gmat problem solving sample questions

GMAT Quant Questions: Problem Solving

Note: GMAT Quant questions cover Problem Solving, and so much more. To get more math practice, try our free GMAT practice test with accurate score prediction and subject-by-subject performance breakdown.

On the GMAT Quantitative section, the Problem Solving questions are just the familiar five-choice multiple choice math problems you have seen on every standardized test since well before puberty.  Here, you have discovered a veritable treasure chest of Problem Solving sample questions.  

Below is a link to thirty-two different articles on this blog, each with at least two Problem Solving questions.  The sample GMAT Problem Solving questions are often at the top of the article, although sometimes they are further down in the text.  The total number of sample Problem Solving problems available from this page is far more than 37, the total number of math questions you will see on a full Quantitative section of the GMAT. 

In each blog, the solutions & explanations to the sample questions are at the ends of the articles.  (If the topic is less than crystal clear for you, you may find the article itself enlightening.)

1. Problems with Averages

https://magoosh.com/gmat/math/gmat-averages-and-sums-formulas/

2. Distance, Rate, Time

https://magoosh.com/gmat/math/word-problems/gmat-distance-and-work-rate-formula/

3. Permutations & Combinations

https://magoosh.com/gmat/math/gmat-permutations-and-combinations/

4. Factors & Prime Factorizations ( five practice PS questions at the bottom of the article )

https://magoosh.com/gmat/math/arithmetic/gmat-math-factors/

5. Advanced Geometric Solids

https://magoosh.com/gmat/math/geometry/gmat-math-advanced-geometric-solids/

6. Estimation questions

https://magoosh.com/gmat/math/the-power-of-estimation-for-gmat-quant/

7. Difficult Dice Questions

https://magoosh.com/gmat/math/basics/gmat-probability-difficult-dice-questions/

8. Difference of Two Squares

https://magoosh.com/gmat/math/algebra/gmat-quant-difference-of-two-squares/

9. Sequences ( five PS practice questions scattered through article )

https://magoosh.com/gmat/math/word-problems/sequences-on-the-gmat/

10. Remainders

https://magoosh.com/gmat/math/basics/gmat-quant-thoughts-on-remainders/

11. Work & Work Rate

https://magoosh.com/gmat/2012/gmat-work-rate-questions/

12. Circle & Line Diagrams

https://magoosh.com/gmat/math/geometry/circle-and-line-diagrams-on-the-gmat/

13. Polygons

https://magoosh.com/gmat/math/geometry/polygons-and-regular-polygons-on-the-gmat/

14. Set Problems, with Double Matrix Method

https://magoosh.com/gmat/math/word-problems/gmat-sets-double-matrix-method/

15. Set Problems, with Venn Diagrams

https://magoosh.com/gmat/math/word-problems/gmat-sets-venn-diagrams/

16. Scale Factor & Percent Change

https://magoosh.com/gmat/math/geometry/scale-factors-on-the-gmat-percent-increases-and-decreases/

17. Standard Deviation

https://magoosh.com/gmat/math/standard-deviation-on-the-gmat/

18. Radicals

https://magoosh.com/gmat/math/algebra/simplifying-radical-expressions-on-the-gmat/

19. Function Notation

https://magoosh.com/gmat/math/arithmetic/function-notation-on-the-gmat/

20. Algebraic Factoring

https://magoosh.com/gmat/math/algebra/algebra-on-the-gmat-how-to-factor/

21. Hard Factorial Problems

https://magoosh.com/gmat/math/arithmetic/gmat-factorials/

22. Backsolving from the answers

https://magoosh.com/gmat/math/gmat-plugging-in-strategy-always-start-with-answer-choice-c/

23. Distance in the x-y plane

https://magoosh.com/gmat/math/geometry/gmat-coordinate-geometry-distance-between-two-points/

24. Pythagoras !

https://magoosh.com/gmat/math/geometry/pythagorean-triplets-to-memorize-for-the-gmat/

25. Lines in the x-y plane

https://magoosh.com/gmat/math/geometry/gmat-math-lines-slope-in-the-x-y-plane/

26. Tricks for Calculating Combinations

https://magoosh.com/gmat/math/gmat-math-calculating-combinations/

27. Parallel & Perpendicular Lines and Midpoints in the x-y plane

https://magoosh.com/gmat/math/geometry/gmat-math-midpoints-and-parallel-vs-perpendicular-lines/

28. Probability: AND & OR Rules

https://magoosh.com/gmat/2012/gmat-math-probability-rules/

29. Probability: “at least” statements

https://magoosh.com/gmat/math/basics/gmat-math-the-probability-at-least-question/

30. Probability: counting problems

https://magoosh.com/gmat/math/gmat-probability-and-counting-techniques/

31. Hard counting problems

https://magoosh.com/gmat/math/word-problems/gmat-counting-with-restrictions/

32. Probability: geometric probability

https://magoosh.com/gmat/math/geometry/geometric-probability-on-the-gmat/

Also check out these GMAT Probability questions .

Other GMAT Practice Questions

Magoosh has practice materials for all of the GMAT question types in GMAT Quantitative  and in GMAT Verbal. Look at the table below, and click the links for more practice!

And make sure you do practice questions that cover the most common GMAT Quant concepts too.

Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as “member of the month” for over two years at GMAT Club . Mike holds an A.B. in Physics (graduating magna cum laude ) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike’s Youtube video explanations and resources like What is a Good GMAT Score? and the GMAT Diagnostic Test .

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this work is fantastics job, i need more of the solved problems

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Free GMAT Test Questions

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Free Official GMAT Practice Questions, with Answers

When you are starting your GMAT preparation , it is essential that you use official GMAT practice questions. Why? The GMAT test exists to assess very particular skills and abilities that predict your success in business school.

The writers of the actual exam questions on the these standardized tests—employees at the non-profit organization ACT—follow very specific guidelines to make sure that the questions are truly assessing what the Graduate Management Admission Council (GMAC) wants them to assess: higher-order thinking (critical thinking, pattern recognition, etc.) and problem-solving skills.

