Being comfortable with common ratios can save you a lot of time on the GMAT. Last week we covered distance/rate problems. Another great application of ratios is work rate problems. An important relation that helps us solve work rate problems is:

Work Done = Rate * Time

This relation will lead a perceptive observer to draw a parallel with another very popular relation most of us have come across:

Distance = Speed * Time

Speed is the same as Rate of work i.e. how fast you cover some distance or how fast you complete some given work. So obviously, if we can use ratios to solve many Distance Speed Time problems, we should be able to solve many Work Rate Time problems using ratios too.

Let’s look at some examples. Try and solve each one of them on your own before you go through the solutions.

Example 1:

A tank has 5 inlet pipes. Three pipes are narrow and two are wide. Each of the three narrow pipes works at 1/2 the rate of each of the wide pipes. All the pipes working together will take what fraction of time taken by the two wide pipes working together to fill the tank?

(A) 1/2

(B) 2/3

(C) 3/4

(D) 3/7

(E) 4/7

We are given that rate of work of 1 narrow pipe : rate of work of 1 wide pipe = 1:2

If we can find the ratio of rate of work of 2 wide pipes : rate of work of all pipes together, then we can easily get the ratio of time taken by 2 wide pipes : time taken by all pipes together. This is because ratio of time taken will be inverse of the ratio of rate of work since work done in both the cases is the same. (For a further explanation of this concept, check out the previous post)

In ratio terms, rate of work of 3 narrow pipes is 1*3 and rate of work of 2 wide pipes is 2*2

Therefore, rate of work of 3 narrow pipes : rate of work of 2 wide pipes = 3:4

Or we can say rate of work of 2 wide pipes : rate of work of all pipes together = 4 : (3+4) = 4:7

Then, time taken by 2 wide pipes : time taken by all pipes together = 7:4 (i.e. inverse of 4:7)

So all the pipes together will take 4/7 th of the time taken by the two wide pipes.

Answer (E)

Example 2:

Working at their respective constant rates, Paul, Abdul and Adam alone can finish a certain work in 3, 4, and 5 hours respectively. If all three work together to finish the work, what fraction of the work will be done by Adam?

(A) 1/4

(B) 12/47

(C) 1/3

(D) 5/12

(E) 20/47

It is given that:

Time taken by Paul : Time taken by Abdul : Time taken by Adam = 3:4:5

Rate of work must be inverse of time taken. But how do you take the inverse when you have a ratio of 3 quantities? Does it become 5:4:3? No. Actually it becomes 1/3 : 1/4 : 1/5 (I will explain the ‘why’ for this when I take variation)

Rate of Paul : Rate of Abdul : Rate of Adam = 1/3 : 1/4 : 1/5

Let’s multiply this ratio by the LCM to convert it into integral form. The LCM of 3, 4 and 5 is 60.

Rate of Paul : Rate of Abdul : Rate of Adam = (1/3)*60 : (1/4)*60 : (1/5)*60 = 20:15:12 (I would like to remind you here that multiplying or dividing each term of a ratio by the same number does not alter the ratio)

So if the total work is 20+15+12 = 47 units, Adam will complete 12 units out of it. Hence the fraction of work done by Adam will be 12/47.

Answer (B)

Example 3:

Machines A and B, working together, take t minutes to complete a particular work. Machine A, working alone, takes 64 minutes more than t to complete the same work. Machine B, working alone, takes 25 minutes more than t to complete the same work. What is the ratio of the time taken by machine A to the time taken by machine B to complete this work?

(A) 5:8

(B) 8:5

(C) 25:64

(D) 25:39

(E) 64:25

In this question, you can think logically to arrive at the answer quickly.

When machine A is working alone, it takes 64 extra minutes. Why? Because there is work leftover after t minutes. The work that would have been done by machine B in t minutes is leftover and is done by machine A in 64 minutes.

Time taken by A : Time taken by B = 64:t ….. (I)

Similarly, when machine B works alone, it takes 25 extra minutes to complete the work that machine A would have done in t minutes.

Time taken by A : Time taken by B = t:25 ……(II)

From (I) and (II) above,

64/t = t/25

t = 40

Time taken by machine A : Time taken by machine B = t:25 = 40:25 = 8:5

Answer (B)

*Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep in Detroit, Michigan, and regularly participates in content development projects such as this blog!*

## 15 thoughts on “Quarter Wit, Quarter Wisdom: Cracking the Work Rate Problems”

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Hello Karishma great blog, congrats i have been following you from quite some times through this blog as well as gmat club post on absolute value ..

I have certain doubts as i’m not able to understand the relation between ratios and Work rate problem .

