ds dstime sd s peed distance

8/9/2019 ds dsTime Sd s peed Distance

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Speed ,Time & Success

• Prologue

•

Speed(S)• Time (T)

• Distance (D)

• Time & Work (In a nutshell)

• Easy Methods

• Soled Pro!lems

• Permutations and com!inations

• Prime "um!er & "um!er Se#uence in $rie%

ust %ollo'ing %rom my preious article on ptitude atio & Proportion and Pro!a!ility*

+ere is the link http--mrunal*org-./01-/0-'rite.'in2result2o%2decem!er2./032competition2%ree2

notes2%or2eeryone2iran2deal2critically2endangered2species2ready2re%erence2ta!le2pressure2

group2ir2economy2polity2and2more*html

ust an aspirant like you, so do %orgie me i% you %ind any mistakes* 4hosen Speed, Time and

distance topic !ecause it5s a %aorite topic %or all pu!lic e6ams

Speed, Time & Distance (STD) are al'ays in a committed triangle relationship

Speed 2 Speed is ho' %ast something is going* nother 'ay to think o% this is as ho' %ar you can

go in a certain amount o% time or ho' %ast I am 'riting this article (!t' it took me 1 hours to

complete this article)* Measured as distance traeled per unit o% time*

E6ample The speed o% these cars is oer 07/ kilometers per hour (07/ km-h)

http://mrunal.org/2014/01/write2win-result-of-december-2013-competition-free-notes-for-everyone-iran-deal-critically-endangered-species-ready-reference-table-pressure-group-ir-economy-polity-and-more.htmlhttp://mrunal.org/2014/01/write2win-result-of-december-2013-competition-free-notes-for-everyone-iran-deal-critically-endangered-species-ready-reference-table-pressure-group-ir-economy-polity-and-more.htmlhttp://mrunal.org/2014/01/write2win-result-of-december-2013-competition-free-notes-for-everyone-iran-deal-critically-endangered-species-ready-reference-table-pressure-group-ir-economy-polity-and-more.htmlhttp://mrunal.org/2014/01/write2win-result-of-december-2013-competition-free-notes-for-everyone-iran-deal-critically-endangered-species-ready-reference-table-pressure-group-ir-economy-polity-and-more.htmlhttp://mrunal.org/2014/01/write2win-result-of-december-2013-competition-free-notes-for-everyone-iran-deal-critically-endangered-species-ready-reference-table-pressure-group-ir-economy-polity-and-more.htmlhttp://mrunal.org/2014/01/write2win-result-of-december-2013-competition-free-notes-for-everyone-iran-deal-critically-endangered-species-ready-reference-table-pressure-group-ir-economy-polity-and-more.html


2/20

ust a passing e%erence di%%erence !et'een speed and elocity

8elocity 2 8elocity is speed 'ith a direction*

So i% something is moing at 7 km-h that is a speed*

$ut i% you say it is moing at 5 km/h westwards that is a velocity*

I% something moes !ack'ards and %or'ards ery %ast it has a high speed, !ut a lo' (or 9ero)

elocity

Time : Precious thing (I guess no need to e6plain time)

Distance is the space !et'een t'o o!;ects or points (So in our e6ample distance is starting point

o% car to its %inish line)

Unit of Measurement

/ ? >/ = 3>// sec

0 km-hr = 0///-3>//

= 7-0@ m-s


3/20

e*g*

Aor simple calc

0 km-hr = /*.B m-s

0 m-s = 3*> km-h

"o' let5s turn our attention %rom more !asic concepts to e6am oriented #uestions and 'ays to

sole it

Some important %ormulas and you can sole any #uestions related to (STD)

0) I% a TI" coers a certain distance at 6 km-ph and an e#ual distance at y km-hr ,the

aerage speed o% the 'hole ;ourney =2 xy

x+ y

Aor e*g* I% a train coers Pune to Mum!ai .7/ km at 6=7/ km-hr and Mum!ai to pune

.7/ km at y=>/ km-hr

Then aerage speed2∗50∗6050+60

= 71*71 km-hr

.) Speed and time are inersely proportional ('hen distance is constant) ⇒ Speed∝

1

time ('hen distance is constant)

