Real GMAT® Problems Ep. 34: Word Translations

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Problem 1: A Team Won 80% of Its First 100 Games

Welcome to Real GMAT® Problems, Episode 34. Doing real GMAT® problems is one of the most tried and true ways to improve your performance on test day, 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 (which is a great resource), there shouldn't be any overlap between the questions 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 extremely similar to what you're going to see on your official GMAT®.

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 example. I recommend pausing after I read through each problem and trying to solve it, or at the very least visualizing how you would solve it. If you're on the go, you might not be able to execute all the steps in your head, but just get as far as you can. It's an excellent mental workout, and you should find that your scrap paper techniques and your strategizing work better when you get back to the desk and the paper. If you're in a situation where you can physically write down what you would on test day, that's definitely ideal.

Let's start with a warm-up problem, and then we'll explore some more complex questions that build on a couple of the central lessons we're going to explore today.

Here's our first problem.

During a certain season, a team won 80% of its first 100 games and 50% of its remaining games. If the team won 70% of its games for the entire season, what was the total number of games that the team played?

Option A is 180. Option B is 170. Option C is 156. Option D is 150. Option E is 105.

I wanted to dig into some longer word translations, deeper dives on word translation specifically, and hopefully you're going to find a lot of value in this, because this type of question is heavy on percent and heavy on words. If you haven't paused and worked through the problem on your own yet, please make an attempt. I'm just going to jump right in.

Percent word problems are some of the most common that appear on the quant section, so it's almost always a good bet to practice those. It's a very good warm-up problem for the lesson, and it's also somewhat long. It's certainly not the longest word problem you're going to see, but it's a good example of what happens to a lot of people on long word problems, and I'm going to try to build on that as we go.

Another reason it's a good warm-up problem: it's pretty intense when it comes to organization. You've probably heard me say in the past that most word problems are more of an organizational challenge than a mathematical challenge, and I think the more of these GMAT® quant word problems you do, the more likely you are to find that to be true.

Writing what's given and asked. The first key to a great solution to a problem like this is just writing what's given and asked clearly. That should come as no surprise if you've been following me and this content for some time, but it doesn't make it any less true. For some of you, that might sound redundant or maybe even overly obvious — but even so, you might be surprised how few people do this well consistently. It's actually not that hard to do things well once in a while, but to do things well 100% of the time is usually really, really, really challenging for people. In fact, if you're struggling to be consistent in general — whether that's with your study time, your approach to questions, or just basics like sleep, diet, exercise, et cetera — we've got an episode on how to be more consistent specifically related to the GMAT® and your GMAT® prep. That's linked below for you. It's not medical advice, I want to be clear about that, but it should be super helpful.

Finding a place to start. Let's assume you've written everything that's given and asked clearly. Where do we go from there? If you're ever unsure where to start once you've got the given information mapped out, look for any places you can combine pieces of information or make simple calculations.

I want to re-emphasize: what I'm about to do is not the only way to do the problem. If you see an obvious place to begin, you don't have to take the advice I'm about to give. Go for what you see and get momentum — however it makes sense to you. I'm more speaking to folks who feel stuck or dead in the water: "I wrote everything down, and now I'm staring at my page. Where do I go from here?"

In this case, I can see that 80% of the first 100 games are wins, and I know I can make that calculation relatively easily. That's a great place to start, and that's what I'm recommending: just look for something you can do to get some kind of momentum. With complicated or organizationally intense problems, if you can just solve one small piece of the problem, that can often unlock the next step.

So I'd start by writing that we've got 80 wins out of the first 100 games. I would probably also write 80 over 100 times 100. That's me, because I like to be super, super hardcore with my scratch work. I really, really hate missing questions I know how to do, and it happens a surprising amount of the time, even at my level. You don't have to do that, just to be clear. If you're never making those kinds of mistakes, I'm never going to criticize your results. But if you're not getting the results you want, then I encourage you to write more down. As simple as it might sound, it should make a very positive difference.

Recognizing when to use rows and columns. You might recall from some of our recent conversations about word problems how useful it is to organize information in simple tables with rows and columns. I don't think that's essential here, but I think you'll probably find it pretty valuable — especially if you're thinking, "OK, I've got 80 out of the first 100 games. Now what? I'm stuck again."

