - XIV ~ (200 - 40) X x2 3 = 200 (50 - 40)25

X23 =2metres25Hence, B can beat C by2metres or 2 seconds,

. C 25 2 di.e. can run 2metres ill secon sC can run 200 metres in {5 X 200 = 32 seconds

2. . B takes (32 - 2), i.e. 30 seconds.and A takes i~x (200 - 40), i.e. 21 seconds.Hence A , B and C take 21 seconds, 30 and 32 seconds respectively to run 200 metres.A, B, and C are three participants in a kilometre race. If A can give B a start of 40 metres and B cangive C a start of 25 metres, how many metres A can give C a start?Here A is the winner (1 )Since B can give C a start, therefore B becomes II and C becomes illin the race.Using the formula,

(L - x1Vx23 = L( x13 - x12)we get (1,000 - 40) x 25 = 1,000 x (x13 - 40)~ 960 x 25 = 1,000 x (x13 - 40) ~ x13 = 64 metresHence, A can give C a start of 64 metres.In a race of 600 metres, A can beat B by 60 metres and in a race of 500 metres, B can beat C by25 metres. By how many metres will A beat C in a 400 metres race?

-l1 Here, length of race is different in each race. So, respective beat distance (given) is to be convertedto the desired length of race L (400 metres).A is the winner (Ist)Since B can beat C, therefore B becomes IInd and C becomes IIIrd in the race

X12 60X 1 2 = - - x L = - x 400 = 40 metresLl 600

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20-6 Quantitative Aptitude for Competitive Examinations /I x23 25X23 = L2 x L = 500 x 400 = 20 metres /

X I 3 = ?Using the formula,(L - x1 2) X v = L (xh - x12 )~ (400 - 40) x 20 = 400 (x1 3 - 40)

360 x 20~ X 1 3 = -4-0-0- + 40 ~ x 1 3 = 58 metre.

{Refer 20.3]

Hence A will beat C by 58 metres in a 400 metres race . ~ .E-12 In a 400 metres race, A gives B a start of 5 seconds and beats him by l',_metres. In another race of400 metres, A beats B by 7t seconds. Find their speeds.

S-12 In a 400 metres race,B takes 7t seconds more time than AIn another 400 metres race,B takes 5 seconds more time and runs 15 metres less distance than A.

B can run in ( 7 t - 5 ) seconds a distance of 1 5 metres.B can run in Iseconds a distance of 4

2-7speed of B is 7 metres/s, i.e. 7 metres

B takes 7t seconds more time than A to run 400 metres.also time = distan~espeeLets the speed of A be ~A metres/sthen 400 400 _ 71speed of B speed of A - 7

400 400 50~ - - - = ~ VA = 8 metres/s7 VA 7Hence, speed of A is 8 metres/soSpeed of B is 7 metres/s

Loser's time Beat time + Start time

I R EG ULA R PR OBLEM S I(1) Ina kilometre race, A beats B by 5 seconds or 40 metres. How long does B take to run the kilometre?

Hint : B is the loser

Winner's distance Beat distance + Start distance

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Races 20-7

(2) A runs 2 1 times as fast as B. If A gives B a s~ of 60 metres~must be the winning post,so that the race ends in a dead heat?Hint: See E - 2 .

(3) P can run a kilometre in 3 minutes 10 seconds and B in 3 minutes 20 seconds. By what distance canA beat B?

Loser's time Beat time +Start timeHint: Winner's distance Beat distance + Start distance(4) A can run one kilometer in half a minute less time than B . In a kilometre race, B gets a start of 100

metres and loses by 100 metres. Find the time A and B take to ~kilometre.Winner's time Loser's timeHint: Loser's distance Winner's distance

A is winner and B is the loser( t - ) t- - - - - - - - - - - - - - - - = 1,000,000 - (1,000 + 100)5 .t = " 2 mmutes

(5) A and B can run 200 metres in 22 and 25 seconds respectively. How far is B from the finishing linewhen A reaches it?

(6) A and B take part in a 100 metres race. A runs at 5 kmIh. A gives B a start of 8 metres and still beatshim by 8 seconds. Find the speed of B.Hint: See E-7.

(7) In a flat race, A beats B by 15 metres, and C by 29 metres. When B and C run over the coursetogether, B wins by 15 minutes Find the length of the course.Hint: Using (L - xd x23 = L (x13 - xl2)Where xl2 = A beats B = 15 metres

x2 3 = B beats C = 15 metresx 13 = A beats C = 29 metres

L = Length of course =? .. (L - 15)15 = L(29 - 15)~ L = 225 metres.

(8) A can give B a start of 20 metres and C a start of 39 metres in a walking race of 400 metres. Howmuch can B give C start?Hint: (L - xl2) X23 = L (x13 - xd

(400 - 20) x23 = 400 (39 - 20). . X z = 20 metres.

(9) A can run 100 metres in 11 seconds, and B 100 metres in 12 seconds. What start can be given to Bto make the race a dead heat?Hint: See E-3. . . .(10) A can run a kilometre in half a minute less time than B. In a kilometre race, B gets a start of 100metres and loses by 25 metres. Find the time A and B take to run a kilometre.-

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20-8 Quantitative Aptitude for Competitive Examinations \11) A and B ran a race which lasted a minute and a half. A gave B a start of 9 metres and beat him b)

1 metre. A ran 40 metres while B ran 39 metres. Find the length of the course.Hint: Let L = length of course

L40L-(9+1)

39 L = 400 metres.U) In a 500 metres race, B gives A a start of 160 metres. The ratio of the speeds of A and B is 2 : 3.

Who wins and by how much?Hint: Let after time t seconds, Breaches 500 metres, then, A reaches XA metres

XA -160500

(Since B gives A a start of 160 m)XA -160500

23

1XA = 493 - metres3

B beats A by (500 - 493 t metres) = 6 ~ metres.(13) Two men A and B ran 100 metres in 11.25 and 12.5 seconds respectively. If they start together, find

how far was B from the finish time, when A completed his 100 metres.Hint: See E-4.