This means as the speed increases the time decreases (or the time taken is less)*

3) I% the ratio o% the speeds o% and $ is a !, then the ratio o% the times taken !y them to

coer the same distance is ! a or1

a 1

b

These are the only %ormulas re#uired %or STD type o% pro!lems*

"o' relating this to train %ormulas as %or train also speed, time and distance logic 'ill !e

same* The only di%%erence 'ill !e 'hen a pole or a man or any C6y9 o!;ect standing*

aymen e*g* The Monorail started in Mum!ai (%rom 4hem!ur to Wadala)* Imagine its

length is 0// m


4/20

0) Time taken !y a train o% length d1 meters to pass a pole or standing man or a signal

post is e#ual to the time taken !y the train to coer d1 meters*ns'er = Time taken %or the train to coer 0// meters

.) Time taken !y a train o% length d1 meters to pass a stationery o!;ect (chem!ur

station= 7/ m) o% length d. meters is the time taken !y the train to coer (d1 + d2)

meters*

ns'er = Time taken %or the train to coer 07/ meters

3) Suppose t'o trains or t'o o!;ects !odies are moing in the same direction at s1 m-s

and s2 m-s, 'here s1 F s2, then their relatie speed is = ( s1 : s2) m-s*

1) Suppose t'o trains or t'o o!;ects !odies are moing in opposite directions at s1 m-s

and s2 m-s, then their relatie speed is = (s0Gs.) m-s*

Easy = opposites attract* nd any kind o% attraction is a positie thing


5/20

7) I% t'o trains o% length d1 meters and m2 meters are moing in opposite directions at s1 m-s and s2 m-s, then

The time taken !y the trains to cross each other = d1+d 2s1+s2 *

ew !ro"lems with super short cut methods

0) monorail coers a distance (d0) o% 7 km 'ith a speed (s0) o% 1 km-hr and ne6t > km'ith a speed o% 3 km-hr in traelling %rom to $* What 'ill !e the train5s aerage speed

during the 'hole ;ourneyH

Solution = I% d0=%irst distance=7 km

nd d.=second distance=> km (gien in a!oe e6ample)nd s0=%irst speed=1 km-hr

nd s.=second speed=3 km-hr

Then aerage speed=( d 1+d2 ) s1 s2(d 1 s2+d 2 s1)

=(∑ of t h e distances)∗( product of t h e speed)

cross multiplication of distance∧speed

Putting the alue o% d0, d., s0 and s. %rom the gien e6ample in the a!oe %ormulaWe get

erage speed=(6+5 )4∗3

((5∗3 )+(6∗4))

erage speed= (006163)- (07G.1)

= (006163)-3

= (0060.)-3

=3*3@ km-hr

4ontinuing 'ith the concept o% Monorail only

.) man reaches Wadala station(to !oard mono rail) %rom home late !y 3/ minutes %rom

his scheduled time i% he 'alks at a speed(s0) o% 7 km-h, !ut i% he 'alks at a speed(speed


6/20

s.) o% > km-h, he 'ill reach his home 7 minutes early* What 'ill !e the distance %rom

station to homeH

ule2 i% a person reaches his destination t0 time early !y 'alking at a speed o% s0 and t. time

later !y 'alking at a speed o% s., then the distance !et'een !oth places is

= s1 s2( t 1+t 2)(s1−s 2)60

= Products of t h e speeds(∑ of time)

(difference of speed)60 "o' e#uate the #uestion gien a!oe to the rule also gien ;ust a!oe*

"o' t0=3/ minutes, t.=7 minutesnd s0=7 km-h, s.= > km-h

=5∗6(30+5)(5−6)60

= 0B*7 km

3) I% a monorail does a ;ourney in J+J hrs, the %irst hal% at Js0J km-hr and the second hal% at

