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TIME AND WORK

Work to be done usually considered as one unit. It may be constructing a wall or road.Filling up or emptying a tank or cistern or eating certain amount of food.

There are some basic assumptions that are made in the problems on time and work.These are taken for granted and are not specified in every problem.

1) If a person (or one member of the workforce) does some work in a certain number of

days then we assume (unless otherwise e!plicitly stated in the problem) that he doesthe work uniformly ie he does the same amount of work everyday.

For e!ample if a person can do some work in 1" days he does 1#1" thof the work in onedayIf a person completes the work in $ days he does 1#$ thof the work on each day andconversely if a person can complete 1#$thof the work in one day he can complete thework in $ days.

%) If there is more than one person (or members of workforce) carrying out the work it is

assumed that each person (or members of the workforce) unless otherwise specifieddoes the same amount of work each day. This means they share the work e&ually.

For e!ample if a tap can fill a tank in %' minutes than in cone minute it can fill 1#%' partof the tankIf two people together can do the work in days it means that one man can do it in 1days. This is turn means each person can do 1#1thof the work per day.

If a man works three times as fast as a boy does the man takes one third of the timethe boy takes to complete the work. If the boy takes 1% days to complete the work then

the man takes $ days to complete the work.This method is known as UNITARY METHOD* ie the time taken per UNIT WORKor number of persons re&uired to complete UNIT WORKor work completed by +unitperson* in +unit time etc is what is first calculated.

We should recollect the fundamentals on variation (direct and inverse) here

The work and men are directly proportional to each other ie if the work increases thenumber of men re&uired increases to complete the work in the same number of days andvice versa.

Men and days are inversely proportional ie if the number of men increases the

number of days re&uired to complete the same work decreases and vice versa.

Wor and days are dire!tly proportionalie if the work increases the number of days

re&uired also increases if the work is to be completed with same number of men and viceversa.

The concept of man days is very important and useful here. The number of menmultiplied by the work number of days that they take to complete the work will give the

number of man days re&uired to do the work. The total number of ,onday-s re&uired tocomplete a specific task will remain a constant. o if other will change accordingly so

that their product will remain constant.(remember from our knowledge of variation twovariables whose product is a constant are said to be inversely proportional to each

other). The two variables men and days are inversely proportional to each other.

E"AM#$E #RO%$EM&

1) If %' men take /' days to complete a 0ob in how many days can %" men complete the0ob

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&ol'

If %' men can complete the 0ob in /' days then the number of mondays re&uiredfor the work is %'2/'3'' mondays

If this work is now is to be done by %" men since the no of mondays will be ''the no of days they will take is ''#%"3%$ days

%) Fifteen men take 1' days to complete the 0ob working 1% hours a day. 4ow may

hours a day should 1' men work to complete the 0ob in %'days

&ol'

ince 1" men take 1' days at 1% hours per day the total men hours re&uired forthe 0ob is 1"21'21%31''When 1' men are re&uired complete the same 0ob in %' days the no of men hoursre&uired for the 0ob will still be the same. 4ence the no of hours they should work perday is 1"21'21%#1'2%'35hours

Therefore 1' men can complete the work in %' days working for 5 hours per day.

(ENERA$ )ORMU$A )*

4ence in general we say thatIf m1 men can do w1 work in d1 days working h1 hours per day and m% men can do w%work in d% days working h% hours per day(where all men work at the same rate)then ,1 61 41#W13 ,% 6% 4%#W%/)7 piece of work can be done by 1 men in days working 1% hours a day. 4ow many

men are needed to complete another work which is three times the first one in %$ daysworking hours a day

&ol'

In the formula above F1 ,1 61 41#W13 ,% 6% 4%#W%We know that w131 and w%3/

,131 h131%d13h%3d%3%$1221%#13m122%$#/3m1 %$ men

)ORMU$A )+

If two persons 7 and 8 can individually do some work in p and & days respectively we

can find out how much can be done by them together in one day. ince 7 can do 1#pwork in one day and 8 can do 1#& work in one day the two of them together do 1#p91#&

work in one day.From this we can find out the no of days that they take to complete the work

If 7 can do a piece of work in p days and 8 can do it in & days then 7 and 8 together can

complete the same in pp9& days.

$)7 can do a piece of work in p days 8 can do the same work in 1% days in how many

days can the work be completed if the 7 and 8 work together

&ol' 7 and 8 together can do the work in 521%#591%31'#%13"21#:days

")7 and 8 together can do 1#1% thof work in 1% days and 7 alone can complete the work in

1 days. 4ow long will 8 alone take to complete the 0ob&ol'

In a day 7 and 8 together can do 1#1%

th

of the work in a day 7 alone can do 1#1

th

ofthe work

Therefore in one day work done by 8 alone is 1#1%;1#131#/o 8 alone can complete the work in / days.