A good way to recognize when it's going to be valuable to use rows and columns: when there's some kind of comparison. Tickets to a show for adults versus tickets for children, 2023 prices versus 2024 prices, 2023 sales versus 2024 sales, revenue versus profit. A recent question we worked through had workers on a day crew and workers on a night crew. Any time you're comparing groups, comparing time periods, or comparing parts of a whole, rows and columns are likely to make a huge, positive organizational effect — and it's going to make questions that are challenging to organize otherwise a lot easier.

Going back to this problem: do you notice anything that might suggest a comparison or parts of a whole? I'm looking at things like wins versus total games, or first 100 games versus the rest of the season. Those are comparisons of time periods or parts of a whole that suggest rows and columns, and they're good cues to look out for in future problems too. Sometimes just solving that one small piece — even when it seems trivial — is enough to trigger the recognition.

Setting up the table. Make a column header that says Wins, and another that says Games. That's the part-to-whole relationship. Under Wins: 80. Under Games: 100. Label that row "First" to indicate that's the first part of the season. Of course, you can organize this a variety of ways — I'm just giving you one example.

Then I like to go back to the problem and ask myself: what given information have I not used yet? It's extremely unusual for quantitative questions on the GMAT® to present you with information you don't need to answer the question. If they're telling you, it's probably essential. If you get stuck, asking what given info you haven't used is often the key to the next step.

We're told that 50% of the remaining games are wins. We're not told how many more games they played, so that's an unknown — which suggests a variable. Let's call it R for remaining games. Make a new row under the first (label it "Remaining"), put R under Games, and 50 over 100 times R under Wins. That captures the wins and total games after the first 100 games.

Building the total row. Next piece of given information: 70% of the games in the entire season are wins. I don't have that attached to anything yet, and it's telling me about total games and total wins. Make another row called Total: 100 + R under Games (the first 100 games plus the unknown R games remaining), and 70 over 100 times (100 + R) under Wins.

At this point, the table makes the final connections way easier. Some problems are just flat-out hard. We can't make them easy, but we can make them easier with good organizational strategies and a consistent approach.

Setting up the equation. Hopefully you can see that you can take the top two entries in the Wins column — 80 for the first 100 games and 50 over 100 R for the remaining games — add those together, and that should equal the total wins for the whole season.

So: 80 + (50 over 100)R = (70 over 100)(100 + R).

Plenty of you probably do not need these rows and columns to make that connection. That's totally cool. I'm never going to argue with your results — if you're getting incredibly good results without anything I'm talking about here, don't stop what you're doing. But my guess is you will probably benefit from a table like this on harder or more complicated questions, where it wasn't as automatic. And for others of you, you'd be totally stuck without the rows and columns, so hopefully that makes the problem a lot more solvable — at the very least, faster.

On feeling slow. It might feel like you're going too slow when you're setting up the problem. But the objective of the exam is not to feel good. There are plenty of other things you can do with your life to feel great that are going to work out a lot better than this — unless you love really hard puzzles, or really difficult standardized exams that just get more and more difficult the more your skills grow. (If you don't know how the scoring algorithm works, we've got a video on our website about that, linked below — it's just thegmatstrategy.com. It'll teach you how to deal with the scoring algorithm and how it's different from other exams: the test is adaptive, and as you get more questions right, the questions it shows you are more difficult.) So maybe we can't make it easy, but we can make it easier — and we can make your score represent your skills better by adding technique, process, and fundamental habits.

Solving. The basic workflow on a lot of these word problems is just what I talked through: recognize the comparison, create rows and columns, start calculating from what's given, figure out a relationship you can use to solve. You'll do those things over and over and over again.

If you execute the algebra well, you'll get R = 50. And this is where having the rows and columns really helps: you can see that 50 does not represent what the question is asking for. You want total games played. As part of our table, we've got 100 + R written in the Total row under the Games column — so the correct answer is 100 + 50 = 150, which corresponds with Option D.

It's great if you wrote down what the question was asking at the beginning, because then you're unlikely to make a mistake there — or to get 50 and think, "Wait a minute, I must have done some math wrong." Sometimes that last connection in the question is the most difficult piece on word problems, just because there can be a lot of steps.