Js.J km-hr* The total distance coered !y the car

=2∗time∗s1∗s2

s1+s2

monorail does a ;ourney in 0/ hrs, the %irst hal% at .0 km-hr and the second hal% at .1 km-hr*Aind the distanceH

ns Distance = (. 6 0/ 6 .0 6 .1) - (.0G.1)

= 0//@/ - 17

= ..1 km*

The 'hole #uestions on Time speed and distance and Train reole around these !asic %ormulas*

$ut practice is needed to sole it more e%%iciently

!ermutations and com"inations

K0) 'hat is com!inationH

We al'ays use the 'ord com!ination in our day to day li%e*e*g "The fruit salad is a combination of apples, grapes and bananas" We donJt care 'hat order

the %ruits are in, they could also !e L!ananas, grapes and applesL or Lgrapes, apples and !ananasL,

its the same %ruit salad*


7/20

e*g* The ./ includes a com!ination o% the largest deeloped and industriali9ed countries, 'hich

make up nearly @7N o% the 'orld5s economy

I% the order doesnJt matter, it is a 4om!ination

K.) What is permutationH

The 'ord permutation is though less used !ut o%ten applied in our eeryday li%e*

e#$#

"The combination to my bank password is 472"(not really just a hypothetical one)* "o' 'e

do care a!out the order* LB.1L 'ould not 'ork, nor 'ould L.1BL* It has to !e e6actly %'*

So, in Mathematics 'e use more precise language

I% the order doesnJt matter, it is a )om"ination*

I% the order does matter it is a !ermutation

So let5s %ocus on permutation %irst (co9 order do matters in li%e)

!ermutation

Permutation is an ordered 4om!ination*

More easy to remem!er O permutation = position (p**p)


8/20

There are !asically . types o% permutation

0) Permutation 'ith repetition

!oe e6ample : I can keep my !ank pass'ord as 1B1 or 111 also (note in this the order matters,

!ut the num!ers are repeated)

Technically speaking

When 'e hae n things to choose %rom *** 'e hae n choices each time

When choosing r o% them, the permutations are

n * n * ### (r times)

(In other 'ords, there are n possi!ilities %or the %irst choice, "D T+E" there are n possi!ilities

%or the second choice, and so on, multiplying each time*)

Which is easier to 'rite do'n using an e6ponent o% rH

n * n * ### (r times+ nr

E6ample in the a!oe !ank pass'ord, there are 0/ num!ers to choose %rom (/, 0,***) and 'e

choose 3 o% them

-. * -. * ### ( times+ -.

-,... permutations

So, the %ormula is simply

.+ !ermutation without 0epetition

In this case, 'e hae to reduce the num!er o% aaila!le choices each time*

In the a!oe !ank pass'ord e6ample*

We hae to create a 3 num!er pass'ord out o% the 0/ num!ers (/,0,.,3,1,7,>,B,@,)* $ut

remem!er repetition is not allo'ed*

So 'e choose num!er 1 (out o% ten options 'e hae chosen one)*

nr

'here n is the num!er o% things to choose

%rom, and 'e choose r o% them(epetition allo'ed, order matters)


9/20

We choose num!er B (as 1 is already taken and repetition is not allo'ed so no' 'e can choose a

num!er out o% nine)

We choose num!er . (as 1 and B is already taken and repetition is not allo'ed so no' 'e canchoose a num!er out o% eight)

'hat i% 'e 'anted to select ;ust 3, then 'e hae to stop the multiplying a%ter 01* +o' do 'e do

thatH There is a neat trick *** 'e diide !y B1 ***

-. * 2 * 3 * ' * 4 ###

-. * 2 * 3 '.

'* 4 ###

The %ormula is 'ritten


%rom, and 'e choose r o% them

("o repetition, order matters)

Some e6amples (It5s ery easy* ust %igure i% repetition is there or not)

0)

Aor e6ample, 'hat order could 0> pool !alls !e

inH

%ter choosing, say, num!er L01L 'e canJt

choose it again*

So, our %irst choice 'ould hae 0> possi!ilities, and our ne6t choice 'ould then hae 07

possi!ilities, then 01, 03, etc* nd the total permutations 'ould !e

-4 * -5 * -% * - * ### .,2,'32,333,...