1.with example one Therefore, rate of work of 3 narrow pipes : rate of work of 2 wide pipes = 3:4

therefore time taken by three narrow pipes:time taken by 2 big pipes wld be in ratios of = 4:3 hence time taken by all the pipes together : 2 big pipes = 3:7 ( i know this is wrong as time taken by all the pipes shld be lesser than 2 big pipes working alone ) but please i request to clarify as to where i’m going wrong

2.Example 3 – i haven’t understood the concept applied in exapmple 3 as to how

Time taken by A : Time taken by B = 64:t and Time taken by A : Time taken by B = t:25 should not it be t+64:t+25 ..

I know these are basics and i’m going terribly wrong some where i did visit all your previous write up and tried understanding the concepts but still could not figure things out .

Great blog and thanks once again

I guess you know the relation between work and rate which is Work = Rate*Time.

Now say, 2 different machines work for 2 hrs each i.e. time they take is the same.

Then, if their rate of work is in the ratio 3:2, what will be the ratio of work done? Can I say it will also be 3:2? Say, if rate of machine 1 is 3 products/hour and rate of machine 2 is 2 products/hour, machine 1 will make 6 products in 2 hrs and machine 2 will make 4 products in 2 hrs. Ratio of work done = 6:4 = 3:2

So ratio of work done will be the same as the ratio of rate provided time taken is the same.

Using the same logic, if work done is the same in 2 cases, ratio of time taken by them will be inverse of the ratio of their rate.

As for your question:

rate of work of 3 narrow pipes : rate of work of 2 wide pipes = 3:4

therefore time taken by three narrow pipes:time taken by 2 big pipes wld be in ratios of = 4:3 (this is correct)

hence time taken by all the pipes together : 2 big pipes = 3:7

How do you get 7 here? Remember, rates are additive, time is not.

To explain, say A takes 2 hrs to complete a work and B takes 3 hrs. Together, will they take 5 hrs? No.

On the other hand, say A’s rate is 3 jobs/hr and B’s rate is 2 jobs/hr. Can we say that together their rate of work is 5 jobs/hr? Yes.

So you cannot add the time taken.

Example 3: You are right that ratio of time should be t+64:t+25 where t+64 and t+25 are the periods of time taken by the machines to complete a particular work. But 64:t is the ratio of periods of time taken to complete a fraction of that. They together complete the work in t mins. A alone takes another 64 mins. Why? because a fraction of the work is leftover. When they both work together, that fraction of work is done by B in t mins. So for that fraction of work, ratio of time taken = 64:t. Similarly we get the ratio of time taken as t:25 for another fraction of work. These ratios must be the same so we equate them.

Why will these ratios be same? Say, time taken by 2 machines for 1 job is in the ratio 3:4. What will be the ratio of time taken for 2 jobs? It will be 6:8 = 3:4. As long as the work done is same for both machines, the ratio of time taken will be the same.

Hi,

I’m not sure about the first answer. I did it in this way.

Suppose it take 10 mins for 1 wide pipe to finish the job.

=> it will take 2 wide pipes 5 mins to finish the job

=> it will take a narrow pipe 20 mins to finish the job

=> it will take 5 narrow pipes 20/5= 4 mins to finish the job

Use the work rate formula it will take all pipes 20/9 mins to finish the job.

20/9 = 5*4/(5+4)

All the pipes working together will take what fraction of time taken by the two wide pipes working together to fill the tank.

=20/9*1/5= 4/9

I realise I misread the question. It should be 3 narrow pipes and not 5! I keep on doing these silly mistakes :(

Hi Karishma,

I did not understand the following :

Then, time taken by 2 wide pipes : time taken by all pipes together = 7:4 (i.e. inverse of 4:7)

=>

So all the pipes together will take 4/7 th of the time taken by the two wide pipes.

Did not understand how the above is deduced..

Kindly help.

Regards,

Sachin

And what is the level of difficulty of the 3rd Question..?

“When machine A is working alone, it takes 64 extra minutes. Why? Because there is work leftover after t minutes. The work that would have been done by machine B in t minutes is leftover and is done by machine A in 64 minutes.

Time taken by A : Time taken by B = 64:t ….. (I)

Similarly, when machine B works alone, it takes 25 extra minutes to complete the work that machine A would have done in t minutes.

Time taken by A : Time taken by B = t:25 ……(II)”

Hi Karishma,

I’m having a hard time understanding how you came up with those two ratios:

Time taken by A : Time taken by B = 64:t ….. (I)

Time taken by A : Time taken by B = t:25 ……(II)

Why is the time taken by A = 64 and time taken by B = t?

Then in the second ratio, time taken by A = t and time taken by B = 25?

Can you make up some easier numbers instead of variables to explain your reasoning? Thank you.

