- INTRODUCTION
- CONCEPT OF UNIT WORK
- USE OF PRODUCT CONSISTENCY TO SOLVE TIME-WORK PROBLEMS
- CONCEPT OF EFFICIENCY
- PIPES AND CISTERNS
Time and Work
When a person performs a certain activity, he does some work. It may be constructing a road, packing certain items, eating food or filling/emptying a tank. As we have discussed in the concept of variation, the number of men required to complete certain work is directly proportional to the amount of work, if other factors such as the number of days remain same. Similarly, the number of men is inversely proportional to the days available, if other factors such as the amount of work remain constant.
Work is generally considered as 1 unit.
In this chapter, we assume (unless otherwise explicitly mentioned) that if a person does some work in a certain number of days, he does the same amount of work on each of those days. For example, if a person takes 7 days to complete some work, then in one day he finishes 1/7th of the work and in 5 days he will complete 5/7th of the work.
In general, if a person takes n days to complete some work, then in one day he finishes 1/nth of the work. This also implies that if a person completes 1/nth of the work in 1 day, then the total number of days taken to finish the work is n. Also, in m days, he will complete m/nth of the work. So, if a person completes 1/3rd of some work in one day, then he takes 3 days to finish the work. Also, in 2 days, he will complete 2/3rd of the work. Hence, the number of days required to complete the work is a reciprocal of the amount of work done in one day.
The number of days to complete the work 
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If A completes 2/3rd of some work in one day, in how many days can he finish it?
Solution: Work done in 1 day = 2/3
Total days needed to complete the work = 1/work done in 1 day = 3/2 = 1.5 days
Hence, A will complete the work in 1.5 days. |
The second assumption considered in this chapter is that, if more than one person is engaged in a work, then each does equal amount of work (unless explicitly mentioned otherwise).
For example, if 4 people can complete a work in 2 days, then 2 people will complete the same work in 4 days and 1 person will complete it in 8 days.
In general, if n persons complete a work in d days, 1 person will complete the same work in nd days, and m persons will do it in nd/m days.
However, many questions do explicitly state that the people involved do different amounts of work. For example, assume that a boy can build a kennel in 12 days and his father can build it in 6 days. Hence, the boy can build 1/12th of the kennel in 1 day; while the father can build 1/6th of the kennel in a day. If they both work together, they can build (1/12 + 1/6) = 1/4th of the kennel in one day. So, they will require 4 days to complete the work.
In general, if two people do some work, and the first can complete it in n days, while the second takes m days to do the same; then in one day they can together do (1/n + 1/m) work; i.e. they can complete the work in [1/(1/n + 1/m)] days. The same technique can also be extended to more than two people.
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If A alone completes a project in 6 days and B alone completes the same project in 4 days, then in how many days will they complete the project if they both work together?
Solution: A alone completes the project in 6 days.
Hence, work completed by A in 1 day = 1/6
B alone completes the project in 4 days.
Hence, work completed in by B in 1 day = 1/4
Work completed by A and B together in 1 day
Number of days to complete the work
Hence, A and B working together can finish the work in 2.4 days. |
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A can complete a piece of work in 4 days. B takes double the time taken by A, C takes double that of B, and D takes double that of C to complete the same task. They are paired in groups of two each. One pair takes two thirds the time needed by the second pair to complete the work. Which is the first pair? [CAT 2001]
(1) A, B(2) A, C (3) B, C(4) A, D
Solution: If A takes 4 days then, B will take 8 days, C will take 16 days and D will take 32 days.
Hence, option 4. |
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A and B together complete some work in 5 days, while A alone takes 15 days to complete it. How much time would B take to complete the work if he is working all alone?
Solution: Let B take b days to complete the work. Hence, work completed by B in one day = 1/b A takes 15 days to complete the work alone. Hence, work completed by A in one day = 1/15
Work done by A and B together in 1 day
= Work done by A in 1 day + Work done by B in one day
Also, A and B can together complete the work in 5 days.
Work done by A and B together in 1 day
Equating (i) and (ii) we get,
Hence, B alone will take 7.5 days to finish the work. |
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Dhruv can complete a piece of work in 8 days while Sameer can complete the same work in 12 days. They work together for 3 days. Then Dhruv quits the work. In how many days will Sameer be able to finish the remaining work?
Solution: Dhruv alone completes the work in 8 days. So he does 1/8th of the work in 1 day. Sameer alone completes the work in 12 days. So he does 1/12th of the work in 1 day.
Work completed together in 1 day
Amount of work that Sameer has to complete alone
Number of days that Sameer will take to complete the work.