$ut may!e 'e donJt 'ant to choose them all, ;ust 3 o% them, so that 'ould !e only

-4 * -5 * -% ,4.


10/20

In other 'ords, there are 3,3>/ di%%erent 'ays that 3 pool !alls could !e selected out o% 0> !alls*

.) +o' many 'ays can %irst and second place !e a'arded to 0/ peopleH

-.1

=

-.1

=

,43,3..

2.

(-.+1 31 %.,.

('hich is ;ust the same as -. * 2 2.)

ote

Instead o% 'riting the 'hole %ormula, people use di%%erent notations such as these

E6ample (-.,+ 2.

)om"inations

There are also t'o types o% com!inations (remem!er the order does not matter no' = %ruit

salad )

0* 0epetition is 6llowed such as coins in your pocket (7,7,7,0/,0/)

.* o 0epetition such as lottery num!ers (.,01,07,.B,3/,33)

)om"inations without 0epetition

This is ho' +ousie game 'orks* The num!ers are dra'n one at a time, and i% 'e hae the luckynum!ers (no matter 'hat order) 'e 'in


11/20

It does not matter i% is coming %irst or last* s 'e complete striking all the num!ers 'e are

'inners (ust an hypothetical e6ample)

The easiest 'ay to e6plain it is to

• assume that the order does matter (ie permutations),

• then alter it so the order does not matter*

oing !ack to our pool !all e6ample, letJs say 'e ;ust 'ant to kno' 'hich 3 pool !alls 'ere

chosen, not the order*

We already kno' that 3 out o% 0> gae us 3,3>/ permutations*

$ut many o% those 'ill !e the same to us no', !ecause 'e donJt care 'hat order

Aor e6ample, let us say !alls 0, . and 3 'ere chosen* These are the possi!ilites

0) Qrder does matter .) Qrder doesnJt matter

0 . 30 3 .

. 0 3

. 3 03 0 .

3 . 0

0 . 3


12/20

So, the permutations 'ill hae > times as many possi!ilities*

In %act there is an easy 'ay to 'ork out ho' many 'ays L0 . 3L could !e placed in order, and 'e

hae already talked a!out it* The ans'er is

1 * * - 4

(nother e6ample 1 things can !e placed in %1 % * * * - % di%%erent 'ays, try it %oryoursel%)

So 'e ad;ust our permutations %ormula to reduce it !y ho' many 'ays the o!;ects could !e in

order (!ecause 'e arenJt interested in their order any more)

That %ormula is so important it is o%ten ;ust 'ritten in !ig parentheses like this


%rom, and 'e choose r o% them

("o repetition, order doesnJt matter)

It is o%ten called Ln choose rL (such as L0> choose 3L)

ote

s 'ell as the L!ig parenthesesL, people also use these notations

78ample

So, our pool !all e6ample (no' 'ithout order) is

-41

=

-41

=

.,2,'32,333,...

54.

1(-4+1 1*-1 4*4,',..,3..


13/20

Qr 'e could do it this 'ay

-4*-5*-%

=

4. 54.

**- 4

So remem!er, do the permutation, then reduce !y a %urther Lr1L

.) We 'ill do it %or the !ank pass'ord e6ample

We hae to choose 3 num!er out o% 0/ (order does matter) = 0/ R R @ = B./

Qut o% the 3 num!ers chosen order does not matter = 3 = >

=720

6 = 0./ (com!inations)

.) 4om!inations 'ith repetition (It5s a !it tough* ou can %ollo' Mrunal Sir article onPermutation and com!ination)

et us say there are %ie %laors o% ice2cream "anana, chocolate, lemon,

straw"erry and vanilla* We can hae three scoops* +o' many ariations 'illthere !eH

etJs use letters %or the %laors !, c, l, s, U* E6ample selections 'ould !e

• c, c, cU (3 scoops o% chocolate)