14) P can run 440 metres in 51 seconds and Q in 55 seconds. By how many seconds will B win if hehas 40 metres start?Hint: See E-S.

t 15) A takes 4 minutes 50 seconds while B takes 5 minutes to complete the race. A beats B by 33~ metres.Find the length of the course.

Loser's time Beat timeBeat distanceHint: Winner's distance

Since Winner's distance = Length of course5x60L :. L = 1,000 metres.

16) P and Q run a kilometre and P wins by 1 minute. P and R run a kilometre, and P wins by 375 metres.Q and R run a kilometre and Q wins by 30 seconds. Find the time taken by P, Q and R to run akilometre.

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Races 20-9,-Answers

1. 125 s 2. 105 m 3. 50 m 4 . 2 m in 21 ID1D 5. 24 m, 26. 4.14 kmIhr 7 . 225 m 8. 20m 9 . 1 10. 31 . 4 .8- m - ID1D ID1D3 2 '

1 1 . 400 m 12. 61 1 3 . 10 m 14. 1 sec 15. 1,000 m316. 150 s, 210 s, 240 s.REAL PROBLEMS

(1) In a 200 metres race, A beats B by 20 metres, while in a 100 metres race, B beats C by 5 metres.A beats C in a kilometre race by(a) 105 metres(d) 145 metres

(b) 25 metres(e) None of these

(c) 150 metres

(2) X runs 1t times as fast as Y. IfX gives Ya start of 300 metres.. how far must X run before he catchesup with Y.(a) 1 km(d) 900 metres

(b) 400 metres(e) 600 metres

(c) 450 metres

(3) X runs 1 times as fast as Y. IfX gives Ya start of 200 metres, how far Y has run before he is caughtby X?(a) 1 km (b) 500 metres (c) 400 metres(d) 800 metres (e) 50 metres

(4) In a flat race, A beats B by 15 metres and C by 29 metres. When B and C run over the course together,B wins by 15 metres. The length of the course is(a) 150 metres (b) 225 metres (c) 275 metres(d) 300 metres (e) 250 metresH int: :. (L- xI-J ~3 = L (x13 - x12)

(5) In a flat race of L metres, A beats B by XI metres, beats C by X3 metres, B beats C by x2 metres, then(a) (L - x3) ~ = L X (x3 - Xl) (b) (L - X3) X 2 _ = L X (Xl - X3)(c) (L - Xl) ~ = L X (X3 - Xl) (d) (L - Xl) ~ = L X (Xl - X3)

(6) In a flat race of L metres, A beats B by Xl metres, beats C by x3 metres but C beats B by ~ metres,then(a) (L - x3) x2 = L X (x3 - Xl) (b) (L - X3) X2 = L X (Xl - x3)(c) (L - Xl) X2 = L X (X3 - Xl) (d) (L - Xl) ~ = L X (Xl - X3)In a flat race of L metres A beats B by Xl metres and beats C by X : 3 metres, then B will beat C, if(a) Xl = x3 (b) Xl > X3 (c) x3 > Xl (d) Xl X3 = 1 (e) Xl + X3 = 0

81 In a flat race of L metres, A beats B by Xl metres, and beats C by x3 metres; then C will beat B if(a) Xl = x3 (b) Xl > X3 (c) x3 > Xl (d) Xl X3 = 1 (e) Xl + X3 = 0

9) J is 1i times as fast as K. If ] gives K a start of 150 metres, how far must be the winning post sothat the race ends in a dead heat?(a) 100 metresCd) 550 metres

(b) 440 metres(e) 1 k:m

(c) 200 metres

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20-10 Quantitative Aptitude for Competitive ExaminationsHint: Race ends in dead heat, i.e. time taken by J and K are same.

J's distance K's distanceK's speed:. use t = J's speed

x-150k . . x = Length of race = 550 metres.

(10) Ina 500 metres race, A beats B by 50 metres or 8 seconds. Find A's time over the course.(a) 2 minutes (b) 1 minutes 12 seconds(c) 2 minutes 35 seconds (d) 1 minutes

(11) In a 300 metre race, A beats B by 5 seconds or 15 metres. A's time over the course is(a) 95 s (b) 100 s (c) 85 s (d) 90 s (e) 105 s

(U) X can run 20 metres while Y runs 25 metres. In a kiIometre race, Y beats X by(a) 150 metres (b) 200 metres (c) 250 metres(d) 175 metres (e) 300 metres

(13) In a 500 metres race, the ratio of speeds of two contestants A and B is 3 : 4. A has a start of 140metres, then (MBA '80)(a) B wins by 20 metres (b) A wins by 20 metres(c) B wins by 40 metres (d) B wins by 25 metres(e) A wins by 40 metres

(14) If A can run a kilometre in 3i minutes and B in 3t minutes, by what distance can A beat B?(a) 100 metres (b) 75 metres (c) 60 metres(d) 50 metres (e) 125 metres

Answers1 . ( d )10. (b) 3 . ( d )12. (b) 4 . ( b )13. (b)

5. (c)14. (d)

6. (b) 7. (c) 8. (b) 9. (d). (d)11. (a)

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