Back to the speed thing: even though it might feel slower to write the question down at the beginning, or to write the rows and columns down, you'll usually see if you time your practice questions that the clock time to finish is lower. You spend a little more time setting up, but that investment pays off on the back end of the problem by making it a lot easier to connect the dots. If you speed through the setup, you might feel like you're going faster, but the total clock time is longer. Experiment with that and make your decision based on the numbers. You don't have to take my word on any of this — test it out for yourself, use the data. Trust but verify, as they say.

The algebra, quickly. Bring back our equation: 80 + (50 over 100)R = (70 over 100)(100 + R). Distribute on the right side: 70 + (70 over 100)R. Subtract 70 from both sides: 10 + (50 over 100)R = (70 over 100)R. Subtract (50 over 100)R from both sides: 10 = (20 over 100)R. Reduce 20 over 100 to 1/5 — we've got an episode on fractions in the Math Basics series linked below if you're not sure what I did there. So 10 = (1/5)R. Multiply both sides by 5: R = 50.

Many, many ways to solve the algebra there — that's just one. If you want to brush up, we've got those Math Basics episodes below. That's where I started: I had to brush up on all this stuff, and I had forgotten tons of math basics when I came around to studying for the GMAT®. It's a totally fine, acceptable place to begin — and that's why we made all that Math Basics content for you.

Putting a bow on this one. As I said, there are other ways to conceptualize the problem. You don't need to use rows and columns, and you don't even need the kinds of equations I just walked you through. But to make this problem as useful as possible for as many people as possible, I chose to focus on those elements — the shortcut methods you might find elsewhere probably make sense to anyone with a lot of experience, but they won't make much sense if you're just starting out. I've tried to be process-neutral and approach-neutral: what I think will be the most usable no matter which steps you decide to take, or where you are in the prep process, beginning, middle, or end. If you find a different method you like, chances are that's going to be a great bet for future problems similar to this one. No need to change what you're doing if you're already having success.

I chose this one as a warm-up because there doesn't seem to be a major pitfall — most of the wrong answers are just evenly distributed between the options. People either know how to get it right and get it right, or they get totally stuck and guess randomly. That's where I thought having some process help could be valuable. If you're struggling with word problems in general — the translations part, the comprehension part, or the speed part — the fundamentals I just discussed, prioritizing good organization, should help quite a bit.

One final mention: it usually takes more than just a handful of problems to build speed with word translations, and it often takes quite a bit of effort. Don't be afraid to start slow. That's where I started, and it took me a lot longer than I wanted it to, to build a good approach and build speed with these. But it's more than worth it. Word translations and word problems are among the most common topics in the quant section, so any time spent there is generally time well spent.

Problem 2: Jack and Bill (Age Problem)

Now that we're warmed up, let's try a couple more examples of word translations. I'm going to stick with that theme for the rest of this lesson, and we'll build on the basics here.

Next problem. The problem says: Jack is now 14 years older than Bill. If in 10 years Jack will be twice as old as Bill, how old will Jack be in five years?

I'll read that again. Jack is now 14 years older than Bill. If in 10 years Jack will be twice as old as Bill, how old will Jack be in five years?

Option A is 9. Option B is 19. Option C is 21. Option D is 23. Option E is 33.

Again, recommend pausing here. Get as far as you can on your own.

A really interesting follow-up to the previous problem, because on the surface this one looks a lot simpler. But it turns out that while 13% of us miss the first one, 20% of us miss this one. Let's dig into that, and let's see if we can make age problems in general a bit easier for you, because I think a lot of people find them pretty painful.

Start with the fundamentals. Just write what's given and asked. We're told Jack is 14 years older than Bill, we're given a relationship between their ages 10 years from now, and then we're asked for Jack's age in five years. Like the previous problem, there are a few different approaches that can work. Some of you might be super comfortable going straight to algebra and using equations. Some of you might be able to reason your way through using logic. Others of you might feel more comfortable plugging in the answers and guessing and testing. All those approaches work well if you're comfortable with them and you execute well. So once again, let's focus on the execution part and build good habits — starting where most of us seem to be going wrong, and using that to build out a system that will minimize the risk.

