'Work' Word Problems Made Easy

Question

‘Work’ Word Problems Made Easy

This post is a part of  GMAT MATH BOOK ]

created by:   sriharimurthy 
 edited by:   bb ,  walker ,  Bunuel

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NOTE:  In case you are not familiar with translating word problems into equations please go through  THIS POST FIRST .

What is a ‘Work’ Word Problem?

It involves a number of people or machines working together to complete a task.
 
  We are usually given individual rates of completion.
 
  We are asked to find out how long it would take if they work together.

Sounds simple enough doesn’t it? Well it is!

There is just one simple concept you need to understand in order to solve any ‘work’ related word problem.

The ‘Work’ Problem Concept

STEP 1: Calculate how much work each person/machine does in one unit of time (could be days, hours, minutes, etc).

How do we do this? Simple. If we are given that A completes a certain amount of work in X hours, simply reciprocate the number of hours to get the per hour work. Thus in one hour, A would complete  \frac{1}{X}  of the work. But what is the logic behind this? Let me explain with the help of an example.

Assume we are given that Jack paints a wall in 5 hours. This means that in every hour, he completes a fraction of the work so that at the end of 5 hours, the fraction of work he has completed will become 1 (that means he has completed the task).

Thus, if in  5  hours the fraction of work completed is  1 , then in  1  hour, the fraction of work completed will be  (1*1)/5

STEP 2: Add up the amount of work done by each person/machine in that one unit of time.

This would give us the total amount of work completed by both of them in one hour. For example, if  A completes  \frac{1}{X}   of the work in one hour and  B completes  \frac{1}{Y}   of the work in one hour, then  TOGETHER , they can complete  \frac{1}{X}+\frac{1}{Y}  of the work  in one hour .

STEP 3: Calculate total amount of time taken for work to be completed when all persons/machines are working together.

The logic is similar to one we used in  STEP 1 , the only difference being that we use it in reverse order. Suppose  \frac{1}{X}+\frac{1}{Y}=\frac{1}{Z} . This means that  in one hour , A and B  working together  will complete  \frac{1}{Z}  of the work.  Therefore, working together, they will complete the work in Z hours.

Advice here would be: DON'T go about these problems trying to remember some formula. Once you understand the logic underlying the above steps, you will have all the information you need to solve any ‘work’ related word problem. (You will see that the formula you might have come across can be very easily and logically deduced from this concept).

Now, lets go through a few problems so that the above-mentioned concept becomes crystal clear. Lets start off with a simple one :

Example 1.   
Jack can paint a wall in 3 hours. John can do the same job in 5 hours. How long will it take if they work together?

Solution: 
This is a simple straightforward question wherein we must just follow steps 1 to 3 in order to obtain the answer.

STEP 1: Calculate how much work each person does in one hour.
Jack → (1/3) of the work
John → (1/5) of the work

STEP 2: Add up the amount of work done by each person in one hour.
Work done in one hour when both are working together =  \frac{1}{3}+\frac{1}{5}=\frac{8}{15}

STEP 3: Calculate total amount of time taken when both work together.
If they complete  \frac{8}{15}  of the work in  1  hour, then they would complete 1 job in  \frac{15}{8}  hours.

Example 2.   
Working, independently X takes 12 hours to finish a certain work. He finishes 2/3 of the work. The rest of the work is finished by Y whose rate is 1/10 of X. In how much time does Y finish his work?

Solution: 
Now the only reason this is trickier than the first problem is because the sequence of events are slightly more complicated. The concept however is the same. So if our understanding of the concept is clear, we should have no trouble at all dealing with this.

‘Working, independently X takes 12 hours to finish a certain work’  This statement tells us that in one hour, X will finish  \frac{1}{12}  of the work.

‘He finishes 2/3 of the work’  This tells us that  \frac{1}{3}  of the work still remains.

‘The rest of the work is finished by Y whose rate is (1/10) of X’  Y has to complete  \frac{1}{3}  of the work.

‘Y's rate is (1/10) that of X‘ . We have already calculated rate at which X works to be  \frac{1}{12} . Therefore, rate at which Y works is  \frac{1}{10}*\frac{1}{12}=\frac{1}{120} .

‘In how much time does Y finish his work?’  If Y completes  \frac{1}{120}  of the work in  1  hour, then he will complete  \frac{1}{3}  of the work in 40 hours.

So as you can see, even though the question might have been a little difficult to follow at first reading, the solution was in fact quite simple. We didn’t use any new concepts. All we did was apply our knowledge of the concept we learnt earlier to the information in the question in order to answer what was being asked.

Example 3.   
Working together, printer A and printer B would finish a task in 24 minutes. Printer A alone would finish the task in 60 minutes. How many pages does the task contain if printer B prints 5 pages a minute more than printer A?

