time and work probs

8/4/2019 Time and Work Probs

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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 explicitly stated in the problem) that he doesthe work uniformly, ie he does the same amount of work everyday.

For example if a person can do some work in 15 days he does 1/15 th of the work in onedayIf a person completes the work in 4 days, he does 1/4 th of the work on each day andconversely, if a person can complete 1/4th of the work in one day, he can complete thework in 4 days.

2) 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 specified,does the same amount of work each day. This means they share the work equally.

For example if a tap can fill a tank in 20 minutes than in cone minute, it can fill 1/20 partof the tankIf two people together can do the work in 8 days it means that one man can do it in 16days. This is turn means, each person can do 1/16th of 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 12 days to complete the work then

the man takes 4 days to complete the work.This method is known as UNITARY METHOD ie the time taken per UNIT WORKor number of persons required to complete UNIT WORK or 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 required 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 required to complete the same work decreases and vice versa.

Work and days are directly proportional ie if the work increases the number of days

required 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 required to do the work. The total number of Mondays required tocomplete a specific task will remain a constant. So 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.

EXAMPLE PROBLEMS

1) If 20 men take 30 days to complete a job in how many days can 25 men complete thejob?


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Sol:

If 20 men can complete the job in 30 days then the number of mondays, requiredfor the work is 20*30=600 mondays

If this work is now is to be done by 25 men since the no of mondays will be 600,the no of days they will take is 600/25=24 days

2) Fifteen men take 10 days to complete the job working 12 hours a day. How may

hours a day should 10 men work to complete the job in 20days?

Sol:

Since 15 men take 10 days at 12 hours per day, the total men hours required forthe job is 15*10*12=1800When 10 men are required complete the same job in 20 days the no of men hoursrequired for the job will still be the same. Hence the no of hours they should work perday is 15*10*12/10*20=9hours

Therefore 10 men can complete the work in 20 days working for 9 hours per day.

GENERAL FORMULA F1

Hence in general we say thatIf m1 men can do w1 work in d1 days working h1 hours per day and m2 men can do w2work in d2 days working h2 hours per day(where all men work at the same rate)then,

M1 D1 H1/W1= M2 D2 H2/W23)A piece of work can be done by 16 men in 8 days working 12 hours a day. How many

men are needed to complete another work, which is three times the first one, in 24 daysworking 8 hours a day

Sol:

In the formula above F1 M1 D1 H1/W1= M2 D2 H2/W2We know that w1=1 and w2=3

M1=16, h1=12,d1=8,h2=8,d2=2416*8*12/1=m1*8*24/3=m1 24 men

FORMULA F2

If two persons A and B can individually do some work in p and q days respectively, we

can find out how much can be done by them together in one day. Since A can do 1/pwork in one day and B can do 1/q work in one day, the two of them together do 1/p+1/q

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

If A can do a piece of work in p days and B can do it in q days then A and B together can

complete the same in pq/p+q days.

4)A can do a piece of work in p days, B can do the same work in 12 days, in how many

days can the work be completed if the A and B work together?

Sol: A and B together can do the work in 9*12/9+12=108/21=5*1/7days

5)A and B together can do 1/12 th of work in 12 days and A alone can complete the work in

18 days. How long will B alone take to complete the job?Sol:

In a day A and B together can do 1/12

th

of the work in a day, A alone can do 1/18

th

ofthe work

Therefore in one day work done by B alone is 1/12-1/18=1/36So B alone can complete the work in 36 days.


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6)Anil and Amit can complete a work in 12 days working together. Amit alone cancomplete it in 16 days. Both of them worked together for 4 days and then amit left.How long will anil will take to complete the remaining work?

Sol:

Work done in 1 day by both together =1/12Work done by both in 4 days =4*1/12=1/3

Remaining work =2/3Work done by anil in 1 day =1/12-1/16=1/48So anil alone takes 48 days to complete the work since only 2/3 of the work is left, anilwill do the remaining work in 2/3*48=32days

7)A and B together can do a piece of work in 12 days, B and C can do it in 15 days and Cand A can do the same work in 20 days . How long would each take to finish the work?

Sol:Work done by A and b in 1 day=1/12

Work done by B and C in 1 day=1/15Work done by C and A in 1 day=1/20

Adding all three we get work done by 2(A+B+C) in 1 day =1/12+1/15+1/20=1/5So A,B and C together finish in 1 day 1/10th of the workWork done by A in 1 day =work done by A,B and C in 1 day-work done by B and C in 1day=1/10-1/15=1/30So A alone can do it in 30 daysWork done by B in 1 day=1/10-1/20 =1/20

So B alone can do it in 20 daysWork done by C in 1 day=1/10-1/12=1/60

So C alone can do it in 60 days

8)A and B can do work in 12 days B and C in 15 days and C and A in 20 days. They all

work together for 6 days and then a left. In how may more days can B and C finish theremaining work?

