Real GMAT® Problems Ep. 44: Work/Rate Problems

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Problem 1: Two Machines, 4 Hours and 3 Hours

Welcome to Real GMAT® Problems Episode 44. Doing real GMAT® problems is one of the most proven ways to make tests that go well, so let's talk through a few examples.

The problems we're going to go through are from the 11th edition of the Official Guide for GMAT® Review and you can web search them if you'd like to follow along visually. Just so you know, the 11th edition of the Official Guide is out of print and that's partly why I'm going through questions from that book — because that way if you purchase this year's Official Guide, there shouldn't be any overlap between the questions that we're going to work through here.

The Official Guide contains retired problems that have appeared on past GMAT® exams. So these are going to be really, really similar to what you're going to see on your official GMAT®. Standardized tests are designed to change very slowly over the decades and the GMAT® is no exception.

I'll give you a chance to work through the solution on your own if you'd like and then I'll discuss my thoughts on what to take away from each question. I recommend pausing after I read through each problem and try to solve it — or at the very least visualize how you would solve it. You may not be able to get through all the steps in your head like you would if you could on test day, but just do your best.

If you are in a situation where you can physically write down what you would if you were actually in the test center, then that's definitely ideal.

We're going to start with a lower difficulty level warm-up problem and then we're going to bump up the difficulty level from there. In fact, we're going to start stretching the difficulty level a bit higher today than we have in the past — quite a bit higher. So don't worry if these questions are a little too difficult for you right now. Feel free to go back into a lower numbered episode of the series. We're on 44 now, so there's 43 other episodes you can do with questions that are steadily increasing in difficulty over time. And work your way back up to where you can feel confident on these.

Today we're going to be talking about work questions — rate and work questions.

Here's our first problem.

The problem says: It would take one machine four hours to complete a large production order and another machine three hours to complete the same order. How many hours would it take both machines working simultaneously at their respective constant rates to complete the order?

Option A is 7/12. Option B is 1 and 1/2. Option C is 1 and 5/7. Option D is 3 and 1/2 and Option E is 7.

Again, recommend pausing and getting as far as you can. I'm just going to jump into my commentary here so we don't have a bunch of dead air for you.

It's a really good warm-up problem for work questions and it's a good way to revisit a concept that we've spoken about actually quite a bit in the past for rates questions, which is the rate chart.

Rate charts can work really well because they're basically a specific way of using rows and columns, which we've talked about a lot in relation to word problems, in relation to visual organization, and in relation to helping you make hard problems easier.

What I think we're going to see as we go through this question is that it's more of an organizational challenge than a mathematical challenge. And that's not true of every work question, but it's true of a good number of them.

So I'll do my best to emphasize the good habits that I think are going to help you the most as you move forward with these.

The first good habit is to become consistent with making the rate chart each time you see a rate question. And a good signal you're looking at a rate question is when you literally see the word "rate" or "rates" like we see here. So as soon as you see that word, you think okay, I could probably use a rate chart here.

The rate chart is just a specific instance of simple rows and columns. We've talked about rows and columns a ton on the pod. So we've done our best to make all these episodes searchable — if you search "word problems" or you search "rates," you should find some good lower difficulty level questions that deal with the rows and columns approach. For now, I'm not going to cover that again. I'm just going to go straight into the rate chart.

Setting up the chart. Your columns will always be the same in rates questions. The first column is Rate. The second column is Time. And the third column is Work (or Distance, if it's a distance problem — we're going to focus on work today). So the very first thing I would write is Rate × Time = Work. Those will function as my three column labels.

Then each row label will be whatever thing is working in the problem — whatever they're telling me is doing work. Usually, each thing that's working gets its own row.

For example, we're told it would take one machine four hours to complete a large production order. So my first row label would just say Machine 1. We're also told another machine takes three hours to complete the same order. So my second row would be Machine 2.

I've got two rows right now: Machine 1 and Machine 2. I can start to fill in the information that they give me about rate, time, and work.

For Machine 1, we're told that it can complete one order. So I would write "1 order" under the Work column. We're told it can complete that order in four hours. So let's write "4 hours" under the Time column.

We're also given information about Machine 2. It can complete the same order. So again, "1 order" under Work. Machine 2 completes that order in three hours. So "3 hours" under Time.

There's a lot to like about the rate chart. One of the main things I really like about it that I think a lot of people underestimate is how simple it is to set up. And what I like about that is it can get you a lot of momentum in these questions where a lot of people kind of freeze on word problems, especially rate problems.

Sometimes the mind can get going and work against us a little bit — telling us things like "hey, you should know how to do this, this isn't that hard, where are you struggling?" And obviously none of that affects your performance positively on the question.

So it's nice to be able to just default to a set of actions that you know is productive and also doesn't take too much brain power. It's quite thoughtless to set up. And I like that at the beginning of what could potentially be an intimidating problem.

Solving for rates. Once you've got all that filled in, you'll often be able to solve for other elements of the chart. And when you can, it's almost always a good idea to do that.

For Machine 1: What rate would I multiply by 4 hours to produce 1 order? Hopefully you're thinking 1/4. So I would write "1/4 orders per hour" in Machine 1's Rate column.

You'll notice I'm being very heavy-handed with my recommendation of writing the actual units into the table. Might seem like a waste of time. Might feel like you don't have time to do it. I would say in 99% of cases that I've ever seen — which is a lot of cases at this point — that helps people a ton. It helps people connect the dots between the pieces of information. It helps avoid missing questions you know how to do.

For Machine 2: What times 3 hours equals 1 order? 1/3. So write "1/3 orders per hour."

Adding rates for combined work. This is a really common move when people, machines, robots — that kind of thing — are working together. If they work together, usually the work gets done faster.

