GMAT Problem Solving: Flexibility is the Key

Of the five different question types you will encounter on the GMAT exam, Problem Solving feels the most familiar to test-takers. After all, most of us have done a considerable number of multiple-choice math questions in our academic careers, so this format is not new. However, what most people don’t understand about these particular “math” questions is that the GMAT does not exist to assess who is the best at math; rather, it exists to assess who is good at creative problem solving, who reads carefully and pays attention to details, who leverages every hint and resource at their disposal, who remains critical and finds contrarian cases – in other words, who possesses the skills required to excel in business school. 

Having the requisite math knowledge and a high level of fluency with the underlying concepts and skills tested in Problem Solving is obviously important, however, scoring even above the 50th percentile on the GMAT quant section requires much more than core math knowledge and fluency.

As I discuss in my previous article, GMAT test writers embed a set of critical thinking and logic skills in GMAT Data Sufficiency that make those questions feel like much more than just math problems. Because GMAT Data Sufficiency questions are more unusual to students, Problem Solving questions feel more like good old-fashioned high school level math problems, but a closer look shows they too involve much more.

Tip #1: Read Carefully! 

As I emphasize over and over with GMAT quantitative questions (both Data Sufficiency and Problem Solving), there is a lot of clever difficulty hiding in the wording and the details of the question. A long career teaching the GMAT has proved to me that people are not good at dealing with this type of difficulty. Overall, people preparing for the GMAT are not used to multiple choice math problems that are designed to “trick” you and elicit certain mistakes unrelated to the math. In business, it really doesn’t matter if you know how many positive factors 441 has, but it sure does matter if you misread a contract or fall for some con or trap set by a competing business!

If you don’t learn to focus on this more devious type of difficulty in your preparation, you will miss countless GMAT quant questions for which you understand the math principles, but still get the question wrong because it is assessing other attributes. Consider a few examples of cleverly worded question stems from official problems (one data sufficiency and two problem solving examples) that show how GMAT question writers assess who reads carefully and who pays attention to details:

• The cost of a certain phone call was $0.75 for the first 3 minutes and $0.20 for each additional minute after the first three minutes. Did the phone call last longer than 15 minutes?

A vast majority of people get this Data Sufficiency question wrong because they interpret the highlighted portion as meaning $0.75 per minute for the first 3 minutes. It doesn’t say that, but you expect it to! If you think the first 3 minutes cost $2.25, you have fallen for the trap and you get the question wrong.

• A certain library assesses fines for overdue books as follows. On the first day that a book is overdue, the total fine is $0.10. For each additional day that the book is overdue, the total fine is either increased by $0.30 or doubled, whichever results in the lesser amount. What is the total fine for a book on the fourth day it is overdue?

Many students miss this question as well, again due to interpretation. People think you are calculating the fine per day and adding it to the previous day, but it is a cumulative fine that you are supposed to be calculating. This problem seems so simple, but it must be difficult somehow and indeed it is…all relating to reading carefully and following the instructions properly.

• A worker carries jugs of liquid soap from a production line to a packing area, carrying 4 jugs per trip. If the jugs are packed into cartons that hold 7 jugs each, how many jugs are needed to fill the last partially filled carton after the worker has made 17 trips?

When people miss this, it is almost never because of the math. After 17 trips, the worker has carried 68 jugs, which divided by 7 gives you a remainder of 5. You feel good about hammering out the math quickly and pick 5 because that is the remainder. But wait a second…the question wasn’t asking for that! It was asking for how many more jugs are needed to fill the carton. That is 2 not 5. Again, you would not believe how many students get this question wrong, mainly because they are being too casual in their approach.

If you want to succeed in GMAT Problem Solving and differentiate yourself from the competition, you must develop an strong focus on wording and recognize when that is the primary difficulty in a question. Always make sure you are answering the proper question and never forget little details or restrictions provided in the question stem. If you do this well, you will be amazed how much higher your quant score can go. Once you become a pro at interpretation and stop falling for wording traps, then you must try to master the hardest part of Problem Solving: choosing the right approach. 

Tip #2: Be Flexible in Your Approach

I believe the most important component to success in GMAT Problem Solving is learning how to choose the best strategy for getting a solution quickly and efficiently – and be flexible in that choice. This decision will always be hard on these multiple-choice questions, but you can get a lot better at it with practice and good coaching.