The parentheses trap. A lot of folks don't struggle with the first piece: if Jack is currently 14 years older than Bill, we can create a variable for each and make an equation like J = B + 14. But then it's clearly very tempting to do the next part of the problem incorrectly by translating that "in 10 years" part wrong.

It says: if in 10 years Jack will be twice as old as Bill. A lot of folks go for J + 10 = 2 × B. And I've seen a lot of folks do J + 10 = 2 × B + 10. That second one seems to be super popular on this problem.

If you take the first equation, J = B + 14, which is correct, and then take J + 10 = 2B + 10, which is not correct, you'll wind up with 33, which is Option E. It looks like about 11% of us go for that one. It's a really popular wrong answer. It has a lot of gravity.

It's a subtle but important detail we need to adjust in that second equation. If we do J + 10 = 2B + 10, we're actually doubling Bill's age first and then adding 10 years. But we want to go the opposite direction: we want to age him into the future 10 years, and only after that do we want to double it. First he ages 10 years, and at that point Jack is twice as old as he is. We can't write 2 × B + 10, because PEMDAS order of operations is going to dictate that we multiply the B by 2 first and add the 10 later, and that gets us the wrong answer.

To get the relationship right, use a set of parentheses: J + 10 = 2 × (B + 10). That is the correct way to translate that piece, because it triggers the right order of operations. If you want to refresh on PEMDAS or what I mean by order of operations, we've got a Math Basics episode on that linked below as well.

Making it easier to see. The next natural question: how can you make it easier to see and set up the correct relationship? A couple of options.

As I've mentioned many times in the past, keeping a list of questions that contain important concepts you want to master, and re-solving some of those questions each time you study, can really help with the kind of pattern recognition we're talking about. So if you're thinking, "Man, next time I see an age problem, I really want to remember that parentheses thing," then file this question away in a list and re-solve it a few times over the coming weeks or months. That'll definitely help.

Another option, which you might suspect is coming: rows and columns. Now, this is not a silver bullet — it won't work without some diligence and ingenuity. It's not going to just solve everything by thinking rows and columns and then the light shines down from the heavens and bestows a great GMAT® score on you. But it definitely is really helpful, and I want to go deep on it here because this is clearly where the issue is.

The time-shift table. You can organize the rows and columns however you like — I'm just going to give one potential example. I'm going to use the time periods in the problem as the column headers. I could do Now, then 5 Years From Now (which is what the question asks), then 10 Years From Now (which is what the question gives me). For me personally, I don't find a lot of value in doing it in chronological order — I just do it in the order they present it in the problem: Now, then 10 years from now, then 5 years from now.

Rows: one for Jack, one for Bill. In Jack's row under the Now header, I can write J. Or another option is to go straight to writing B + 14 — either one works, whichever makes the most sense to you. Under Bill, Now: B. As basic as it might sound, under the In 10 Years header, I would write J + 10 and B + 10. Under the In 5 Years column, I would write J + 5 = ? — something like that to remind me what my objective is. What's my North Star? What am I looking to solve for?

From there, a lot of the math is going to be exactly the same as what we just did. But having that very organized setup really helps cut down on the algebra mistakes — like missing the parentheses piece, or doing 2B but forgetting to age Bill 10 years into the future at all. That's really common too.

It's quite a similar tip to the one I gave around Episode 26, when I recommended writing some half-math, half-English as your notes after you read a word problem but before you set up an equation. This rows-and-columns thing is kind of a version of that — a bridge between the logic of the question and the equations you're going to set up. Building that bridge in advance really, really helps nail down the right equations, because it's really hard to separate the logic that's happening in word problems from the math that's happening in word problems. If you try to rush into the equation, you often make a mistake in the logic.

That's what I was saying on the previous question: it often feels slower to do a good setup, but it often pays quite a bit in the net gain on time. J + 10 = 2B + 10 is super close to being right — that's just a small difference in logical understanding. Having slightly increased odds that you've got the right logical understanding could be the difference between getting this question right and wrong. It's those small little setup differences that compound as you go through the question.

The solution, quickly. If we set it up properly with J = B + 14 and the correct equation J + 10 = 2 × (B + 10), we can do some simple order of operations and algebra. Distribute the 2 on the right-hand side: 2B + 20. Subtract the 10 from J + 10 on the left and combine like terms: J = 2B + 10.