Understand that the GMAT exam is not a math test, a grammar test, or a reading test:

• Reading comprehension in the verbal section of the GMAT is not about pure literacy: rather, it is a test of verbal reasoning .
• Likewise, the quantitative section is not evaluating you on mathematical rules, but on your quantitative reasoning .

Take a look at our guides to answering official GMAT practice questions by topic to learn how to prepare yourself for the GMAT exam !

Sample GMAT Questions by Topic

• GMAT Problem Solving Questions
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The GMAT Test Is a Standardized Entrance Exam Tailored for MBA Programs

As such, the goal is not to assess how well you know obscure content or difficult math concepts, but rather how efficiently and creatively you apply core knowledge to solve different types of problems. It evaluates your critical reasoning as well as your problem-solving skills.

Instead of focusing on the right answer when looking for GMAT practice tests, learning the correct approach and how to train your thinking, is what will guide you to the right answer choice.

To practice with official test material—and gain expert advice on how to approach each example problem on the actual test day— check out our pages covering multiple free GMAT practice questions.

Enroll in our GMAT prep course to effectively prepare for the exam with official practice questions and expert analysis.

Danger! Avoid Unofficial GMAT Practice Questions

GMAT prep is crucial for success at the actual exam, but when you are looking for a GMAT practice test, you should be aware of some of the issues with many of the GMAT practice questions out there.

A majority of unofficial free GMAT practice tests made by test prep companies miss this mark in one way or another, particularly on the verbal side of the exam. When you prepare with too many unofficial practice questions, you develop bad habits and you don’t prepare for the type of difficulty you will actually see on the exam.

There is a reason that official GMAT sample questions cost on average more than $2500 per question to make: expert item writers use the institutional knowledge of GMAC and ACT to create consistently “unique” questions assessing certain abilities, and then they vet the questions tirelessly to make sure they are perfect.

In a word, these questions are brilliant—they are difficult to attack initially yet generally simple in retrospective analysis, a quality that is challenging for test prep companies to replicate.

Writing Questions to Test Higher-Order Thinking Is No Easy Task

Additionally, I would guess that of the 1000 official questions that appear as scored items on the actual exam over a certain time period, no more than 5 of those end up having marginal issues that lead to their removal as “unfair” or flawed. For unofficial questions, I would guess that number is literally 200 per 1000 (mostly subtle issues on verbal questions) with many of the other “valid” questions not really mimicking the type of difficulty for which you should be preparing.

Having written as many unofficial GMAT sample questions as anyone in the test prep industry, I know how painstaking it is to create “perfect” questions, and most question writers working for the big test prep companies simply do not spend the time necessary to capture the real exam questions (mainly because the writers are financially incentivized to write them quickly, or because they don’t really understand how to build questions that test higher-order thinking).

Knowing this is a crucial step in deciphering the answer choices on the GMAT.

A Curriculum Built On GMAT-Official Practice Questions 

Our 5-week GMAT prep course has “Refresh Modules” to help you remember the algebra, arithmetic, and logic you need to solve GMAT problems, and after that, we have our students practice exclusively with official GMAT practice questions. In both our courses and one-on-one GMAT tutoring sessions , we then thoroughly deconstruct these sample questions to show students why they are truly missing them.

Only through this detailed analysis of GMAT question types and actual GMAT questions can you really gain the skills and strategies needed to achieve a high score.

To highlight some of the difficulties present in real GMAT questions and the strategies required to solve them, let’s get a taste of the Menlo Coaching curriculum and deconstruct a GMAT sample question from each section represented on the GMAT test, by following these links to some free GMAT practice questions :

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GMAT Sample Questions

Want a preview of the question types you'll face on the GMAT? Try your hand at the GMAT practice questions below. Then, check your answers against our in-depth explanations to see how you did.

We pulled these GMAT sample questions from our book Cracking the GMAT and from our test prep course materials. For more GMAT practice, take a full-length practice test with us held under the same testing conditions as the real thing. Find out how you'd score, and get  a personalized score report from us that shows your strengths and weaknesses.

• GMAT Verbal Questions 
• GMAT Math Questions
• GMAT Integrated Reasoning Questions 
• Essay Prompt 

Below you'll find sample GMAT questions covering the three question types you'll encounter on the Verbal section: Sentence Correction , Critical Reasoning, and Reading Comprehension.

GMAT Sentence Correction Questions

1. In order to better differentiate its product from generic brands, the cereal company first hired a marketing firm that specializes in creating campaigns to build brand awareness and then retools its factory to produce a variety of different shapes of cereal. (A) then retools its factory to produce a variety of different shapes of cereal (B) retools its factory to produce a variety of different shapes of cereal (C) then retooled its factory to produce a variety of different shapes of cereal (D) then will retool its factory to produce a variety of different shapes of cereal (E) then produces a variety of different shapes of cereal through retooling its factory

Answer: (C) The actions of the cereal company are not in parallel form. First the company hired then it retools . Eliminate choice (A). Choice (B) still has the same error. Choice (D) changes the verb form incorrectly to the future tense. Choice (E) rewrites the sentence but retains the error.

[+] See the Answer

2. Believed to be one of the first widely read female authors of the Western world, Christine de Pizan's masterwork The Book of the City of the Ladies , was written in 1405 and is a history of the Western world from the woman's point-of-view. (A) Believed to be one of the first widely read female authors of the Western world (B) Written by one of the first widely read female authors of the Western world (C) One of the first widely read female authors of the Western world, as some believe (D) Written by what some believe as one of the first widely read female authors of the Western world (E) Believed by some as one of the first works by a widely read female author in the Western world

Answer: (B) As written, this sentence has a misplaced modifier error: the book, The Book of the City of the Ladies isn't believed by anyone to be an author— Christine de Pizan is. Choices (A) and (C) repeat that error and can be eliminated. Choices (B) and (D) both change the introductory phrase to clearly refer to a written work, but choice (D) uses the incorrect idiom believe as instead of the correct form, believe to be . Choice (E) repeats that idiom error.