What can be less intuitive is that you simply add the individual rates to get the combined rate. Think of it this way: imagine one robot can build 4 cars in an hour and another robot can build 7 cars in an hour. If they work simultaneously, how many cars can they produce together in one hour? Hopefully you're thinking 11. Same concept here — the only difference is we have fractional rates.

So make a "Together" row (or "M1 + M2"). Write "1 order" under Work. Then add the rates: 1/4 + 1/3. Common denominator of 12 gives us 3/12 + 4/12 = 7/12 orders per hour.

Finding the answer. Now: 7/12 orders per hour × how many hours = 1 order? That's 12/7 hours. Convert to a mixed number: 1 and 5/7.

That gets us to our correct answer of Option C.

Why 7/12 is the most common wrong answer. The most common incorrect answer here is Option A — 7/12. That's the combined rate, not the time. It's really easy to mix up rate and time, especially if you're taking an algebraic approach. You get to the end, see 7/12, and it's very tempting to pick it.

That's where the chart comes in. If everything is clearly labeled — "7/12 orders per hour" — you're much less likely to pick a number that represents the wrong thing. About 12% of test takers end up going for A. Think about how painful that is: they probably knew exactly how to do the problem strategically and mathematically, and they still got it wrong.

If there's one skill that's the most foundational for getting work questions right, it's good organization habits.

Problem 2: Machine Y and Machine Z

Two machines, Y and Z, work at constant rates, producing identical items. Machine Y produces 3 items in the same time Machine Z produces 2 items. If Machine Y takes 9 minutes to produce a batch of items, how many minutes does it take for Machine Z to produce the same number of items?

Option A is 6. Option B is 9. Option C is 9 and 1/2. Option D is 12. And Option E is 13 and 1/2.

This one's a good follow-up — not too much harder than the previous one, but definitely more challenging. About 50% more people miss this one. It takes a lot of the same concepts and adds more complexity.

Setting up the chart. Once again, the word "rates" appears — that's your signal. Rate × Time = Work across the top. Make a row for Machine Y and a row for Machine Z.

We're told Y produces 3 items and Z produces 2 items in the same time. Write "3 items" under Work for Y and "2 items" under Work for Z.

Using variables. Instead of giving us a specific time, the problem says "in the same time." When you see that, use a variable. Write T under Time for both machines. We don't know the time, but we know it's the same for both.

Solving for rates. For Machine Y: What times T equals 3? That's 3/T items per minute. For Machine Z: What times T equals 2? That's 2/T items per minute.

Adding new rows. We're told Y takes 9 minutes to produce a batch. That's new information — it conflicts with the original row for Y. Just make a new row for Machine Y below the others. Write "9 minutes" for Time and bring down the rate we already calculated: 3/T items per minute.

Now: 3/T × 9 = 27/T items. That's how many items Y produces in 9 minutes.

Solving for Z. The question asks how long Z takes to produce the same number of items. Make a new row for Z. Write "27/T items" under Work and bring down Z's rate: 2/T items per minute. Solve: 2/T × P = 27/T. Divide both sides by 2/T (multiply by T/2). The T's cancel: P = 27/2 = 13 and 1/2.

That gets us to Option E.

Where people get stuck: (1) No system for organizing the data. (2) Not knowing when to introduce a variable for T. (3) Not knowing how to handle the second piece of information about 9 minutes — just add more rows to the chart.

Problem 3: Pumps A, B, and C

Pumps A, B, and C operate at their respective constant rates. Pumps A and B operating simultaneously can fill a certain tank in 6/5 hours. Pumps A and C operating simultaneously can fill the tank in 3/2 hours. And Pumps B and C operating simultaneously can fill the tank in 2 hours. How many hours does it take Pumps A, B, and C operating simultaneously to fill the tank?

Option A is 1/3. Option B is 1/2. Option C is 2/3. Option D is 5/6. And Option E is 1.

By the numbers, this one is quite a bit harder — only 30% get it right. About 70% of us get the first one right, and about 70% of us get this one wrong.

Setting up the chart. Rate × Time = Work across the top. Rows: A+B, A+C, B+C, and A+B+C (what we're solving for). Fill in Work as "1 tank" for every row. Fill in the times: 6/5 hours, 3/2 hours, 2 hours, and "?" for the combined row.

Solving for rates. A+B: What times 6/5 equals 1? That's 5/6 tanks per hour. A+C: What times 3/2 equals 1? That's 2/3 tanks per hour. B+C: What times 2 equals 1? That's 1/2 tanks per hour.

The shortcut. You could set up three equations and solve for A, B, and C individually. That works but it's a lot of steps and each step is a chance for a fraction error.

Instead, notice that if you add the rows: (A+B) + (A+C) + (B+C) = 2A + 2B + 2C. Add the rates: 5/6 + 2/3 + 1/2. Common denominator of 6: 5/6 + 4/6 + 3/6 = 12/6 = 2 tanks per hour.

That's the rate for 2A + 2B + 2C. Divide by 2: A + B + C = 1 tank per hour. If the combined rate is 1 tank per hour, the time to fill 1 tank is 1 hour.

That gets us to Option E.

Closing Thoughts

Work and rate questions generally require more practice to get comfortable with than other problem types. It was true for Isaac back in the day — it took a lot longer to get comfortable with rates than he wanted. So don't stress if it's not coming naturally or quickly.

Come back to this episode a few times in the coming days. Re-solving problems you've already seen can be surprisingly helpful — a lot of people don't think it would be, but you'll find a lot of value in it.

If you're struggling with fraction math, check out the Math Basics episodes on our podcast — fractions, remainders, and algebra. They'll help across many question types.

If your studies are progressing at exactly the pace you want, keep doing what you're doing. We'll be back soon to support your success for free right here.