Solution: 
This problem is interesting because it tests not only our knowledge of the concept of word problems, but also our ability to ‘translate English to Math’

‘Working together, printer A and printer B would finish a task in 24 minutes’  This tells us that A and B combined would work at the rate of  \frac{1}{24}  per minute.

‘Printer A alone would finish the task in 60 minutes’  This tells us that A works at a rate of  \frac{1}{60}  per minute.

At this point, it should strike you that with just this much information, it is possible to calculate the rate at which B works: Rate at which B works =  \frac{1}{24}-\frac{1}{60}=\frac{1}{40} .

‘B prints 5 pages a minute more than printer A’  This means that the difference between the amount of work B and A complete in one minute corresponds to 5 pages. So, let us calculate that difference. It will be  \frac{1}{40}-\frac{1}{60}=\frac{1}{120}

‘How many pages does the task contain?’  If  \frac{1}{120}  of the job consists of  5  pages, then the  1  job will consist of  \frac{(5*1)}{\frac{1}{120}} = 600  pages.

Example 4.   
Machine A and Machine B are used to manufacture 660 sprockets. It takes machine A ten hours longer to produce 660 sprockets than machine B. Machine B produces 10% more sprockets per hour than machine A. How many sprockets per hour does machine A produce?

Solution: 
The rate of A is  \frac{660}{t+10}  sprockets per hour;
The rate of B is  \frac{660}{t}  sprockets per hour.

We are told that B produces 10% more sprockets per hour than A, thus  \frac{660}{t+10}*1.1=\frac{660}{t}  -->  t=100  --> the rate of A is  \frac{660}{t+10}=6  sprockets per hour.

As you can see, the main reason the 'tough' problems are 'tough' is because they test a number of other concepts apart from just the ‘work’ concept. However, once you manage to form the equations, they are really not all that tough.

And as far as the concept of ‘work’ word problems is concerned – it is always the same!

More on Work/Rate Problems Under the Spoiler 
  17. Work/Rate Problems 
     Theory   
 Work Rate Problems - All in One Topic! 
 Work Word Problems Made Easy 
 How to Solve Relative Rate of Work Questions on the GMAT 
     Questions   
 DS Questions 
 PS Questions

On other subjects:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

asocialnot · Answer

Bunuel ,

I have a question regarding Example 2.

Why isn't the rate of work for X calculated as follows:

12hrs ---- 2/3 work 
1 hr --- 1/18 work

Therefore, rate of work for Y should be (1/18)*(1/10) = (1/180).

1/180 work --- 1hr
1/3 work --- 180*(1/3) = 60hrs.

Please do clarify.

Thanks.

Bunuel · Answer

We are told that X takes 12 hours to finish a certain job, not 2/3 of the job. So:
12 hours = 1 job, not 2/3 of the job.

SivaKumarP · Answer

Can anyone help me how to use this formula for work problems?

EgmatQuantExpert · Answer

Hi  SivaKumarP ,

The formula is an extension of the basic concept of work rate problem. Let me explain it with an example.

Assume two persons A & B working alone who complete a work  'W'  in  5  and  6  days respectively. We need to find the time taken by A & B working together to complete the same work  W .

We know the amount of work to be done (i.e.  W ) and are asked to find the time taken. From the work rate equation   Work = Rate * Time  , the only variable remaining is the rate, so let's find out the rates of persons A & B and put in the equation to calculate the time taken.

Rates of work done by A & B  
Work done by A =  W 
Time taken by A to do W amount of work =  5  days

Since Rate = Work/time

Hence rate of work done by A =  \frac{W}{5}

Similarly rate of work done by B =  \frac{W}{6}

Putting the values of rates of A & B in the Work Rate equation to find out the time taken when both A & B work together:

Work = Rate * Time

W = (\frac{W}{5} + \frac{W}{6}) * t   (since A & B would work for the same time 't' to complete the work when working together)

W = \frac{11W}{30} * t  i.e.  t = \frac{30}{11}  days =  \frac{(5*6)}{(5+6)}  days

Time taken when A & B work together to do the same work = Time taken by A * Time taken by B/(Time taken by A + Time taken by B)

On similar lines, if we need to find the work done by A & B working together for 30 days, we can calculate it using the rate of work of A & B working together which is  \frac{11W}{30} .

Work done by A & B working together for 30 days =  \frac{11W}{30} * 30 = 11W = 6W + 5W  i.e. work done by A & B respectively working alone for 30 days.

Work done by A & B together in time t = Work done by A in time t + Work done by B in time t

Hope its clear!

Regards
Harsh

JRAppz · Answer

Can someone help with this question?
If it takes 4 machines, working at the same constant rate, 2 hours to complete 2 jobs, how long will it take 3 machines, working at the same constant rate, to complete 3 jobs?"