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)7nil and 7mit can complete a work in 1% days working together. 7mit alone cancomplete it in 1 days. 8oth of them worked together for $ days and then amit left.4ow long will anil will take to complete the remaining work

&ol'

Work done in 1 day by both together 31#1%Work done by both in $ days 3$21#1%31#/

o of days 7 worked 3/9/3o total work done by 7321#1%31#%

The remaining ? of the work is done by 8 in / dayso complete work will be done by 8 in /2%#13days

1')7 and 8 together can do a piece of work in 1$ %#"days 8 and = together can do thesame work in 1% days. 7fter 7 worked for days 8 for 1% days = takes up and finishedit alone in days. In how many days will each of them do the work working alone&ol'

Work done by 7 and 8 in 1 day3" :#1%Work done by 8 and = in 1 day 31#1%

7 worked for days 8 worked for days to complete the work

a91%b9c31 where

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abc are the work0 done in one day by 78= respectively(a9b)9($a9$c)9%c31("#:%'9$(1#1%)9%c31$'#:%9$#1%9%c31

%c31;($#:%)3#:% 3$#:%31#1so = alone can complete the work in 1 days

work done by 8 alone in 1 day3(b9c);c3"#:%;1#/31#%$so 8 alone can do it in / days and 7 alone can do it in %$ days.

11)To do a certain work = alone takes twice as long as 7 and 8 together 7 would take /times as 8 and = together. 7ll the three complete the work in " days. 4ow long wouldtake separately&ol'

/ times 7-s daily work3 (89c-s) 1 days work adding 1 time 7-s daily work to bothsides we get $ times 7-s daily work3 (7989=-s)daily work31#"

7-s daily work 31#%'% times c-s daily work 3(798)-s daily work adding 1 times =-s daily work to both sides

we get / times =-s daily work3 (7989=)-s daily work 31#"

o =-s work 31#"@ = takes 1" days to do the worko 8-s daily work3 1#"A1#%'91#1"B31#1%o a alone can do the work in %' days 8 alone in 1% days and = alone in 1" days

1%)$ men or " women can construct a wall in % days. 4ow long will it take " men and $

women to do the same&ol'

Civen $m3"w where m is work done by one man in one day and w is work done byone women in one day

3D1m3"w#$>ow "m9$w3"A"w#$B9$w3$1w#$if "w can do the work in % days $1w#$ can do in

"w2%2$#$1w3$' days

1/)If 5 men and 1% boys can do a work in $ days and $men and 1 boys can do the same

work in days how long will men and %$ boys take to complete the same work

&ol'

Civen 5m91%b can do the work in $ days and $m91b can do the same work in dayso $(5m91%b) 3($m91b) 3Dm3$b>ow 5m91%b35($m)91%b3$b

We want m9%$b3$bo men and %$ boys also take $ days to do the work.

1$)

7 certain no of men can do the work in %' days. If there were $ more men the work canbe done in " days less. 4ow many men were initially

&ol'

Eet the initial no of men be p>o of days is %'@ so no of man days is %'pIf there are $ more men that is (p9$) it is completed in 1" dayso %'p3(p9$)1"3Dp31%o initially there were 1% men

1")

is three times as fast as G and is able to complete the work in $' days less then G.find the time in which they can complete the work together

&ol'

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If G does the work in / days does it in 1 day that is difference is % days.8ut the actual difference is $' days.If the difference is % days takes 1 day and G takes / days. If difference is $' days(that %' times)

takes %' days G takes ' dayso the time taken together 3 %'2'#%'9' 3 1" days

1)

ita can do work in 1% days working $ hours a day. Cita can finish the same in 1" daysworking / hours a day. 4ow many days can they finish it working together at $ ? hoursa day&ol'

>o of hours taken by sita31%2$3$>o of hours taken by gita3 1"2/ 3 $"o together they will do$"9$#$"2$3/1#:%' th work in one hour

Together they will take 3 :%'#/1 hours or :%'#/1 days working 1 hour a dayso working $ ? hours a day they can complete the work in :%'#/125#%3:%'2%#/1253 "

"#/1 days

1:)7 alone can do the work in 1% days and 8 alone in 1 days. If = takes twice as long as 7and 8 together how long wills 8 and = together take to complete the same work&ol'

7 does the work in 1% days @ 8 does the same in 1 days. 4ence when they worktogether they take

1%21#1%913/#" days.= take twice the time 7 and 8 together take

That is = take the %2/#"3:%#" days to do the workIf 8 and = work together work done per day is 3 1#19"#:%35#:%31# hence 8 and =

can do in days

1)7 and 8 each working alone can do a work in 1' and 1" days respectively. They startedthe work together but 8 left after sometime and 7 finished the remaining work in " days.

