Work rate problems sometimes cause more anxiety than they need to. First off, work rates aren’t that common. Often, you can take an entire GRE math section without seeing a single work rate problem. Even if you see a work rate problem, by following a simple formula, you should be able to get most of them correct.

First, try the problem below.

Jonas takes 5 hours to paint a fence. Mark takes twice as long to paint the same fence. Working together, how long will it take them to complete the fence?

(A) 2 hrs

(B) 5/2 hrs

(C) 10/3 hrs

(D) 5 hrs

(E) 7.5 hrs

First off, we want to note that Mark takes twice as long as Jonas, so he takes 10 hours to paint the fence alone. With this information, we next need to find how much of a fence each can paint in one hour. By getting this hourly rate, we simply add up the amount of fence they paint in one hour. This number tells us how much of the fence they paint together in one hour. So, letโs do the math up until this point.

1/5 the amount of fence Jonas paints in one hour.

1/10 the amount Mark paints in one hour.

To find the work rate, we must first add the two independent hourly rates: 1/5 + 1/10 = 3/10, which is the amount of fence they paint together in one hour. At a rate of 3/10 of a fence together, how long is it going to take them to paint an entire fence? One approach is to set up a simple equation:

3/10 x = 1, where 1 stands for the entire job. Solving for x, or the combined work rate, we get 10/3, Answer C, or the reciprocal of 3/10.

A good rule of thumb is that whatever the rate is in one hour, in this case 3/10 of a fence, just take the reciprocal of that fraction to find how long it would take them to paint an entire fence.

An even quicker way is to set up a fraction. In the numerator, you will multiply their respective rates, in this case 10 x 5, and in the denominator you will add their rates, 10 + 5. This gives you 50/15, which, when reduced, equals 10/3. This quick method of finding work rate applies only when you are working with two rates.

To recap: to find the work rate, first find the hourly rates for each individual. Then, add these two rates together, and then flip, or take the reciprocal of, that fraction. Itโs that easy. As they say, itโs nothing to get worked up about!

Magoosh Test Prep ExpertFebruary 11, 2018 at 12:36 pm#

Hi Bishop,

Yes, that’s an easy way to look at it! This might be more difficult for students who have a hard time recognizing the “big picture” of the problem. You’ve essentially combined the last few steps to create one equation, which is easier to solve if you can see how to set it up! One thing I would like to mention, however, is that I would caution you against converting these numbers into decimals–that is an easy way to make a silly mistake, particularly if using the calculator!

Thanks Chris! This is really helpful. I’ve been dreading this topic for as long as I can remember. 3 days to test day and I feel better now. Glad I found this ๐

Running at their respective rates Machine A takes 4 days longer to produce x widgets than Machine B. At these rates, if the two machines together produce 2x widgets in 3 days, how many days would it take Machine A alone to produce 5/2x widgets? I’m currently learning rates and seem to be stuck with this. Any help?

First, you’ve got your individual rates for Machines A and B, Ra and Rb respectively, both producing x widgets. Using Work = Rate x Time: Ra= x/t Rb= x/(t-4) <— the problem tells you Machine A's time is 4 days longer so Ta=t and Tb= t-4

The combined rate for A and B will be Ra+Rb. You also know that this rate will produce 2x widgets in 3 days so Ra+Rb = 2x/3

You get the following equation:

x/t + x/(t-4) = 2x/3

(xt + xt + 4x)/(t^2 – 4t) = 2x/3

2x (t+2) / (t^2 – 4t) = 2x/3

Divide each side by 2x

(t+2)/(t^2 – 4t) = 1/3

cross multiply

3t + 6 = t^2 – 4t

0 = t^2 – 7t -6

Solving the quadratic equation you get (t – 6) (t-1) = 0 and roots t= 6 and t = 1 We know t = 1 won't work (If Time for A is 1 day, time for B can't be 1-4= -3 as time can't be negative). So machine A takes 6 days to produce x widgets

Now we have all the information for Machine A to solve the question: rate=x/6 and we wan't to produce 5x/2 at that rate

T = (5x/2) / (x/6) T = 5x(6)/2x T= 30/2 T= 15 days

I don't know if there's a faster way to solve this problem but there you go

Magoosh Test Prep ExpertApril 23, 2016 at 11:28 am#

Hi Remya,

Happy to help! ๐

The per hour rate at which each man works is just R = 1/(3 x 8) = 1/24 houses per hour. It is weird to talk about houses this way, but this matches how we have dealt with the rate in the post above here, and it is going to allow us to compare in a moment.

The rate for 5 men is just 5R. The work is 1 total house. This means that our equation gives us 1 = 5/24T.

Chris, Thanks for making this understandable, and even fun! The lessons overwhelmed me, but your blogs have given me that “Aha!” moment. Wish I had been studying your blogs these last three months, my score would be even better.

Thank you for the question, Chris. There is a much faster way to do this problem given the choices above.

Since the faster of the two can finish it alone in 5 hrs, together they should finish it in less than 5 hrs but more than 2.5 hrs (other person is slower). Voila, the answer is C

Great solution! That’s exactly how the GRE wants you to think (the spread of the answer choice is anything but arbitrary).

So, yeah, 2.5 is too low. If two people working at 5 hrs worked together, they would finish in 2.5, therefore the answer is (C).

With my quick method of combined work rate equals xy/x + y, which is 50/15 can get you the answer even faster! It seems that the GRE mixes things up to thwart such an approach. They will use something like 45 minutes and 2.5 hours. So in general your intuitive approach will help you test day :).