Hence, Sameer can finish the remaining work in 4.5 days. |
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2 men and 2 boys complete a task in 4/3 days while 3 men and 1 boy complete the same task in 8/7 days. In how much time will 4 men and 8 boys complete the task?
Solution: Let one man take m days and one boy take b days to complete the task. Task completed by a man in 1 day = 1/m Task completed by a boy in 1 day = 1/b.
2 men and 2 boys complete the work in 4/3 days.
3 men and 1 boy complete the work in 8/7 days.
Solving the two equations simultaneously, we get x = 1/4 and y = 1/8 Hence, m = 4 and b = 8 Work done by a man in 1 day = 1/4
Work done by a boy in 1 day = 1/8
Hence, 4 men and 8 boys can complete the work in half a day. |
ALTERNATE METHOD (USING PERCENTAGES RATHER THAN FRACTIONS)
Many students prefer working with percentages; if you’ve mastered the “alternative method” given in the Percentages chapter which allows you to calculate percentages almost mentally, then we suggest you use the percentages method for Time and Work problems as well. This is because by not using fractions, this method lets you avoid finding the LCMs at each stage of calculation, thereby saving valuable time. Moreover, as certain problems give the work done as a percentage; it is generally more convenient to maintain it as a percentage rather than convert it to a fraction.
Let us consider an example. If X completes the entire work (i.e. 100%) in n days, and Y completes the same work in m days; then the percentage of work each complete in 1 day will be (100/n)% and (100/m)% respectively.
Hence, if they work together, they will complete
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If a man does 40% of a work in 12 days, then how many days will he take to complete the same work 5 times?
Solution: Let the man take m days to complete the work. Then, in one day he does (100/m)% of the work.
It is given that the man does 40% of the work in 12 days.
Thus, doing 100% of the work takes 30 days. Doing the work 5 times means doing (5 This will take 5 |
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There's a lot of work in preparing a birthday dinner. Even after the turkey is in oven, there are still the potatoes and gravy, yams, salad, and cranberries, not to mention setting the table. Three friends Asit, Arnold, and Afzal, work together to get all of these chores done. The time it takes them to do the work together is six hours less than Asit would have taken working alone, one hour less than Arnold would have taken, and half the time Afzal would have taken working alone. How long did it take them to do these chores working together? [CAT 2001]
(1) 20 minutes(2) 30 minutes (3) 40 minutes(4) 50 minutes
Solution: Suppose Asit, Arnold and Afzal finish the work together in x hours. Then Asit, Arnold and Afzal will take x + 6, x + 1 and 2x hour respectively.
3x2 + 7x
(x + 3)(3x
Hence, option 3. |
6 = 0
Number of hours cannot be negative.
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Three friends, Larry, Jimmy and Jay, were working on a report and all that was left now was to type it out. They had been asked to type out 5 copies of the same report for their college submissions. The first 3 copies, each friend typed individually, and it was seen that Jay took an hour less than Larry to finish. The fourth copy, all three friends typed together, and they took 15 minutes to type 25% of the report. Then Jay fell seriously ill had to be rushed to the hospital. So, Larry and Jimmy started to type out the fifth copy together. An hour after they had begun, Larry went to the hospital to check up on his friend; while poor Jimmy continued typing for another 24 hours and finished the last copy. How much percent of a single report does Jay type in an hour?
Solution: Jay takes one hour less than Larry to complete typing. So, let Larry, Jimmy and Jay take x, y and (x – 1) hours respectively to type the whole report.
Thus, in one hour, Larry, Jimmy and Jay will type (100/x)%, (100/y)% and (100/(x – 1))% of the report respectively.
It is given that the 3 friends complete 25% of the work in 15 minutes. So, in one hour they will complete 100% of the work. Hence,
Since, Larry and Jimmy working together for an hour and then Jimmy working alone for 24 hours, results in 100% work being done,
Comparing equations (i) and (ii), we get,
Substituting the above in equation (ii), we get,
x = 3/8 or x = 8/3
However, x cannot be less than 1, as the number of hours Jay takes to finish, which is (x – 1), will become negative. So, x = 8/3
Thus, in one hour, Jay will type (100/(x – 1)) = (100/(5/3)) = 60% of the report. |
We can regard work done as the product of the rate at which work is done and the time taken to finish it.
i.e. Total Work Done = Work Rate 
Time
Now, there are three ways of looking at this; in the first case, we consider work rate as the work done per unit time (work done per day, for example) and in the second case, we consider work rate as the number of workers. You will generally need to choose one of these two ways of looking at a situation based on the information given in a question. The third case, however, is completely different from the first two; here, work done is the volume of work. Now, let’s have a closer look at each of the three cases:
- This method is generally used when in a given group of men, each man works at his own pace; i.e. the work rate is different for each worker. Here,
Work rate 
work done per unit time
Time
number of hours/days/weeks/months/years etc.