Then solve for B in the original equation, J = B + 14: subtract 14 from both sides to get B = J − 14. In substitution, I want the B to disappear, because I'm solving for J — we've got a whole Math Basics episode on that, linked below, called Systems of Equations: how to solve multiple equations at the same time. It might have been a while for you.

Substitute J − 14 for B: J = 2 × (J − 14) + 10. Distribute the 2: J = 2J − 28 + 10, so J = 2J − 18. Subtract J from both sides and add 18: J = 18.

Now, they were nice to us and didn't put 18 as one of the answer options. But on a harder version of this question, they might have. That's why you also want J + 5 = ? in your chart. Jack's age in five years is 23 — which also happens to be Option D.

They'll often throw that kind of wrinkle in at the end just to test your organizational skills, your fortitude, your mental stamina, your attention to detail — all the things most of us would say we admire in great business leaders. I'm not saying this test measures your leadership potential perfectly. It definitely doesn't. But they're at least trying to measure those types of skills, and it's a good way to reframe the exam: instead of just having it be this evil thing in the way of you and your dreams, it's an opportunity to build those skills even before you start business school. I know that might not resonate with a lot of you. But if it's between feeling horrible about it and feeling like it's benefiting you in some way, personally, I would rather look at it the way that makes me feel positive about it. Maybe you thrive on negativity — that's cool. But that's not the majority of us.

Problem 3: Towel Pricing (Inferring the Original Situation)

Last problem for you. The problem says: A store currently charges the same price for each towel that it sells. If the current price of each towel were to be increased by $1, 10 fewer of the towels could be bought for $120, excluding sales tax. What is the current price of each towel?

Option A is $1. Option B is $2. Option C is $3. Option D is $4. Option E is $12.

Again, go ahead and pause. See how far you can get on your own.

A good follow-up, in my opinion, to our previous problems — and actually a great follow-up to one of the questions we did together in our previous episode about quadratic equations. Hopefully you'll be able to use both of those discussions to your advantage here.

Inferring the original situation. First things first: probably comes as no surprise that I recommend writing what's given and asked, and noticing we've got a comparison here — an old price and old quantity, and a new price and new quantity. What's tricky for a lot of folks with this question is that the old price and quantity aren't explicitly mentioned in the problem. You'll need to infer that because there's a new price-quantity relationship for $120, there must be an old price-quantity relationship for $120 as well. That's not necessarily easy to do, and I've noticed a lot of people get stuck right at the beginning of this problem on exactly that.

What can make that a lot easier: rows and columns. Just experimenting with hammering a technique here for you — I haven't done this in past episodes, so definitely give us feedback. If you're like, "Hey, that's really annoying, please, please stop," just let us know. We're super open to feedback, and you definitely can't hurt our feelings. We're at The GMAT® Strategy on the current social channels, and you can email us at contact@thegmatstrategy.com.

Notice that this question asks for the current price of each towel — that's a hint that there's a current situation as well as a new situation. There's also a hint earlier in the problem, when we're told "if the current price of each towel were to be increased." So there's a current price, and then an increased price of $1 more. That's your cue that rows and columns would work well — there's a comparison, or a change over time.

Setting up Current vs. New. Column headers: Current, and New. Row one, price per towel: under Current, make a variable, P, since that's unknown. Under New: P + 1. The problem says the current price were increased by a dollar — so I don't know what the current price is, and if there's an unknown in the question, that's usually the signal to make a variable for it and operate on that unknown.

Row two, number of towels: using the same technique from the first problem — going back to the given information and asking what have I not used yet — it says 10 fewer towels can be bought for $120. So the current number of towels you can buy for $120 is also an unknown: T. Under New: T − 10.

That Current vs. New table helps you backtrack the inference: you need a variable and a relationship for the current situation, and then you can operate and figure out what the new situation is. Again, it's not a silver bullet — it's not going to fix all your problems or make that easy to do 100% of the time — but it'll very highly likely make it easier to spot, especially if you didn't see it on your own.

If you want, you could make a Total row at the bottom and put 120 at the bottom of each column: $120 originally and $120 in the new situation. Whatever makes the most sense to you — I like to do that kind of thing. The more I write down, the better. But your mileage may vary there.