• !, l, U (one each o% !anana, lemon and anilla)

• !, , U (one o% !anana, t'o o% anilla)

(nd ;ust to !e clear There are n5 things to choose %rom and 'e choose r o% them*

Qrder does not matter, and 'e can repeat)

"o', I canJt descri!e directly to you ho' to calculate this, !ut I can sho' you a special

techni9ue that lets you 'ork it out*


14/20

Think a!out the ice cream !eing in !o6es, 'e could say Lmoe past the %irst !o6, then take 3 scoops, then moe along 3 more

!o6es to the endL and 'e 'ill hae 3 scoops o% chocolate

So it is like 'e are ordering a ro!ot to get our ice cream, !ut it

doesnJt change anything, 'e still get 'hat 'e 'ant*

We could 'rite this do'n as (arro' means move, circle means scoop)*

In %act the three e6amples a!oe 'ould !e 'ritten like this

c, c, cU (3 scoops o% chocolate)

!, l, U (one each o% !anana, lemon and anilla)

!, , U (one o% !anana, t'o o% anilla)

QV, so instead o% 'orrying a!out di%%erent %laors, 'e hae a simpler #uestion Lho' many

di%%erent 'ays can 'e arrange arro's and circlesHL

"otice that there are al'ays 3 circles (3 scoops o% ice cream) and 1 arro's ('e need to moe 1

times to go %rom the 0st to 7th container)*

So (!eing general here) there are r ! (n#) positions, and 'e 'ant to choose r o% them to hae

circles*

This is like saying L'e hae r ! (n#) pool !alls and 'ant to choose r o% themL* In other 'ords it

is no' like the pool !alls #uestion, !ut 'ith slightly changed num!ers* nd 'e 'ould 'rite itlike this

'here n is the num!er o% things to choose%rom, and 'e choose r o% them

(epetition allo'ed, order doesnJt matter)

Interestingly, 'e could hae looked at the arro's instead o% the circles, and 'e 'ould hae then !een saying L'e hae r ! (n#) positions and 'ant to choose (n#) o% them to hae arro'sL, and

the ans'er 'ould !e the same ***


15/20

So, 'hat a!out our e6ample, 'hat is the ans'erH

(5:-+1

=

'1

=

5.%.

5

1(5-+1 1*%1 4*%

!rime num"er and um"er Se9uences in short

!rime um"ers

prime num!er can !e diided, 'ithout a remainder, only !y itsel% and !y 0* Aor e6ample, 0B

can !e diided only !y 0B and !y 0*

Some %acts

• The only een prime num!er is .* ll other een num!ers can !e

diided !y .*

• I% the sum o% a num!erJs digits is a multiple o% 3, that num!er can !e diided !y 3*

• "o prime num!er greater than 7 ends in a 7* ny num!er greater than 7 that ends in a 7

can !e diided !y 7*

• ero and 0 are not considered prime num!ers*

• E6cept %or / and 0, a num!er is either a prime num!er or a composite num!er*

composite num!er is de%ined as any num!er, greater than 0, that is not prime*

To proe 'hether a num!er is a prime num!er, %irst try diiding it !y ., and see i% you get a

'hole num!er* I% you do, it canJt !e a prime num!er* I% you donJt get a 'hole num!er, ne6t trydiiding it !y prime num!ers 3, 7, B, 00 ( is diisi!le !y 3) and so on, al'ays diiding !y a