GMAT Critical Reasoning Questions

1. One food writer wrote that reducing the amount of animal products in one's diet can contribute to better health and well-being. Based on this claim, some people are completely eliminating meat from their diets in order to be healthier. The argument above relies on which of the following assumptions?

Answer: (B) The argument states that some people are eliminating meat from their diets because reducing the amount of animal products in one's diet can lead to better health. Meat is only one type of animal product, however. The argument assumes that by eliminating meat, the people are reducing the total amount of animal products in their diets. Choice (A) addresses increasing the amount of vegetables and grains, but the argument just deals with animal products. Choice (B) correctly addresses the people who are eliminating meat and states that those people are not increasing their consumption of dairy, which is another instance of using animal products. Thus, these people are actually reducing the amount of animal products in their diets. Choice (C) addresses most food writers, who are irrelevant to this argument. Choice (D) addresses health lifestyles, which are irrelevant to this particular argument. Choice (E) addresses the reasons behind not eating animal products, which is irrelevant to the argument.

2. Studies reveal that a daily exercise regimen helps stroke survivors regain dexterity in their extremities. Being given an exercise routine and having a consultation with a doctor about the exercise routine have been shown to be effective mechanisms to get patients to exercise daily. From the above information, which of the following statements can be reasonably inferred? (A) A stroke survivor that is given a detailed exercise plan and consults her physician about the plan will regain full dexterity in her extremities. (B) If a stroke survivor is not given an exercise plan and does not consult with a doctor, she will not regain dexterity in her extremities. (C) Stroke survivors who are given an exercise routine and consult with a doctor about that routine will sometimes regain dexterity in their extremities. (D) Being given an exercise routine and having a consultation with a doctor about the routine is the best way to help a stroke survivor regain dexterity in their extremities. (E) Only being given an exercise routine is necessary to regenerate dexterity in the extremities of seniors who have suffered a stroke.

Answer: (C) This is an inference question, so evaluate the passage and then look for an answer choice that can be reasonably inferred from the information. The passage states that a daily exercise regimen helps stroke survivors regain dexterity in their extremities and that survivors who are given an exercise routine and who have a consultation with a doctor about the routine have been shown to be effective at getting patients to exercise daily . So it can be inferred that if a survivor is given a routine and consults with a doctor, they are more likely to exercise daily, which will help them regain dexterity. Choice (A) is an example of extreme language. The phrasing will regain full dexterity is not promised in the information in the passage, as the passage only states that a routine and consultations may help a survivor exercise more. Eliminate (A). Choice (B) is also an example of extreme language. There is no way to discern from the information provided that a strong survivor would not regain dexterity without an exercise routine and a consultation, so eliminate (B). Choice (C) is a reasonable inference to make from the information in the passage so keep (C). Choice (D) also contains the extreme language best way . The information does not compare this method with any other method so eliminate (D). Choice (E) is recycled language and does not address consulting with a doctor so eliminate (E). The correct answer is (C).

GMAT Reading Comprehension Questions

Although oft-maligned in modern culture, the pigeon once stood not only for speed and reliability but also for grace and beauty. Darwin himself became a pigeon fancier after beginning to work with the humble Columbia livia , discovering them to be more fascinating than he had formerly believed. During the Victorian age, in fact, raising show pigeons was a popular hobby, with new breeds continuously arising as amateur (and not-so-amateur) ornithologists crossed animals in the hopes of creating ever more fantastic creatures. One of the most sought-after varieties was known as the Almond Tumbler, a name presumably derived from the color of the birds combined with the distinctive flight style. Over the course of many generations, this bird was so manipulated as to have a beak so small as to prevent the adult birds from feeding their offspring. And yet, it was wildly popular, drawing high prices at auctions and high prizes at competitions. How then did an animal once so well-loved come to be so loathed? As recently as World War II, the military used pigeons to carry messages but today, many people would kick a pigeon before they would feed one. Perhaps it is just a problem of population density - a lack of esteem for that which is ubiquitous. Pigeons have become our constant urban companions and, as such, have been transformed from symbols of peace, plenty, and prosperity, to representatives of disease and decay.

1. The primary purpose of this passage is to (A) convince the reader of the nobility of the pigeon, based on its history as a symbol of virtue (B) dissuade the reader from mistreating a once-majestic animal that has fallen from favor (C) rebut claims that the pigeon carries disease any more frequently than do other domestic animals (D) promote a renewal of pigeon fancying and a resurgence of breeds such as the Almond Tumbler (E) suggest that there might be more to the story of some urban wildlife than is commonly known

Answer: (E) The passage gives a brief description of the pigeon's place in recent human history and then goes on to contrast that with modern perspectives of the birds. Choice (A) goes too far—the author doesn't give any indication of believing the pigeon to be noble. Choice (B) focuses too specifically on a side comment in the second paragraph. Choice (C) also focuses too specifically on a side comments—the passage is not primarily about disease. Choice (D) is too strong—the passage isn't really promoting any specific action. Choice (E) remains neutral and informational, as does the passage.

2. The case of the Almond Tumbler is most analogous to which of the following? (A) a strain of wheat that can be grown in plentiful quantities but loses much of its nutritional value in the process (B) Arabian horses that are able to run at phenomenal speeds due to centuries of careful breeding designed to enhance those physical attributes (C) vitamins that were purported to provide all of the necessary nutrients but have since been found not to be very effective (D) the dachshund, a popular breed of dog that is nonetheless prone to severe back problems, due to weaknesses exacerbated by targeted breeding (E) the wild rock doves that are most commonly found nesting in the faces of cliffs far from human habitation

Answer: (D) The Almond Tumbler is described as a breed of pigeon that was very popular during the Victorian era. The passage also mentions that the selective breeding used to create that particular kind of bird also led to tiny beaks that kept parent birds from feeding their babies. Therefore, the best analogy would be another animal that is popular even though it has problems due to its design. Choice (A) is incorrect because it leaves out the aspect of popularity. Choice (B) is only positive and you need something that's also negative. Choice (C) is not about something that has been bred for a specific purpose, nor does it deal with popularity. Choice (D) correctly refers to a popular animal with a common health problem. Choice (E) does not refer to pigeons that have been bred by humans.