Work done
either 1 or 100%
- This method is generally used when in a given group of men, all men work at the same pace. Here,
Work rate
number of workers
Time
number of hours/days/weeks/months/years etc.
Work done
number of ‘worker-days’ or ‘worker-months’ or ‘worker-hours’ etc., depending on
the units for time
- This method is used generally when work done is characterized as volume of work. For example, if the dimensions of a rectangular object is given (in terms of length, breadth and height), the following formula can be used:
The above formula could be altered and used for any other unit of time per day (not just number of hours per day).
The following few examples will throw some more light on when to use each of the above cases:
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Three sisters, Lizzy, Meg and Mary, together want to play an entire book of piano sheet music over a period of a few days as part of a festival. Mary, being the most proficient of the three, can play the entire book in 8 days; while Meg and Lizzy each take 12 and 16 days respectively to play the same. During the festival, in order to not exhaust themselves, they decide to take turns playing: Mary plays on day 1, Meg on day 2, Lizzy on day 3, then Mary again on day 4, and so on until the entire book is played. How many days does it take for the whole piano book to be played?
Solution: Mary, Meg and Lizzy take 8, 12 and 16 days respectively to finish playing the entire piano book.
Hence, in one day, Mary, Meg and Lizzy finish playing
(These are their work rates)
Now, 1 – 13/16 = 3/16 portion of the book is remaining, and it is Mary’s turn to play. So, on the 10th day, Mary will play 1/8th of the book. At the end of this day, 3/16 – 1/8 = 1/16th of the book is remaining.
On the 11th day, it is Meg’s turn. Since her work rate is 1/12 and the work to be done is 1/16, hence,
Hence, it takes the 3 sisters 10.75 days to play the entire piano book. |
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Mr. Singh, who had been facing acute water shortage, decided to drill a bore-well in his backyard. He decided that a depth of 640 metres should be sufficient, and he hired some men to drill him such a well. His friends had warned him that the workers were not honest and had a tendency to prolong as much as possible; so he drew up a strict schedule telling them how many metres they were to dig each day. Mr. Singh planned the schedule so that at the end of it, the men would have exactly drilled to the requisite depth. Four days after they started work, Mr. Singh noticed some men whiling away the time. However, on closer inspection, he realized that they were sticking to the schedule. So, from the fifth day onward, he asked them to dig 5 metres deeper per day than what was in the original schedule. The men then followed this, and one day prior to the originally scheduled finish date, they had dug 15 metres more than necessary. According to the original schedule, how many metres were they supposed to dig each day?
Solution: Let the number of metres the men were to dig per day according to the original schedule be x. This can be regarded as the work rate. Hence, the work rate from the fifth day onward in the implemented schedule will be (x + 5) metres per day. Since work rate is in ‘metres per day’, we will regard this question to be of Case 1.
Let the number of days for which the drilling should have lasted according to the original schedule be t. Hence, according to the implemented one, the time will be (t – 1). Although this question is somewhat based on Case 1, here the work done cannot be taken as 1. In fact, we will consider the amount of metres drilled as the work done. So, the work done according to the original schedule should have been 640 metres, while that in the implemented schedule was 640 + 15 = 655 metres.
Now, Work Done = Work Rate Hence, for the original schedule, we have:
640 = xt For the implemented schedule, the work rate for the first 4 days was x metres/day, while that for the remaining [(t – 1) – 4] days was (x + 5) metres/day. Hence, we have:
655 = 4x + (t – 5)(x + 5) = 4x + xt + 5t – 5x – 25 = 5t – x + xt – 25
However, xt = 640. Hence, we have, x = 5t – 40 Putting t = 640/x, we have,
Since the work rate cannot be negative, hence x = 40 metres/day. |
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It takes 6 technicians a total of 10 hours to build a new server from Direct Computer, with each working at the same rate. If six technicians start to build the server at 11 am and one technician per hour is added beginning at 5 pm, at what time will the server be complete? [CAT 2002]
(1) 6:40 pm(2) 7:00 pm (3) 7:20 pm(4) 8:00 pm
Solution: 6 technicians
In the hour after 5 pm,
Hence, option 4. |
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If a group of 15 stylists work for 6 hours a day, they can create a wedding collection in 30 days. If they have only 20 days, and they decide to put in 9 hours per day, then how many more stylists would they need to include in the group?