The answer is 4 hours but I have no idea how to arrive at the answer, hopefully someone can help.

Bunuel · Answer

STEP BY STEP:

4 machines, 2 hours to complete 2 jobs;
1 machine, 8 hours to complete 2 jobs;
1 machine, 4 hours to complete 1 job;
3 machines, 4/3 hours to complete 1 job;
3 machines, 4 hours to complete 3 jobs.

JRAppz · Answer

Bunuel - Could you possibly structure that in an algebraic manner?

Bunuel · Answer

Rate*Time = Job done.

(4*rate of one machine)*2 = 2;

rate of one machine = 1/4 job/hour.

Rate*Time = Job done.

(3*rate of one machine)*x = 3;

(3*1/4)*x = 3;

x = 4 hours.

ENGRTOMBA2018 · Answer

You can follow the method mentioned by Bunuel above .

The only thing you need to remember is that the work rate problems are similar to distance-speed-time wherein Distance is similar to job, speed is similar to rate and time is constant.

Thus, as distance = speed X time

Job or work = rate x time

Additionally, as with more machines, the total time required will reduce for the same amount of work, combined rates of 'n' machines will be n*rate of 1 machine.

GMATinsight · Answer

CONCEPT :  \frac{(Machine_Power * Time)}{Work} = Constant

i.e.  \frac{(M_1 * T_1)}{W_1} = \frac{(M_2 * T_2)}{W_2}

i.e.  \frac{(4 * 2))}{2} = \frac{(3 * T_2)}{3}

i.e.  T_2 = 4

dave13 · Answer

pushpitkc , Bunuel VeritasKarishma if in denominator always is given time (unless is reversed ) \frac{1}{120} then how can 120 be amount of work ? what does 5*1 mean ? and why are we dividing by 1/120 to get WORK DONE as per FORMULA ---> WORK = RATE * TIME pls explain

KarishmaB · Answer

Imp: Ensure that you always have your eye on the units. When you are flipping a quantity, know why.

A and B finish a task in 24 mins. If work done has to be 1 (complete work), their rate of work is (1/24)th of the work per min. (It is usually the case that we consider work to be 1 work)

Work = Rate*Time 
1 work = Rate* 24 mins
Rate = (1/24)th of work per min

Similarly, A's rate of work = (1/60)th of work per min

Since rates are additive, we get that B's rate must be (1/40)th of work per min.

So B does (1/40)th of the work in a min while B does only (1/60)th of the work. We are given that this difference of (1/40)th work and (1/60)th work is 5 pages. 
So 1 complete work is 600 pages. 
Now the rate of work can be expressed in terms of pages/min too. 
B prints (1/40)*600 = 15 pages/min
A prints (1/60)*600 = 10 pages/min

jyii07 · Answer

Can someone please explain this in Example 2:
‘In how much time does Y finish his work?’ If Y completes 1/120 of the work in 1 hour, then he will complete 1/3 of the work in 40 hours.

GDT · Answer

VeritasKarishma

Can you pls explain me why in statement 2 we are dividing combined time with individual time and also if there is any easier method of solving this

Thanks in advance!

KarishmaB · Answer

A group of 5 craftsmen, working together at the same rate, can finish a job in 3 hours. How long will it take a group of 4 apprentices working together to do the same job?
(1) Each apprentice works at 2/3 the rate of a craftsman.
(2) The 5 craftsmen and the 4 apprentices working together will take 45/23 hours to finish the job.

Time taken by 5 craftsmen (group 1) = 3 hrs
Rate of work of group 1 = 1/3

Time taken by 5 craftsmen (group 1) + 4 apprentices (group 2) = 45/23 hours
Rate of work of group1 + group 2 = 23/45 hrs

Rate of work of group 2 = 23/45 - 1/3 = 8/45
Time taken by group 2 = 45/8 hrs

atsdabhay · Answer

6 men can finish a piece of work in 2 hours. How long will 9 men take to finish the work if they can work only at 4/5th the capacity of the initial workers?

avigutman · Answer

You have 3/2 as many men, working at 4/5 the capacity. Multiplying the two ratios gets us 6/5. Since we have 6/5 as much “workforce”, the job will take just 5/6 as long. 5/6 of two hours is an hour and 2/3, or an hour and 40 minutes.

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BLTN · Answer

Nice shortcut, I have utilized proportion:
 \frac{#of workers}{Rate1}  =  \frac{#New of workers}{Rate2} 
in which rates can be similar or different.

Thus,
 \frac{7}{3/8 }  =  \frac{2}{1/time}

\frac{7*8}{3 }  =  \frac{2*time}{1}

\frac{7*8}{3*2 }  = time =  \frac{28}{3}  = 9.3h

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