3. The passage suggests that (A) pigeons were once known for flying with celerity (B) the Almond Tumbler was the most beautiful breed of pigeon (C) Darwin was infatuated with his fancy pigeons (D) modern pigeons are dirtier than the fancy pigeons of yore (E) only scientists should breed new kinds of animals

Answer: (A) For a question this open-ended, it's usually best to check each of the answers against the passage. Choice (A) appears to match the opening line of the passage, which states that the pigeon once stood not only for speed and reliability. Choice (B) goes too far—although many Victorians seems to have loved the Tumbler, there's no evidence that it was definitively the most beautiful. Choice (C) also goes too far—the passage mentions that Darwin was fascinated by his pigeons, not that he was infatuated. Choice (D) draws an incorrect assumption—the passage comments that the common opinion has changed, not the pigeon itself. Choice (E) is not supported by the passage, which states that amateurs, as well as trained individuals, bred pigeons.

Below you'll find GMAT sample questions covering the two question types you'll encounter on the Quantitative section: Problem Solving and Data Sufficiency.

Problem Solving Questions

1. A certain company sells tea in loose leaf and bagged form, and in five flavors: Darjeeling, earl grey, chamomile, peppermint, and orange pekoe. The company packages the tea in boxes that contain either 8 ounces of tea of the same flavor and the same form, or 8 ounces of tea of 4 different flavors and the same form. If the order in which the flavors are packed does not matter, how many different types of packages are possible? (A) 12 (B) 15 (C) 20 (D) 25 (E) 30

Answer: (C) Begin by figuring out how many different ways you can package the tea in boxes that contains 8 ounces of tea, all of the same flavor. There are five flavors, each flavor can come in either loose leaf or bagged form, so 5 flavors x 2 forms = 10 different ways to package the tea in boxes that contain only one flavor each. Now find the number of different ways to package 4 different flavors of the same form per box. In this case, you must choose 4 of 5 possible flavors, and order does not matter, so the formula is 5 x 4 x 3 x 2 ⁄ 4 x 3 x 2 x 1 = 5 different ways to combine the 4 flavors. Each combination can come in either loose leaf for bagged form, so you have 2 different forms x 5 different combinations = 10 total possible ways to combine the 4 flavors in either bagged or loose-leaf form. Thus, the total number of combinations is 10 + 10 = 20 total combinations. The answer is choice (C).

2. Karen sold her house at a loss of 25 percent of the price that she originally paid for the house, and then bought another house at a price of 30 percent less than the price she originally paid for her first house. If she sold the first house for $225,000, what was her net gain, in dollars, for the two transactions? (A) $15,000 (B) $25,000 (C) $60,000 (D) $75,000 (E) $90,000

Answer: (A) If Karen sold her first house for $225,000 and at a loss of 25 percent, then 25 percent of the original price equals $225,000. 75 ⁄ 100 x = 225,000, so x, or the price she originally paid, equals $300,000. Thus, Karen lost $75,000 on the sale of her first house. If she bought a second house for a price of 30 percent less than $300,000, then the second house cost $210,000, so she gained $90,0000. $90,000 - $75,000 = $15,000, so the answer is choice (A).

Sample Data Sufficiency Questions

1. In a certain company, at least 200 people own manual transmission vehicles. If 12 percent of the people who own manual transmission vehicles also own automatic transmission vehicles, do more people own automatic transmission vehicles than own manual transmission vehicles? (1) 5 percent of the people who own an automatic transmissions vehicle also own a manual transmission vehicle. (2) 15 people own both an automatic transmission vehicle and a manual transmission vehicle. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH Statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Answer: (A) According to statement (1), 5 percent of the people who own an automatic transmission vehicle also own a manual transmission vehicle. The question also indicates that 12 percent of the people who own a manual transmission vehicle also own an automatic transmission vehicle. Both figures relate to the total number who own both, so that means that 5 percent of the automatic transmission owners = 12 percent of the manual transmission owners. The overlap in ownership makes up a smaller percent of those who own automatic transmission vehicles, so there must be more people who own automatic transmission vehicles. Statement (1) is sufficient, so you can eliminate choices (B), (C), and (E). Statement (2) indicates that 15 people own both an automatic transmission vehicle and a manual transmission vehicle, so you know that 12 percent of the people who own a manual transmission is equal to 15 people. 12 ⁄ 100 = 15, so x = 125. Thus, there are 125 people who own a manual transmission vehicle. However, you have no further information to allow you to calculate the number of people who own automatic transmission vehicles, so statement (2) is insufficient. The answer is choice (A).

2. What is the value of x ⁄ 2 ? (1) x is 1 ⁄ 5 less than 9 ⁄ 10 (2) x is between 2 ⁄ 5 and 4 ⁄ 5 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH Statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Answer: (A) Statement (1) allows you to find the value of x, so you can answer the question. (If x is 1 ⁄ 5 less than 9 ⁄ 10 , then 9 ⁄ 10 - 1 ⁄ 5 = x. 1 ⁄ 5 = 2 ⁄ 10 , so x equals 9 ⁄ 10 - 2 ⁄ 10 = 7 ⁄ 10 . If x equals 7 ⁄ 10 , then x ⁄ 2 = 7 ⁄ 10 divided by 2, or 7 ⁄ 20 .) Statement (1) is sufficient, so eliminate choices (B), (C), and (E). According to statement (2), x is between 2 ⁄ 5 and 4 ⁄ 5 . That means that one possible value for x is 3 ⁄ 5 , but another possible value is 7 ⁄ 10 . Statement (2) is insufficient, so the answer is choice (A).

Below you'll find examples of how you'll be asked to use a chart, graph, or table to answer questions on the Integrated Reasoning section.