Solution: Initially, we have, Work rate = 15 stylists Time = 6 hours per day Work done = (15 Later, we have, Work rate = x stylists Time = 9 hours per day (15 Hence, x = 15 stylists ∴ They would need to include no more stylists. |
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A swarm of 70 worker-bees are capable of building a hive in 60 hours. If, 20 hours after they start, 10 more bees join in, then how many more hours will the bees take to build the rest of the hive?
Solution: Total work to be done = 70 bees
Work remaining after 20 hours = (70
Now, Work rate = 70 + 10 = 80 bees Work to be done = (70
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10 funeral-workers work for 8 hours per day and
create a casket of dimensions 80 inches
Solution: Initially, we have,
Let the required number of days be x.
Hence, the 5 funeral-workers working for 10 hours a
day will build a casket of dimensions 85 inches
It is important to observe that since work done is
the product of work rate and time, when there is an increase in the amount
of work done, there will be an equivalent increase in the product of work
rate and time. For example, if work done increases by 50%, then (work rate
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A contract is to be completed in 50 days and 105 men were set to work, each working 8 hours a day. After 25 days, 2/5th of the work is finished. How many additional men be employed so that the work may be completed on time, each man now working 9 hours a day? [SNAP 2009]
(1) 34(2) 36 (3) 35(4) 37
Solution: Since 2/5th of the work is completed in the 25 days, remaining 3/5th of the work is to be completed in 25 days. Let x men work for 25 days to complete 3/5th of the work.
Hence, option 3. |
If A is twice as efficient as B, it implies that A takes half the time as B.
So, if A is n times as efficient as B, then.
However, if A is two times more efficient than B, it implies that A takes 1/3rd the time that B takes. In other words, it is equivalent to saying that A is 3 times as efficient as B.
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If A takes 5 days to complete a job and B is twice as efficient as A, then in how many days can they finish the job together?
Solution: Number of days taken by A to finish the job = 5 B is twice as efficient as A. So, B will take half the number of days as A.
Job completed together in 1 day=1/5+2/5=3/5 Number of days to complete the job = 5/3 = 1.667 days Hence, A and B working together can finish the job in 1.667 days. |
REMEMBER:
- As efficiency increases, the number of days taken/required to complete the work decreases.
The concept of pipes and cisterns is an extension of the concept of work. Pipes are of two types – inlet and outlet. Inlet pipes fill the tank, while outlet pipes empty the tank. Work done by an inlet pipe is treated as positive work and that done by an outlet pipe is treated as negative work.
Total tank filled = Tank filled by inlet pipes – Tank emptied by outlet pipes.
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A tank has to be filled with pipes A and B. Pipe A can fill the tank in 6 hours and pipe B can fill it in 10 hours. In how much time will these pipes fill up the tank if both are opened simultaneously?
Solution: Pipe A can fill the tank in 6 hours. So, tank filled by A in 1 hour = 1/6
Pipe B alone fills the tank in 10 hours. So, tank filled by B in 1 hour = 1/10
Number of hours taken to fill the tank = 15/4 = 3.75 hours
Hence, pipes A and B can fill the tank in 3.75 hours. |
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An empty tank is connected to pipes A, B and C. Pipes A and B are inlet pipes and they fill the tank in 4 hours and 3 hours respectively while pipe C is an outlet pipe and it empties the tank in 2 hours. Find the time in which the tank will fill up if all the pipes are opened simultaneously?
Solution: Pipe A alone fills the tank in 4 hours. So, tank filled by A in 1 hour = 1/4
Pipe B alone fills the tank in 3 hours. So, tank filled by B in 1 hour = 1/3
Pipe C alone empties the tank in 2 hours. So, tank emptied by C in 1 hour = 1/2
Tank filled in 1 hour = Tank filled by A + Tank filled by B – Tank emptied by C
Number of hours to fill the tank = 12/1 = 12 hours Thus, when all three pipes are opened simultaneously, the tank gets filled in 12 hours. |
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A can built up a structure in 8 days and B can break it in 3 days. A has worked for 4 days and then B joined to work with A for another 2 days only. In how many days will A alone built up the remaining part of the structure? [SNAP 2009]
(1) 10 days(2) 9 days (3) 12 days(4) None of these
Solution: A can build the structure in 8 days.
Now, both A and B work together for 2 days.
So, the fraction of structure built in 2 days
If A takes x days, to build up the remaining structure, then
Hence, option 4. |