Sol:Work done by A, B and C in 1 day

=1/10(As calculated in previous sample)Work done by A,B and C in 6 days= 6*1/10=3/5

So B and C can do the remaining 2/5 of the work in 2/5*15 = 6 days

9)A can do a work in 12 days. When he had worked for 3 days B joined him. If theycomplete the work in 3 more days in how many days can be alone finish the work?

Sol:Work done by A In 1 day =1/12No of days A worked =3+3=6So total work done by A=6*1/12=1/2

The remaining of the work is done by B in 3 daysSo complete work will be done by B in 3*2/1=6days

10)A and B together can do a piece of work in 14 2/5days, B and C together can do thesame work in 12 days. After A worked for 8 days, B for 12 days C takes up and finishedit alone in 6 days. In how many days will each of them do the work, working alone?Sol:

Work done by A and B in 1 day=5 7/12Work done by B and C in 1 day =1/12

A worked for 8 days, B worked for 6 days to complete the work

8a+12b+6c=1 where


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a,b,c are the workj done in one day by A,B,C respectively(8a+8b)+(4a+4c)+2c=18(5/720+4(1/12)+2c=140/72+4/12+2c=1

2c=1-(64/72)=8/72 =4/72=1/18so C alone can complete the work in 18 days

work done by B alone in 1 day=(b+c)-c=5/72-1/36=1/24so B alone can do it in 36 days and A alone can do it in 24 days.

11)To do a certain work C alone takes twice as long as A and B together, A would take 3times as B and C together. All the three complete the work in 5 days. How long wouldtake separately?Sol:

3 times As daily work= (B+cs) 1 days work adding 1 time As daily work to bothsides we get 4 times As daily work= (A+B+Cs)daily work=1/5

As daily work =1/202 times cs daily work =(A+B)s daily work adding 1 times Cs daily work to both sides

we get 3 times Cs daily work= (A+B+C)s daily work =1/5

So Cs work =1/5; C takes 15 days to do the workSo Bs daily work= 1/5[1/20+1/15]=1/12So a alone can do the work in 20 days, B alone in 12 days and C alone in 15 days

12)4 men or 5 women can construct a wall in 82 days. How long will it take 5 men and 4

women to do the same?Sol:

Given 4m=5w where m is work done by one man in one day and w is work done byone women in one day

=>1m=5w/4Now 5m+4w=5[5w/4]+4w=41w/4if 5w can do the work in 82 days, 41w/4 can do in

5w*82*4/41w=40 days

13)If 9 men and 12 boys can do a work in 4 days and 4men and 16 boys can do the same

work in 6 days, how long will 6 men and 24 boys take to complete the same work?

Sol:

Given 9m+12b can do the work in 4 days and 4m+16b can do the same work in 6daysSo 4(9m+12b) =6(4m+16b) =>m=4bNow 9m+12b=9(4m)+12b=48b

We want 6m+24b=48bSo 6 men and 24 boys also take 4 days to do the work.

14)

A certain no of men can do the work in 20 days. If there were 4 more men, the work canbe done in 5 days less. How many men were initially?

Sol:

Let the initial no of men be pNo of days is 20; so no of man days is 20pIf there are 4 more men that is (p+4) it is completed in 15 daysSo 20p=(p+4)15=>p=12So initially there were 12 men

15)

X is three times as fast as Y and is able to complete the work in 40 days less then Y.find the time in which they can complete the work together?

Sol:


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If Y does the work in 3 days, X does it in 1 day that is difference is 2 days.But the actual difference is 40 days.If the difference is 2 days, X takes 1 day and Y takes 3 days. If difference is 40 days(that 20 times)

X takes 20 days Y takes 60 daysSo the time taken together = 20*60/20+60 = 15 days

16)

Sita can do work in 12 days working 4 hours a day. Gita can finish the same in 15 daysworking 3 hours a day. How many days can they finish it working together at 4 hoursa day?Sol:

No of hours taken by sita=12*4=48No of hours taken by gita= 15*3 = 45So together they will do45+48/45*48=31/720 th work in one hour

Together they will take = 720/31 hours or 720/31 days working 1 hour a dayso working 4 hours a day they can complete the work in 720/31*9/2=720*2/31*9= 5

5/31 days

17)A alone can do the work in 12 days and B alone in 18 days. If C takes twice as long as Aand B together, how long wills B and C together take to complete the same work?Sol:

A does the work in 12 days ; B does the same in 18 days. Hence, when they worktogether they take

12*18/12+18=36/5 days.C take twice the time A and B together take

That is C take the 2*36/5=72/5 days to do the workIf B and C work together work done per day is = 1/18+5/72=9/72=1/8 hence B and C

can do in 8 days

18)A and B each working alone can do a work in 10 and 15 days respectively. They startedthe work together but B left after sometime and A finished the remaining work in 5 days.