prime num!er (see ta!le !elo')*

+ere is a ta!le o% all prime num!ers up to 0,///

. 3 7 B 00 03 0B 0 .3


16/20

. 30 3B 10 13 1B 73 7 >0 >B

B0 B3 B @3 @ B 0/0 0/3 0/B 0/

003 0.B 030 03B 03 01 070 07B 0>3 0>B

0B3 0B 0@0 00 03 0B 0 .00 ..3 ..B

.. .33 .3 .10 .70 .7B .>3 .> .B0 .BB

.@0 .@3 .3 3/B 300 303 30B 330 33B 31B

31 373 37 3>B 3B3 3B 3@3 3@ 3B 1/0

1/ 10 1.0 130 133 13 113 11 17B 1>0

1>3 1>B 1B 1@B 10 1 7/3 7/ 7.0 7.3

710 71B 77B 7>3 7> 7B0 7BB 7@B 73 7

>/0 >/B >03 >0B >0 >30 >10 >13 >1B >73

>7 >>0 >B3 >BB >@3 >0 B/0 B/ B0 B.B

B33 B3 B13 B70 B7B B>0 B> BB3 B@B BB

@/ @00 @.0 @.3 @.B @. @3 @73 @7B @7

@>3 @BB @@0 @@3 @@B /B 00 0 . 3B

10 1B 73 >B B0 BB @3 0 B

)ommon um"er !atterns

Numbers can have interesting patterns.

Here we list the most common patterns and how they are made.

6rithmetic Se9uences


17/20

n rithmetic Se#uence is made !y addin$ some alue each time*

Example:

0, 1, B, 0/, 03, 0>, 0, .., .7, ***

This se#uence has a di%%erence o% 3 !et'een each num!er*

The pattern is continued !y addin$ to the last num!er each time, like this

Example:

3, @, 03, 0@, .3, .@, 33, 3@, ***

This se#uence has a di%%erence o% 7 !et'een each num!er*

The pattern is continued !y addin$ 5 to the last num!er each time, like this

The alue added each time is called the ;common difference;

What is the common di%%erence in this e6ampleH

0, .B, 37, 13, ***

ns'er The common di%%erence is 3

The common di%%erence could also !e negatie

Example:

.7, .3, .0, 0, 0B, 07, ***

This common di%%erence is


18/20

=eometric Se9uences

eometric Se#uence is made !y multiplyin$ !y some alue each time*

Example:

., 1, @, 0>, 3., >1, 0.@, .7>, ***

This se#uence has a %actor o% . !et'een each num!er*

The pattern is continued !y multiplyin$ "y each time, like this

Example:

3, , .B, @0, .13, B., .0@B, ***

This se#uence has a %actor o% 3 !et'een each num!er*

The pattern is continued !y multiplyin$ "y each time, like this

Special Se9uences

Triangular Numbers

0, 3, >, 0/, 07, .0, .@, 3>, 17, ***

This Triangular "um!er Se#uence is generated %rom a pattern o% dots 'hich %orm a triangle*

$y adding another ro' o% dots and counting all the dots 'e can %ind the ne6t num!er o% the

se#uence


19/20

S9uare um"ers

/, 0, 1, , 0>, .7, 3>, 1, >1, @0, ***

They are the s#uares o% 'hole num!ers

/ (=/X/)

0 (=0X0)1 (=.X.)

(=3X3)

0> (=1X1)

etc***

)u"e um"ers

0, @, .B, >1, 0.7, .0>, 313, 70., B., ***

They are the cu!es o% the counting num!ers (they start at 0)

0 (=0X0X0)

@ (=.X.X.)

.B (=3X3X3)

>1 (=1X1X1)

etc***

i"onacci um"ers

/, 0, 0, ., 3, 7, @, 03, .0, 31, ***

The Ai!onacci Se#uence is %ound !y adding the t'o num!ers !e%ore it together*

The . is %ound !y adding the t'o num!ers !e%ore it (0G0)

The .0 is %ound !y adding the t'o num!ers !e%ore it (@G03)

The ne6t num!er in the se#uence a!oe 'ould !e 77 (.0G31)


20/20

Tricks o% 4oding Se#uence

0) E, , Q, T, = 7, 0/, 07, ./, .7 (alpha!et and its e#uialent num!ers* ust remem!er

EQ is in Third year )

The normal English alpha!et contains .> letters in all, as sho'n a!oe

(

ds dstime sd s peed distance

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