7fter how may days form the start did 8 leave&ol'

7-s work for " days 3"21#'31#% workThe remaining half of the work was done by 7 and 8 together

Work done by 7 and 8 in a day is 31#1'91#1"31#o no of days they worked together is 3 ? # 1# 3 / days4ence 8 left / days form the start of the work.

15)7 contractor decided to complete the work in $' days and employed ' men at the

beginning $' men additionally after 1' days and got the work completed as perschedule. If he had not employed the additional men how may e!tra days would he

have to complete the work&ol'

1'' men did the remaining work in ($';1')3/' days 3D ' men did it in "' dayso e!tra days needed is 3 "';/' 3 %' days

%')

7 group of /" men is employed to complete some work in $ days. 7fter // days "more men are employed and the work is finished 1 day earlier@ if " more men were not

employed how may more days would it have taken beyond the e!pected period&ol' $' men will do it in $'21$#/"31 days

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H!tra time taken 3 1;1"31 day behind the schedule

%1)7 and 8 working separately can do the work in and 1% day. They work on alternate

days starting with 7 on the first day. In how many days will the work be completed&ol'

ince they are working on alternate days. Eet us consider the time period of two daysin which 7 does

ne day-s and 8 does the ne!t days work. in a period of two days work done by 7 and 8together31#91#1%31#$ince they complete 1#$thof the work in a period of % days to complete the work theyneed $ such periods of % days ie $2%3 days

%%)7 and 8 working separately and do a work in and 5 days they work on alternate days

starting with 7 on the first day. In how many days will the work be done&ol'

ince they are working on alternate days let us consider a period of two daysIf a period of two days work done by 7 and 8 31#91#53"#1

If we consider / such time periods of % days (we are considering / periods because inthe fraction of "#1 the numerator " goes / times in the denominator 1)Work done 3 /2("#1) 31"#1ow it is 7-s turn since / whole no of periods are overTime taken by 7 to finish 1#1thof the work is one day

o total time taken 3/2%913 : days

%/)

7 and 8 working separately can do a work in 1% and 1" days. They work on alternatedays starting with 7 on the 1stday. In how many days will he work be completed

,&ol'

In a period of % days work done 3 1#1%91#1" 3 /#%'In such time periods of % days ie 1% days (we are considering periods because in

the fraction /#%' the numerator / goes to times in the denominator %')Work done is 1#%'

ow it is 8- turn to do 1#'thof work he takes 1#' # 1#1" 3 1#$thof a dayTotal no of days taken 3 (2%) 9191#$ 3 1/ J days

%$)7 and 8 working together separately can do a work in %' and %$ days. They work onalternate days starting with 8 on the first day. In how many days will the work becompleted

&ol' Work done by 7 and 8 in a period of % days3 1#%'91#%$311#1%'

If we consider 1' such tike periods (we are considering 1' periods because in thefraction 11#1%' the numerator 11 goes 1' times in the denominator 1%')Work done 3 1'211#1%'311#1%ow it is 8-s turn and he can complete 1#%$thof the work in 1 day so he cannot completethe work in 1 day. o he cannot complete the balance 1#1%thof the work. o after 1 days

Work done by 8 balance work31#1%;1#%$31#%$>ow it is 7-s turn and he can complete 1#%'thof the work in one day

o to complete 1#%$thof work he will take %'#%$3"#thof the day.Total tike taken 3 (1'2%) 919"# 3 %1 "# days

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NOTE&

When the people doing some work earn some money together for doing the work then

this money has to be shared by all the people doing the work.In general money earned should be shared by people doing the work together in the

ratio of the TT7E W

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The 1stman can do 1#1thwork in one day and since he works for days he does21#131#%th work. o he should get 1#%th amount 3

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7983 hourso =31#/;1#31#o = alone can fill the tank in hours

/")7 tank has a leak which would empty it in hours. 7 tap is turned on which fills at the

rate of $ litres per minute and the tank is now emptied in 1% hours. Find the capacity ofthe tank

&ol'

Work done by leak and filling tap in 1 hour 31#1%Work done by filling tap 3 1#91#1%931#%$Tank can be filled in %$ hours =apacity of tank 3 %$2'2$ 3":' litres