After how may days form the start did B leave?Sol:

As work for 5 days =5*1/0=1/2 workThe remaining half of the work was done by A and B together

Work done by A and B in a day is =1/10+1/15=1/6So no of days they worked together is = / 1/6 = 3 daysHence, B left 3 days form the start of the work.

19)A contractor decided to complete the work in 40 days and employed 60 men at the

beginning 40 men additionally after 10 days and got the work completed as perschedule. If he had not employed the additional men how may extra days would he

have to complete the work?Sol:

100 men did the remaining work in (40-10)=30 days => 60 men did it in 50 daysSo extra days needed is = 50-30 = 20 days

20)

A group of 35 men is employed to complete some work, in 48 days. After 33 days, 5more men are employed and the work is finished 1 day earlier; if 5 more men were not

employed how may more days would it have taken beyond the expected period?Sol:

40 men will do it in 40*14/35=16 days


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Extra time taken = 16-15=1 day behind the schedule

21)A and B working separately can do the work in 6 and 12 day. They work on alternate

days starting with A on the first day. In how many days will the work be completed?Sol:

Since they are working on alternate days. Let us consider the time period of two daysin which A does

One days and B does the next days work. in a period of two days work done by A and Btogether=1/6+1/12=1/4Since they complete 1/4th of the work in a period of 2 days to complete the work theyneed 4 such periods of 2 days ie 4*2= 8 days

22)A and B working separately and do a work in 6 and 9 days they work on alternate days

starting with A on the first day. In how many days will the work be done?Sol:

Since they are working on alternate days, let us consider a period of two daysIf a period of two days, work done by A and B =1/6+1/9=5/18

If we consider 3 such time periods of 2 days (we are considering 3 periods because inthe fraction of 5/18, the numerator 5 goes 3 times in the denominator 18)Work done = 3*(5/18) =15/18Remaining work = 3/18=1/6Now it is As turn since 3 whole no of periods are over

Time taken by A to finish 1/16th of the work is one day

So total time taken =3*2+1= 7 days

23)

A and B working separately can do a work in 12 and 15 days. They work on alternatedays starting with A on the 1st day. In how many days will he work be completed?

\Sol:

In a period of 2 days work done = 1/12+1/15 = 3/20In 6 such time periods of 2 days, ie 12 days (we are considering 6 periods because in

the fraction 3/20, the numerator 3 goes to 6 times in the denominator 20)Work done is 18/20

Remaining work = 1-18/20=2/20=1/10Now it is As turn but a does only the 1/12 th of the work in 1 day. So balance work after

A works for one day (Ie the 13th) = 1/10-1/12=2/120=1/60Now it is B turn to do 1/60th of work he takes 1/60 / 1/15 = 1/4th of a dayTotal no of days taken = (6*2) +1+1/4 = 13 days

24)A and B working together separately can do a work in 20 and 24 days. They work onalternate days starting with B on the first day. In how many days will the work becompleted?

Sol:Work done by A and B in a period of 2 days= 1/20+1/24=11/120

If we consider 10 such tike periods (we are considering 10 periods because in thefraction 11/120 the numerator 11 goes 10 times in the denominator 120)Work done = 10*11/120=11/12Remaining work =1-11/12=1/12Now it is Bs turn and he can complete 1/24th of the work in 1 day so he cannot completethe work in 1 day. So he cannot complete the balance 1/12th of the work. So after 1 days

Work done by B balance work=1/12-1/24=1/24Now it is As turn and he can complete 1/20th of the work in one day

So to complete 1/24th of work he will take 20/24=5/6th of the day.Total tike taken = (10*2) +1+5/6 = 21 5/6 days


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NOTES

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 TOTAL WORK done by each of them.

For example if A 2/5th of the work, then he should 2/5 th of the total earnings for thework. In the remaining 3/5th of the work is done by B and C in the ratio 1:2 then theremaining 3/5th of earnings(after paying A) should be shared by B and C in the ration of1:2 suppose Rs 500 is paid to A,B and C together for doing the work, then A will get Rs200(which 2/5 of 500)B will get Rs 100 and C Rs 200 (because remaining 300 afterpaying to A is to be divided in the ratio of 1:2 between B and C.When people work for the same no of days each, then ratio of the total work done will bethe same as the work done by each of them PER DAY. Hence if all the people involved

work for the same no of days, then the earnings can directly be divided in the ratio ofthe WORK DONE PER DAY by each of them.