I think this question is pretty representative of what I'd call medium-difficulty algebraic translation questions. You'll probably see harder stuff, but you'll definitely see easier stuff as well. A lot of times, the bottleneck is getting organized, number one, and number two, having to backtrack to something that wasn't explicitly mentioned. Again — just a good visual organization system. Hate to sound like a broken record, but it usually really helps.

The algebra and the pitfall. Once you've got the table filled out, you can either set up a couple of equations or start guessing and testing with the answer choices. Both of those processes work really well — I encourage you to experiment with both and pick the one you like best.

Just to make sure the algebraic approach is clear, and to discuss one of the most common wrong answers: the original situation gives us P × T = 120 — price times quantity equals total cost. The new situation is (P + 1)(T − 10) = 120. (P plus 1, my apologies — that was a little typo in our notes. We're increasing the price per towel by $1, which decreases the quantity of towels I can buy for $120 by 10.)

The question asks for P, the current price of each towel. Using substitution — isolate T in the first equation and substitute it in; we've got that Math Basics episode on substitution linked below if you're fuzzy on that — you'll wind up with a quadratic equation. If you do the math out, you get P squared minus blah, blah, blah equals zero. That's something we were just discussing last week, and we've got a Math Basics episode on quadratic equations linked below too — what a quadratic equation means, how to solve them, how to factor them.

The solving piece is a little long, so let's just say you do all the algebra right and you know how to factor quadratic equations. You would eventually get down to (P + 4)(P − 3) = 0. From there, two possible solutions make that equation work: negative 4 and positive 3. Positive 3 is the right answer to this question — C, as in Charlie.

Now let me read the options again so you can guess what the most popular wrong answer is. A is 1. B is 2. C is 3. D is 4. E is 12. Remember, the equation we got was (P + 4)(P − 3) = 0. If you're thinking the most popular wrong answer is D — $4 — you're right. About 11% of us go for that one.

And that really hurts. That really, really, really hurts, because at that point you've done all the math exactly right — you've just switched the sign on the factored form of (P + 4)(P − 3).

So beyond rows and columns, I wanted to put a little additional attention on the make-sure-you're-solving-for-the-right-thing idea, which is a big, big knife in the side if you're not super careful with it. Writing the question at the top of your scratch work really helps — put a box around the question, or circle it. Double-check each step of your computation. I've talked about that a ton in the past, and I know it might sound redundant, but I'm saying it a lot because, for me personally, I generally need to be reminded of things way more than I think I do in order for them to become a habit — to get to the place where I don't need to put effort into executing on them. So if you find yourself making those kinds of mistakes — choosing D instead of C because of the sign — make it a habit to double-check every step of your algebra and arithmetic every single time, 100% of the time. And if you struggle to do things 100% of the time, check out that episode on how to be more consistent — it's called How To Score High (Consistently).

Difficulty check. A couple of reasons this one might seem tougher than the previous two. For sure, the math is a bit more intricate — quite a few more steps if you do the algebra out. The setup is also less intuitive: you have to infer the original situation even though it's not explicitly mentioned. About 25% of us miss this one, versus 13% on the first and 20% on the second. So just by the numbers, this one's a little tougher. But hopefully what we talked about here, the quadratics discussion from last episode, and the Math Basics episodes I recommended — if you put all that together, you've got really, really good odds of success on this one.

One final time, I recommend you pause here and make a note to self if there's anything you want to remember from our discussion. And we'll be back soon with more tips, more strategies, more advice, more walkthroughs, more breakdowns to put you on the fastest possible path to success. That's our objective for you here.

If you have questions or feedback, you can reach us anytime at The GMAT® Strategy on the current social channels, or contact@thegmatstrategy.com for email. We're happy to help.

In the meantime, if you want more tips and strategies for optimizing your performance on the exam, head to our website, thegmatstrategy.com, which is linked below as well. And check out our free video that I mentioned earlier on how you can reach your dream GMAT® score in half the normal time — even just understanding how the algorithm is different from typical exams will help tremendously.

In the meantime, this is a regular show, so please subscribe. And please, stay positive and stay consistent with your studies, everybody. Talk to you all soon.