Sample Integrated Reasoning Questions

Item 1: Andre is buying gifts for his office staff. He wants to spend exactly $280 and he can buy either sweatshirts, which cost $22, or baseball caps, which cost $26. In the table below, choose the number of sweatshirts and the number of baseball caps that Andre should buy.

Answer: Sweatshirts, 8; Baseball caps 4 To solve this question, systematically test out the answer choices. The equation you need to solve is 22s + 26h = 280, in which both s and h are integers and s represents the number of sweatshirts and h represents the number of baseball caps. So, start with plugging in 4 for sweatshirts and see if the number of baseball caps is an integer. 22(4) +26h = 280 h = 7.38 Since the number of baseball caps is not an integer, Andre could not have bought 4 sweatshirts. Keep trying more sweatshirts one by one until you find an answer that will you an integer value for baseball caps. 8 sweatshirts will give you 4 baseball caps.

Question 2-1 The ratio of the U.S. population in 2000 to the U.S. population in 1900 is closest to __. (A) 1 to 4 (B) 2 to 7 (C) 2 to 1 (D) 3 to 1 (E) 11 to 3

Answer: (E, 11 to 3) According to the graph, the U.S. population in 2000 was a little bit more than 275 million, and the U.S. population in 1900 was a little over 75 million. Since the question asks what the ratio is "closest to," these numbers are good enough to approximate. 275 to 75 can be reduced by 5 to get 55 to 15, which can be reduced by 5 again to get 11 to 3. Alternatively, you could reduce 275 to 75 by 25 to get this same ratio.

Question 2-2 The U.S. population in 1950 was approximately __ of the U.S. population in 1850. (A) 800% (B) 600% (C) 200% (D) 85% (E) 15%

Answer: (B, 600%) The question asks what percent the U.S. population in 1950 is of the U.S. population in 1850. To get this you need to calculate population 1950 ⁄ population 1850 x 100. Since the U.S. population in 1950 is higher, you want something that is greater than 100%. Eliminate 85% and 15%. Since the sentence says "approximate" and also since the remaining answer choices are not close to each other, you can estimate the values. According to the chart, the population in 1950 was about 150 million and the population in 1850 was about 25 million. Therefore, you need to calculate 150 ⁄ 25 x 100 = 6 x 100 = 600%.

Question 2-3 The U.S. population increased by approximately __ from 1900 to 1950. (A) 25% (B) 33% (C) 50% (D) 100% (E) 200%

Answer: (D, 100%) To get percent increase, you need to use the formula difference ⁄ original x 100. The population in 1900 was about 75 million, and the population in 1950 was about 150 million. The difference between the two figures is 75 million. Therefore, the percent increase is 75 ⁄ 75 x 100 = 100%.

Below you'll find a sample Analytical Writing Assessment (AWA) question. On the GMAT you'll have 30 minutes to write a critique of the argument.

Analysis of an Argument

The following appeared as part of a medical advertisement in a magazine.

A new medical test that allows the early detection of a particular disease will prevent the deaths of people all over the world who would otherwise die from the disease. The test has been extremely effective in allowing doctors to diagnose the disease six months to a year before it would have been spotted by conventional means.

Discuss how logically convincing you find this argument. In explaining your point of view, be sure to evaluate the line of reasoning and the use of evidence in the argument. For example, it may be necessary to consider what questionable assumptions underlie the thinking and what other explanations or counterexamples might weaken the arguments conclusion. You can also discuss what kind of evidence would strengthen or refute the argument, what changes in the argument would make it more logically persuasive, and what, if anything, would enable you to better evaluate its conclusion.

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GMAT Math : Problem-Solving Questions

Study concepts, example questions & explanations for gmat math, all gmat math resources, example questions, example question #1 : gmat quantitative reasoning.

A fair coin is flipped successively until heads are observed on 2 successive flips. Let x denote the number of coin flips required. What is the sample space of x ?

{ x  : x = 2, 3, 4 . . .}

{ x  : x = 2, 3, 4, 5, 6}

{ x : x is a real number}

not enough information

{ x  : x = 0, 1, 2, 3, 4 . . .}

We need to flip a coin until we get two heads in a row. The smallest number of possible flips is 2, which would occur if our first two flips are both heads. This eliminates three of our answer choices, because we know the sample space must start at 2. 

This leaves us with { x  : x = 2, 3, 4 . . .} and { x : x = 2, 3, 4, 5, 6}. Let's think about { x  : x = 2, 3, 4, 5, 6}. What if I flip a coin 6 times and get 6 tails? Then I have to keep flipping beyond 6 flips until I get two heads in a row; therefore the answer must be { x : x = 2, 3, 4 . . .}, because we don't have an upper limit on the number of flips it will take to produce two successive heads.

Example Question #2 : Gmat Quantitative Reasoning

Example Question #1 : Problem Solving Questions

525 is a multiple of all three of the integers 3, 5, and 7:

Mark will hire 5 of the 8 job applicants he interviews. In how many different ways can he do this?

Since order doesn't matter here, set this up as a combination:

Example Question #5 : Gmat Quantitative Reasoning

Example Question #1 : Understanding Sets

What is the median of the following number set?

In order to find the median, the set needs to be written in numerical order:

Example Question #7 : Gmat Quantitative Reasoning

In a group of 30 freshman students, 10 are taking Pre-calculus, 15 are taking Biology, and 10 students are taking Algebra. 5 Students are taking both Algebra and Biology, and 7 students are taking both Biology and Pre-calculus. There is no student taking both Algebra and Pre-Calculus. If none of the students take the three classes together, how many of the students don't take any of the three classes?

Example Question #1 : Sets

Set B contains all prime numbers. Set C contains all even numbers. How many numbers are common to both sets? 

Impossible to determine from the information provided

All real numbers

Prime numbers are numbers with no other factors than themselves and one. Two is the first prime number and the only even prime number. Other examples are 5, 7, 11, etc.

Even numbers are numbers divisible by 2. Set C includes all numbers ending in 0, 2, 4, 6, or 8.

Thus, there is one number common to both sets: 2.