25)

A, B and c can do a piece of work in 4,5 and 7 days respectively. They got Rs 415 for thework. What is As share?Since they work for the same no of days the ration in which they share the money is theratio of the work done per dayThat is :1/5:1/7=35:28:20Hence As share is (35/83)*415= Rs 175.

26)A, B and c can do a work for Rs 4500. A and B together complete 3/5th of the work and

then C takes over the finished the work. What is the amount got by C?Work done by C=1-3/5=2/5

So Cs amount =2/5*Rs 4500=Rs 1800

27)Wages for 40 women and for 30 days are RS 21,600. How many men must work for 25

days to earn Rs14, 400 if the daily wages for a man is double that of a women?

Sol:Wages of women per day = 21600/30*40= Rs 18

Wages of a man= 2*18 = Rs 36So no of men = 14400/25*36 = 16

28)

A, B and C can together earn Rs 1,620 in 9 days. A and C can earn Rs 600 in 5 days;where as B and C in 7 days can earn Rs 910 find the daily earnings of C?

Sol:

Daily wages of A+B+B = 1620/9=180----- (1)Daily wages of A+C = 600/5 = 120 ----- (2)

Daily wages of B+C = 910/7=130----- (3)[(2) + (3)] ------ (1) gives[A+B+2C]- [A+B+C] =C= (120+130)-180=70

29)

Two men undertake a work for Rs 480; they can each do the work in 16 and 24 days. Ifthey complete the work in 8 days with the help of a boy, how should they divide the

money?

Sol:


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The 1st man can do 1/16th work in one day and since he works for 8 days, he does8*1/16=1/2th work. So he should get 1/2th amount =Rs 240The 2nd man can do the work in 1/24th work in one day and he also worked for 8 days, hedoes 8*1/24=1/3th work. So he should get 1/3th amount = Rs 160

Since the 2 men earned Rs 400, the balance of Rs 80 will be paid to the boy.

PIPES AND CISTERNS

There can be pipes of the taps filling or emptying tanks with water. Times taken bydifferent taps fill or empty the tank are different. Problems related to these can also bedealt with in the same manner as the foregoing problems on work have been dealt with.These are only one difference between the problems on regular work. (of the type seenearlier)and those in pipes and cisterns. In pipes and cisterns, a filling pipe or tap does apositive work and an emptying pipe or a leak does negative work.

30)

Two pipes A and B can fill a tank in 12 and 18 minutes. If both pipes are openedsimultaneously, how long will it take to fill the tank?

Sol: part of tank filled by A in a minute= 1/12Part of tank filled by B in a minute=1/18Part of tank filled the pipes in one minute = 1/12+1/18= 5/36So the tank can be filled in 36/5=7 minutes

31)

Pies A can fill a tank in 12 minutes and pipe B in 18 minutes and pipe C and empty a fulltank in 36 minutes. If all of them work together, find time taken to fill the empty tank?

Sol:Work done by the three pipes together in 1 minute = 1/12+1/18-1/36=1/9

So the empty tank will be filled in 9 minutes.

32)Two pipes A and B can fill a tank in 20 and 30 minutes. There is an outlet C. if all 3pipes are opened the tank will be filed in 24 minutes. How much time will it take for Calone to empty the full tank?

Sol:

Work done by c in one minute = (1/20+1/30)-1/24=1/12-1/24=1/24So C can empty tank in 24 minutes.

33)Two pipes A, B fill a tank in 20 and 30 minutes. if both pipes are opened at once, afterhow much time should A be closed so that the tank is filled in 15 minutes?

Sol:Pipe woks for 15 minutes. In 1 minutes B fills 1/30th of the tank ie the 1/5th minutes itfills 15*1/30=1/2The remaining is filled by A since A fills the tank fully in 20 minute. It takes 10

minutes to fill of the tank. Hence A worked for 10 minutes. So A should be closedafter 10 minutes.

34)Three tapes A, B and C together can fill a tank in 3 hours. After 1 hour C is closed andthe tank is filled in 4 more hours. Find the time in which C alone can fill the tank?

Sol:

Work of A+B+C IN 1 hour =1/3Remaining part of the tank = 1-1/3=2/3

Time taken by (A+B)to fill this 2/3rd of the tank = 4 hoursA and B together fill the tank in 6 hours

Now we know that A+B+C= 3 hours


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A+B= 6 hoursSo C=1/3-1/6=1/6So C alone can fill the tank in 6 hours

35)A tank has a leak, which would empty it in 8 hours. A tap is turned on which fills at the

rate of 4 litres per minute and the tank is now emptied in 12 hours. Find the capacity ofthe tank?

Sol:

Work done by leak and filling tap in 1 hour =1/12Work done by filling tap = 1/8+1/12+=1/24Tank can be filled in 24 hours Capacity of tank = 24*60*4 =5760 litres

time and work probs

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