There are numerous ways to solve GMAT problem solving questions: you can use algebra or other “traditional” math approaches; you can pick numbers to remove abstraction in percent or variable word problems; you can use conceptual thinking or logic (what I call the Leather Chair approach); you can actively leverage the answer choices (i.e. backsolving). 

When GMAT Problem Solving questions are created, they are designed to push people who always approach questions the same way. Business schools want creative, flexible problem solvers, not rote math machines! As an example: a hard 700+ level Problem Solving question on the exam will make you use fairly hard math (algebra, arithmetic, etc.) to go most of the way to the answer, but then the only way to get it correct at the end is to leverage the answers. The problem can’t be solved without answer choices to leverage, yet the “math machine” student will knock his or her head against the computer trying to solve the problem in a vacuum without actively using the answers. More commonly, questions are created in which one approach is extremely difficult or time consuming, while another approach is quite simple. 

Before you jump headfirst into a problem, always take the time to consider which approach will work best. You will not always make the right choice initially, but your instincts will become better and better as you complete more GMAT official practice problems. Remember: You can’t be a one-trick pony on this test if you want to score highly!

To become more flexible in your approach, you must practice and master all the different ways available to solve a GMAT Problem Solving question:

• When there are variables in answer choices or the original number does not matter (for example, in percent questions), it is often much easier to pick your own numbers, solve, and then find the correct answer using those numbers. Are you fluent with this technique and have you practiced it with numerous questions?
• When using the answers (i.e., backsolving) is required or makes the problem much easier, do you recognize this and use the answers actively?
• Are you good at solving problems conceptually from your leather chair when such an approach is available? 
• Are you ready to just do the math when that is the best or only way to solve a problem?

Mastering all these techniques and training yourself to be flexible with them is essential to success and requires active practice during your preparation. 

Tip #3: Know What You Need to Know

After reading the question carefully and considering the best approach for solving it, you then need to execute some type of underlying math calculations or apply particular math concepts. As I mentioned earlier, it is essential that you have a thorough understanding of these concepts and a high level of fluency with the necessary calculations. Most students need to do considerable work on this in their preparation, but it is important to remember that the underlying math concepts used to create these questions are not particularly difficult – most people just haven’t used them in a while so the concepts are rusty.

The best way to improve this fluency is to do full official GMAT problems so that you practice three separate things: reading carefully and interpreting wording correctly, choosing the best approach to the problem, AND solving the underlying math components quickly and efficiently. After every official GMAT Problem Solving question that you do, ask yourself three questions:

1. Why did I really get this question wrong? Was it because I didn’t understand a math concept or because I fell for some other trap?  Did I choose the right approach?

2. If I was slow on the problem, what led to the pacing issue and how can I improve my speed next time?

3. Overall, what repeatable takeaways exist from this problem and how can I apply those to future questions?

Remember: Don’t Mistake Activity for Achievement. You can do a lot fewer problems in your preparation if you always isolate the proper takeaways. To do this effectively, make sure to use official problems that help you develop all the different skills required in GMAT Problem Solving, not simply the underlying math.

GMAT Problem Solving Best Practices

Let’s summarize a set of GMAT Problem Solving best practices and then apply them to one full problem:

1. Very carefully assess the question, looking out for wording traps and noting any important restrictions or details in the problem (for instance, if it says “x is a positive even integer,” focus on how that changes the answer to the question compared to the case in which x was not restricted to being an integer, positive, and even).

2. Look out below! After analyzing the question stem, look carefully at the answers and consider how those answers will dictate your approach to the problem and what action is actually required to choose one of them. What hints do the answer choices give you for how to best attack the problem? 

3. Spend a considerable amount of time deciding what approach you should use. Is number picking a valid option? How actively can I leverage the answers? Are there any conceptual shortcuts? Do I just need to grind out some math? As a rule, people jump into questions too quickly and your first inclination for how to solve most GMAT Problem Solving questions will be wrong. 

4. Make sure you are leveraging every little hint provided by both the question stem and the answers. Often these questions are almost impossible to solve if you don’t leverage all the given clues. As an example, maybe it is impossible to actually solve for the correct answer, but fairly easy to prove that 4 answers are incorrect. 