Example Question #9 : Gmat Quantitative Reasoning

Every senior not enrolled in physics is also not enrolled in calculus.

Every senior enrolled in calculus is also enrolled in physics.

No senior is enrolled in both French IV and calculus.

No senior is enrolled in both French IV and physics.

Every senior enrolled in physics is also enrolled in calculus.

Example Question #10 : Gmat Quantitative Reasoning

Choose the statement that is the logical opposite of:

"John is a Toastmaster but not an Elk."

If John is not an Elk, then he is not a Toastmaster.

John is neither a Toastmaster nor an Elk.

John is an Elk but not a Toastmaster.

John is a Toastmaster and an Elk.

If John is not a Toastmaster, then he is an Elk.

the logical opposite of this is that John belongs to the shaded set in the diagram:

In plain English, if John is not an Elk, then John is not a Toastmaster.

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Problem Solving (PS)

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Problem Solving Practice Test 1

The GMAT Problem Solving questions will test your ability to evaluate information and solve numerical problems. Our practice problems are designed to be very challenging in order to prepare you for the harder-level questions found on the GMAT. Answers and detailed explanations are include with each problem. Start your test prep now with our free GMAT Problem Solving practice test.

Directions: Solve the problem and select the best of the answer choices given.

Next Practice Test: Problem Solving Practice Test 2>>

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How to Master GMAT Problem Solving

Stacey Koprince - Manhattan Prep

Stacey Koprince is an mba.com Featured Contributor and the content and curriculum lead and an instructor for premier test prep provider Manhattan Prep .

The GMAT™ exam feels like a math test, especially GMAT Problem Solving problems. They read just like textbook math problems we were given in school; the only obvious difference is that the GMAT Quant section gives us five possible answer choices.

It’s true that you have to know certain math rules and formulas and concepts, but actually, the GMAT is really not a math test. First of all, the test doesn’t care whether you can calculate the answer exactly (e.g., 42). It cares only that you pick the right answer letter (e.g., B)—and that’s not at all the same thing as saying that you have to calculate the answer exactly, as you did in school.

More than that, the GMAT test-writers are looking for you to display quantitative and critical reasoning skills (the section is literally called Quantitative Reasoning ); in other words, they really want to see whether you can think logically about quant topics. They’re not interested in testing whether you can do heavy-duty math on paper without a calculator. And here’s the best part: They build the problems accordingly and you can use that fact to make GMAT Problem Solving problems a whole lot more straightforward to solve. I’ll show you how in this article!

GMAC’s team (aka, the people who make the GMAT) gave me three random problems to work through with you. I had no say in the problems; I didn’t get to choose what I liked. Nope, these three are it, and every single one illustrates this principle: The GMAT is really a test of your quantitative reasoning skills, not your ability to be a textbook math whiz.

GMAT Quant is not a math test

Okay, let’s prove that claim I just made. Grab your phone and set the timer for 6 minutes. (If you’ve been granted 1.5x time on the GMAT, set it for 9 minutes. If you’ve been granted 2x time on the GMAT, set it for 12 minutes.)

Do the below 3 problems under real GMAT conditions:

• Do them in order. Don’t go back.
• Pick an answer before you move to the next one. (Don’t just say you’re not sure and move on. Make the guess, as you have to do on the real test.)
• Have an answer for all the problems by the time your timer dings—even if your answers are random guesses.

Problem #1: Fellows in the org

According to the table above, the number of fellows was approximately what percent of the total membership of Organization X? 

(A) 9% (B) 12% (C) 18% (D) 25% (E) 35%

Problem #2: Yolanda and Bob

One hour after Yolanda started walking from X to Y, a distance of 45 miles, Bob started walking along the same road from Y to X. If Yolanda’s walking rate was 3 miles per hour and Bob’s was 4 miles per hour, how many miles had Bob walked when they met?

(A) 24 (B) 23 (C) 22 (D) 21 (E) 19.5

Problem #3: Oil cans

Two oil cans, X and Y, are right circular cylinders, and the height and the radius of Y are each twice those of X. If the oil in can X, which is filled to capacity, sells for $2, then at the same rate, how much does the oil in can Y sell for if Y is filled to only half its capacity? 

(A) $1 (B) $2 (C) $3 (D) $4 (E) $8

Time’s up! Do you have an answer for each problem? If not, make a random guess—but do choose an answer for every problem.

You probably want me to tell you the three correct answers so you’ll know whether you got them right. But I’m not going to.

We’re going to review these in the same way that I want you to review them when you’re studying on your own—and that means *not* looking up the correct answer right away. 

• How confident are you about this problem?
• Did/do you have another idea for how to solve? Try it now.
• Were you straining to remember some rule or formula? Look it up and try again.
• Still stuck? Okay, look at the correct answer. Does knowing that give you any ideas? Push them as far as you can. 
• Stuck again? Start to read the explanation. Stop as soon as the explanation gives you a new idea. Push it as far as you can before you come back to the explanation again.

Basically, push your own thinking and learning as far as you can on your own. Use the correct answer and explanation only as a series of hints to help unstick yourself when you get stuck.

Okay, let’s dive in!

GMAT Problem Solving #1: Estimate

We’re going to use the UPS solving process: Understand, Plan, Solve. (A mathematician named George Polya  came up with this.) Use this rubric to approach any quant-based problem you ever have to figure out in your life!

The basic idea is this: Don’t just jump to solve. (That’s panic-solving! We’ve all been there. It does not end well.) Understand the info first. Come up with a plan based on what you see. Only then, solve. 

And if you don’t understand or can’t come up with a good plan? On the GMAT, bail! Pick your favorite letter and move on. UPS can help you know what to do and what not to do.

Glance at the answers. Yes, before you even read the problem! 

The answers indicate that this is a percent problem and they’re also pretty decently spread apart. One is a little less than 10% and another is a little greater than 10%, so that’s one nice split. The remaining three are a little less than 20%, exactly 25%, and about 33%, otherwise known as one-third. Those are all “benchmark,” or common, percentages, so now I know I can probably estimate to get to my answer. Excellent.