Sure.

Say, Mike and Molly start painting a fence together. They paint at their own respective speeds. They take 2 hrs to finish the job together. If Mike works alone, he takes 3 hrs to finish the job. What can we say about the ratio of their time taken?

Why does Mike take 1 hr extra when he works alone? Because the part of the fence that Molly was painting in 2 hrs is not getting painted now. So after Mike paints for 2 hrs, he has to paint the rest of the fence too. We are told that he takes 1 hr to paint that ‘rest of the fence’. So whatever work was getting done by Molly in 2 hrs is now getting done by Mike in 1 hr. So what is the ratio of their time taken?

Mike:Molly = 1:2

We can say that if you give them equal work (painting some fence), time taken by Mike: time taken by Molly will be in the ratio 1:2

Similarly, if we say that Molly working alone takes 6 hrs to paint the fence, it means that she works for 2 hrs first (the way she was doing with Mike) and then works another 4 hrs to complete the work that Mike was doing in 2 hrs.

Ratio of time taken by Mike:Molly will still be 1:2.

We use this logic to find the ratio of time taken in two different scenarios in this question.

Hi Karishma,

Thank you so much for taking your time to help me understand this concept. I love how you use my name in the example to make it more personal. You are AWESOME! :o)

My Pleasure!

I’m sorry, but I think that #3 is dead wrong.

There are 2 ways to interpret the last sentence of the question “What is the ratio of the time taken by machine A to the time taken by machine B to complete this work?”:

1) It’s asking what is the ratio of the time machine A spends to the the time machine B spends while working TOGETHER.

2) It’s asking what the ratio of the time machine A spends working alone to the time machine B spends working INDIVIDUALLY

Let’s start off by saying we all agree that t = 40

If we interpret the question like 1):

The answer of this is just going to be 1. They are working together and will spend t = 40 minutes working on it. Neither of them works for longer than the other. They will both work 40 minutes and so the ratio is 40:40 = 1.

While there times will be the same, the amount of WORK they do will differ. The amount of work they do is (rateA + rateB)*t = (1/104 + 1/65)*40 = 40/104 + 40/65 = 5/13 + 8/13.

Here we see that the ratio of the amount of WORK they individually do is in fact 8:5, but the amount of TIME does not differ between the two. It is always 40.

Now if we interpret the question like 2):

Rate of A = 1/(40 + 64) = 1/104, so the time will be 104 to complete on its own

Rate of B = 1/(40 + 25) = 1/65, so the time will be 65 to complete on its own.

Therefore the answer under this assumption is simply 104/65, which is not a choice.

Hi karishma,

Thanks for the brilliant post as always.

Could you mind to tell how to solve the below problem with such an approach?

Tom, working alone, can paint a room in 6 hours. Peter and John, working independently, can paint the same

room in 3 hours and 2 hours, respectively. Tom starts painting the room and works on his own for one hour.

He is then joined by Peter and they work together for an hour. Finally, John joins them and the three of them

work together to finish the room, each one working at his respective rate. What fraction of the whole job was

done by Peter?

Tom’s rate = 1/6

Peter’s rate = 1/3

John’s rate = 1/2

In 1 hr, Tom completes 1/6th of the work and you have 5/6 leftover.

In the next hr, Tom and Peter do 1/6 + 1/3 = 1/2 of the work and now 5/6 – 1/2 = 1/3 of the work is leftover.

Rate of all three together = 1/6 + 1/3 + 1/2 = 1 work per hour. So they complete 1/3th of the work in 1/3 of an hour.

Peter worked for a total of 1 + 1/3 = 4/3 hrs at a rate of 1/3.

Work done by Peter = Rate*Time = 1/3 * 4/3 = 4/9th of the work.

Got it!! rates of A:B:C= 1/6:1/3:1/2= 1:2:3

rate of a= 1/(1+2+3)=1/6

rate of b= 2/6

rate of c=3/6

so in 2 hrs. work done by a=1/6*2=1/3

In the second hour ,work done by b =1/3*1=1/3

So total work left=1-2/3=1/3

now a b & c have to do this together.

now a would do 1/6 th of the remaining work in the same time as b would do 1/3rd of the remaining work in same time and c would do 1/2 of the remaining work

So work done by b(i.e. peter)= 1/3 + 1/3*1/3=4/9

Just one doubt , why should i take rate as 1/6 , while rate could be (1/6)k where k is any real number. please reply!!

thanks in advance

We say rate of A is 1/6 when A completes a work in 6 hrs. The reason is the the unit of rate is work/time = 1 work/6 hrs. So basically, when we say that A’s rate is 1/6, it implies that A completes 1/6th of the work in one hour.