5. After choosing an approach, be ready to pivot quickly if that approach is not working. It is hard to always choose the proper approach initially, and sometimes the best approach only becomes clear once you have dug into a problem.

6. Execute any basic calculations or conceptual math quickly and efficiently. Avoid unnecessary calculations and learn how to avoid careless errors when doing computations (mental math and focusing on the given answers is often helpful).

7. Before selecting your answer and hitting “Next,” do two things: (a) reread the question stem and make sure you are answering the proper question and (b) ask yourself if there is truly a reward in your answer (i.e., is it tricky enough to differentiate yourself from other smart test takers). In doing these two simple things, you will avoid trap answers and false negatives on the exam – questions you should have gotten correct but didn’t.

Sample GMAT Problem

Consider the following problem and try to solve it yourself in a few minutes using the best practices above:

During the four years that Mrs. Lopez owned her car, she found that her total car expenses were $18,000. Fuel and maintenance costs accounted for 1/3 of the total and depreciation accounted for 3/5 of the remainder. The cost of insurance was 3 times the cost of financing, and together these two costs accounted for 1/5 of the total. If the only other expenses were taxes and license fees, then the cost of financing was how much more or less than the cost of taxes and license fees

(A) $1,500 more

(B) $1,200 more

(C) $100 less

(D) $300 less

(E) $1,500 less 

Explanation:

Let’s go through the question applying the best practices in the same order as described above:

1.   With so much convoluted wording in this question stem, you should recognize that proper interpretation is key and wording tricks will surely be present. The first thing you should notice is that it says “3/5 of the remainder” not the “total” in the 2nd line, so you will need to account for that in your calculations. Additionally, you should note that there are many components to this question, so you better slow down and execute each part carefully.

2.   The answer choices don’t provide too many hints, but there a few takeaways: you will not be able to backsolve (they are asking for a difference) and it must be easy to make computational mistakes with that difference, since 2 answers say “more” and three say “less”. Make sure you calculate the difference carefully.

3.   For the approach, I have already noted that backsolving is not an option nor is number picking because you must work with the given total of $18,000. This question will require an algebraic approach and setting up those equations and/or calculations properly will be key.

4.  Carefully using all the provided information, let’s execute the math:

Variables				Steps		Values
Fuel and maintenance costs (let’s use M as a variable)		=		⅓ of $18,000 = $6000		M = $6000
Depreciation (let’s use D as a variable)		=		⅗ of the remainder = ⅗ of $12,000 = $7200		D = $7200
Insurance + Financing (let’s use I and F as variables)		=		⅕ of total (not the remainder) = ⅕ of $18,000 = I+F = $3600		Since I = 3F we can substitute and say 3F+F = $3600 so F = $900 and I = $2700

The last step is to figure out the amount of taxes and licensing fees (let’s use T for that sum):

• So far, we have accounted for the portion of the $18,000 total made up of M, D, I and F
  • M = $6000, D = $7200, and I + F = $3600.
• That is $16,800, which leaves $1200 for T.
• The question is asking for the difference between T and F, so you can see that F ($900) is $300 less than T ($1200).

Correct answer is thus D.

5. (and 6/7) There was no need to pivot in your approach at any point since you must just do the calculations carefully in this problem. It is very important that you re-read the question and you double check that no careless errors were made in the calculations to get there. People get this question wrong because there are so many steps and thus many opportunities to make calculation mistakes or interpretation mistakes. Note: all of the calculations in this problem can easily be done mentally, so if you are writing much down beyond the totals for each component in this problem, you should work on your calculation fluency.

Final Thoughts

To succeed in Problem Solving you must learn how to be flexible and creative in your approach to each problem, and you must focus on precision in wording. Everyone understands that you must possess the requisite math knowledge to do well, but in general people spend too much time on raw math and not enough time on math + everything else that makes these questions difficult (and relate more to what GMAC is really trying to assess with this exam). 

While it is smart to start your studying process with some raw math skills work, make sure you quickly move into full official GMAT problems, so you can work not just on the underlying math, but also on “everything else.” With the right type of preparation, you can master all the different elements required to succeed in GMAT Problem Solving questions and knock your quant score out of the park!