And then the problem actually includes the word approximately ! Definitely going to estimate on this one.

Start building a habit of glancing at the answers on every single Problem Solving problem during the Understand phase, before you even think about starting to solve. (And yes, I really do glance at the answers before I even read the question stem!)

Here are some examples of the types of answer-choice characteristics that indicate there’s a good chance you’ll be able to estimate at least a little:

• The answers are really spread out (e.g., 10, 100, 300, 600, 900)
• Some are positive and some are negative
• Some are less than 1 and some are greater than 1
• They’re spread out on a percent scale (0 to 100) or on a probability scale (0 to 1)—less than half, greater than half, etc.

Next, there’s a table with a bunch of categories and each category is associated with a specific number. What does the question ask?

It wants to know the Fellows as a percent of the total. That’s a fraction with fellows on the top and the total of all members on the bottom:

The Fellows category is already listed in the table. Great, that’s the numerator.

What about the total? That means adding up all the numbers in the table without a calculator or Excel. Rolling my eyes. And that’s how I know that I will not be doing “textbook math” here. Pay attention to those feelings of annoyance! There’s some other easier, faster path to take. Use your Plan phase to find it.

I need the Total. I can estimate. Look at the collection of numbers. Can you group any into pairs that will add up to “nicer” numbers—numbers that end in zeros?

Here’s one way: 

• Honorary is a tiny number compared to the rest. Ignore it. 
• Fellows are a little under 10,000 and Members are a bit over 35,000. Group them. 
• Associates are a little less than 28,000 or a little more than 2,000 away from 30,000. And Affiliates are a little over 2,000! Combine those two groups.

We’re already spilling into the solve stage on this one. Fellows and Members together are about 45,000. Associates and Affiliates together are about 30,000. Altogether, there are 75,000 members:

That goes on the bottom of the fraction. Fellows go on top. They’re about 9,200, so let’s call that 9,000. Make a note on your scratch paper that you’re underestimating —just in case you need to use that to choose your final answer. I use a down-arrow to remind myself.

How to simplify 9 out of 75? Both of those numbers are divisible by 3.

Ok, 3 out of 25: what percent is that? We normally see percentages as “out of 100.” Hmm. 

If you multiply the denominator by 4, that gets you to “out of 100.” And whatever you do to the denominator, you have to do to the numerator, so the fraction turns into 12 out of 100, or 12%.

12% is in the answers; the next closest greater value (since we slightly underestimated) is 18%. That’s too far away, so the only answer that makes sense is (B).

Notice how the numbers looked really ugly to start out, but as soon as you started estimating, they combined and simplified really nicely? It’s not just luck. The test-writers know you don’t have access to a calculator, so they’re building the problems to work out nicely if you use these types of approaches. They actually want to reward you for using the kind of quantitative reasoning that you’d want to use at work and in business school.

You can certainly solve GMAT Problem Solving problems using traditional textbook math approaches. You’ll just do a lot more work that way. And using textbook approaches won’t actually help train your brain for the kind of analytical thinking about quant that you’ll need to do in business school or in the working world.

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GMAT Problem Solving #2: Logic (and draw!) it out

One hour after Yolanda started walking from X to Y, a distance of 45 miles, Bob started walking along the same road from Y to X. If Yolanda’s walking rate was 3 miles per hour and Bob’s was 4 miles per hour, how many miles had Bob walked when they met? 

The answers are real values and on the smaller side. They’re pretty clustered, so probably won’t be estimating on this one. Four of the five are integers. I wonder whether I can work backwards on this one (i.e., just try some of the answers)?

This problem is what I call a Wall of Text—a story problem. Get ready to sketch this out. Take your time understanding the setup; if you don’t “get” the story, you’ll never find the right answer. (And if you don’t get the story, that’s your clue to guess and move on.)

There are two people, 45 miles apart, and they’re walking towards each other. Normally, I’d only write initials for the two people, but annoyingly, Yolanda shares her initial with one of the locations.

The first sentence has a critical piece of info that’s easy to gloss over: Yolanda starts first, an hour before Bob. 

It’s super annoying that they don’t start at the same time. I don’t know what to do about that yet, but I’m noting it because I want to think about that when I get to my Plan stage. Again, pay attention to whatever annoys you about the problem! That’s why I put START FIRST in all-caps on my scratch paper.

Next, Yolanda walks a little slower than Bob. Add that to your diagram.

Finally, the problem asks who walked further by the time they meet—and how far that person walked. If Yolanda and Bob had started at the same time, then I’d know Bob walked farther, since he’s walking faster, but Yolanda started first, so I can’t tell at a glance. Still annoyed by that detail.

The two people have to cover 45 miles collectively in order to meet somewhere in the middle. Glance at the answers again. There are two sets of pairs that add to 45: (A) 24 and (D) 21 and (B) 23) and (C) 22. 

On a problem like this one, the most common trap answer is going to be solving for the wrong person (in this case, Yolanda instead of Bob). So the correct answer is going to fall into one of those pairs, because then the most common trap answer will also be built into the problem. The other pair will represent some common error when solving for Bob—and then also mistakenly solving for Yolanda instead. But answer (E) 19.5 doesn’t have a pairing, so it has no built-in trap. If you have to guess, don’t guess the unpaired answer, (E).

Once I subtract the 3 miles that Yolanda walked alone, the two of them together have 42 more miles to cover before they meet. I did note the extra 3 miles she walked off to the side just in case.

Bingo. Now I know how I’m going to solve this problem, because now it’s a more straightforward rate problem.

From here, you can do the classic “write some equations and solve” approach to rates problems. But I’m going to challenge you to keep going with this Logic It Out approach we’re already using—both because it really is easier and because it’s what you would use in the real world. You’re not getting ready to take the GMAT because you want to become a math professor. You’re doing this to be able to think about quant topics in a business context. So make your GMAT studies do double-duty and get you ready for b-school (and work!) as well.