Magoosh blog comment policy: To create the best experience for our readers, we will only approve comments that are relevant to the article, general enough to be helpful to other students, concise, and well-written! ๐ Due to the high volume of comments across all of our blogs, we cannot promise that all comments will receive responses from our instructors.

We highly encourage students to help each other out and respond to other students' comments if you can!

If you are a Premium Magoosh student and would like more personalized service from our instructors, you can use the Help tab on the Magoosh dashboard. Thanks!

These types of problems can also be solved with a SIMPLE formula of

A * B/A + B

1) Jonas’ Time * Mark’s Time / Jonas’ Time + Mark’s Time =

2) 10 * 5 / 10 + 5

3) 50 / 15 = 10/3

Done.

That is an excellent shortcut, Bishop, and it really will work for problems of this type. Thanks for sharing. ๐

Very simple if you look at it like this:

1/(Jonas time) + 1/(Mark’s time) = 1/(combined time)=your answer

STEP 1: 1/5 + 1/10 = 0.3

STEP 2: 1/0.3 = 3.3 or 10/3 hrs

done

Hi Bishop,

Yes, that’s an easy way to look at it! This might be more difficult for students who have a hard time recognizing the “big picture” of the problem. You’ve essentially combined the last few steps to create one equation, which is easier to solve if you can see how to set it up! One thing I would like to mention, however, is that I would caution you against converting these numbers into decimals–that is an easy way to make a silly mistake, particularly if using the calculator!

Thanks Chris! This is really helpful. I’ve been dreading this topic for as long as I can remember. 3 days to test day and I feel better now. Glad I found this ๐

Running at their respective rates Machine A takes 4 days longer to produce x widgets than Machine B. At these rates, if the two machines together produce 2x widgets in 3 days, how many days would it take Machine A alone to produce 5/2x widgets?

I’m currently learning rates and seem to be stuck with this. Any help?

Is the answer 15/(x^2)

The answer is 5

First, you’ve got your individual rates for Machines A and B, Ra and Rb respectively, both producing x widgets. Using Work = Rate x Time:

Ra= x/t

Rb= x/(t-4) <— the problem tells you Machine A's time is 4 days longer so Ta=t and Tb= t-4

The combined rate for A and B will be Ra+Rb. You also know that this rate will produce 2x widgets in 3 days so Ra+Rb = 2x/3

You get the following equation:

x/t + x/(t-4) = 2x/3

(xt + xt + 4x)/(t^2 – 4t) = 2x/3

2x (t+2) / (t^2 – 4t) = 2x/3

Divide each side by 2x

(t+2)/(t^2 – 4t) = 1/3

cross multiply

3t + 6 = t^2 – 4t

0 = t^2 – 7t -6

Solving the quadratic equation you get (t – 6) (t-1) = 0 and roots t= 6 and t = 1

We know t = 1 won't work (If Time for A is 1 day, time for B can't be 1-4= -3 as time can't be negative). So machine A takes 6 days to produce x widgets

Now we have all the information for Machine A to solve the question: rate=x/6 and we wan't to produce 5x/2 at that rate

T = (5x/2) / (x/6)

T = 5x(6)/2x

T= 30/2

T= 15 days

I don't know if there's a faster way to solve this problem but there you go

Post the problem with necessary information.

1. Where are the answer choices.

2. Does the question require formula to be derived?.

Answer depends on whole information you going to provided.

Thanks

Arun

Whoa! You just saved me so much time. This trick is so much easier than the Magoosh video explanation in the lesson.

Hi Chris!

What is the shortest way to solve below problem:

It takes 3 men 8 hours to paint a house. How long will it take 5 men to paint the same house?

Hi Remya,

Happy to help! ๐

The per hour rate at which each man works is just R = 1/(3 x 8) = 1/24 houses per hour. It is weird to talk about houses this way, but this matches how we have dealt with the rate in the post above here, and it is going to allow us to compare in a moment.The rate for 5 men is just 5R. The work is 1 total house. This means that our equation gives us

1 = 5/24T.T = 24/5 = 4.8 hours

I hope that helps! ๐

Thank you Chris!

You’re very welcome, Remya! ๐

Chris,

Thanks for making this understandable, and even fun! The lessons overwhelmed me, but your blogs have given me that “Aha!” moment. Wish I had been studying your blogs these last three months, my score would be even better.

Thanks Mary!

I’m glad I’ve been making some of these seemingly inscrutable problems much easier ๐

Chris

But the actual GRE work rate problems cannot be that simple. Right?

Thank you for the question, Chris. There is a much faster way to do this problem given the choices above.

Since the faster of the two can finish it alone in 5 hrs, together they should finish it in less than 5 hrs but more than 2.5 hrs (other person is slower). Voila, the answer is C

^_^

Great solution! That’s exactly how the GRE wants you to think (the spread of the answer choice is anything but arbitrary).

So, yeah, 2.5 is too low. If two people working at 5 hrs worked together, they would finish in 2.5, therefore the answer is (C).

With my quick method of combined work rate equals xy/x + y, which is 50/15 can get you the answer even faster! It seems that the GRE mixes things up to thwart such an approach. They will use something like 45 minutes and 2.5 hours. So in general your intuitive approach will help you test day :).

Hey Chris, thanxx man !

Thanks a lot Chris. I love Magoosh blogs. You make intimidating things so easy and approachable. ๐

You are welcome! My motto is GRE math doesn’t have to be horrible :).