Back to Bob and Yolanda. They’re 42 miles apart and walking towards each other. Every hour, Yolanda’s going to cover 3 miles and Bob’s going to cover 4 miles, so they’re going to get 7 miles closer together. Together, they’re walking 7 miles per hour.

When two people (or cars or trains) are moving directly towards each other, you can add their rates and that will tell you the combined rate at which they’re getting closer together. (You can do the same thing if the two people are moving directly away from each other—in this case, the combined rate is how fast they’re getting farther apart.)

One more thing to note: The distance still to cover is great enough (42 miles) compared to their combined rate (just 7 mph) that Bob is going to “overcome” the 3 miles that Yolanda walked on her own first. So Bob covered a greater distance than Yolanda did. The answer is going to be one of the two greater numbers in the pairs: (A) 24 or (B) 23.

So Yolanda and Bob are getting closer together at a rate of 7 miles each hour and they have a total of 42 miles to cover until they meet. How long is it going to take them?

Divide 42 by 7. They’re going to meet each other after 6 hours on the trail. At this point, Bob has spent a total of 6 hours walking, but not Yolanda! She started first, so she spent a total of 6 + 1 = 7 hours walking. The question asks how far Bob walked: 4 miles per hour for 6 hours, or a total of (4)(6) = 24 miles. 

The correct answer is (A).

If you’d solved for Yolanda first, you’d have gotten (3 miles per hour)(7 hours) = 21 miles. That’s in the answer choices, but it’s less than half of the total distance, so she wasn’t the one who walked farther. In other words, answer (D) is a trap.

Even if you do know how to solve the problem, it’s important to have done that earlier thinking to realize that the answer must be (A) or (B). That way, when you solve for Yolanda, you won’t accidentally fall for answer (D), since Yolanda’s distance is in the answer choices.

When the problem talks about two people or two angles in a triangle or two whatevers and the problem also tells you what they add up to, the non-asked-for person/angle is almost always going to show up in the answer choices as a trap. You do the math correctly, but you accidentally solve for x when they asked you for y . We’ve all made that mistake. 

Noticing that detail earlier in your process is a great way to avoid accidentally falling for the trap answer during your Solve phase.

(Have Polya and I sold you yet on using the UPS process? I hope so.)

Should I Retake the GMAT?

Should you retake the GMAT, and does retaking the GMAT look bad? Manhattan Prep’s Stacey Koprince answers the most common retake the GMAT questions.

GMAT Problem Solving #3: Draw it out; Do arithmetic, not algebra; Choose smart numbers

Glance at the answers. Small integers. Kind of close together, so estimation might not be in the cards, but perhaps working backwards (try the answer choices) could work, depending on how the problem itself is set up. (I don’t know yet because I haven’t actually read the problem.)

Now I’m part-way into the first sentence and see the word cylinders . Overall, I’m not a fan of geometry and I really dislike 3D geometry in particular. So as soon as I see that word, part of my brain is thinking, “If this is a hard one, I’m out.”

But I’m going to finish reading it before I decide. Let’s see. Two cylinders, and then they give me some relative info about the height and radius. They’re probably going to ask me something about volume, since the volume formula uses those measures, and scanning ahead: yep, volume.

So now I know I need to jot down the volume formula and I’m also going to draw two cylinders and label them.

I’m going to make sure I note really clearly what I’m trying to solve for. On geometry problems in particular, it’s really easy to solve for something other than the thing they asked you for. And on this one, I’m also making an extra note that the larger cylinder is only half full. I both wrote that down and drew little water lines in the cylinders to cement that fact in my brain.

This is a complex problem, so just pause for a second here. Do you understand everything they told you, including what they asked you to find? If not, this is an excellent time to pick your favorite letter and move on.

If you are going to continue, don’t jump straight to solving. Plan first. (And if you can’t come up with a good plan, that’s another reason to get out.)

The thing that’s annoying me: They keep talking about the dimensions for the two cylinders but they never provide real numbers for any of those dimensions. And boom, now I know how I’m going to solve. When they talk about something but never give you any real numbers for that thing, you’re allowed to pick your own values. Then you can do arithmetic vs. algebra—and we’re all better at working with real numbers than with variables.

My colleagues  and I call this Choosing Smart Numbers. The “Smart” part comes from thinking about what kinds of numbers would work nicely in the problem—make the math a lot less annoying to do.

We usually avoid choosing the numbers 0 or 1 when choosing smart numbers because those numbers can do funny things (e.g., multiplying with a 0 in the mix will always return 0, regardless of the other numbers involved).

And if we have to choose for more than one value, we choose different values. Finally, as I mentioned earlier, we’re looking to choose values that will work nicely in the problem. (Most of the time, this means choosing smallish values.)

Finally, before I start solving, I’m going to ask myself two things: What am I solving for and how much work do I really need to do?

I’m trying to figure out how much oil is in the larger (but only half-full!) cylinder. I know that the full capacity of the smaller cylinder costs $2 and that the oil is charged at the same rate for the larger one. So if I can figure out the relative amount of oil in the larger cylinder, I can figure out how much more (or less) it will cost. For example, if it turns out that the larger cylinder contains twice as much oil as the smaller one, then the cost will also be twice as much.

In the volume formula, the radius has to be squared while the height is only multiplied, so I want to make the radius a lower value. I’m going to choose r = 2 and h = 3.

Use those values to find the relative volumes of the two cylinders. Reminder yet again: The larger cylinder is only half full, so multiply that volume by one-half:

What’s the relative difference between the two? They both contain pi, so ignore that value. The difference is 12 to 48—if you multiply 12 by 4, you get 48.

So the money will also get multiplied by 4: Since the oil in the smaller cylinder costs $2, the oil in the larger one costs (2)(4) = $8. The correct answer is (E).

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Understand, plan, solve on GMAT Quant

Whenever you solve any GMAT Problem Solving (PS) or Data Sufficiency (DS) problem, follow the Understand, Plan, Solve process. Print out this summary and keep it by you when you’re studying:

• Glance at the answers (on PS) or the statements (on DS) and the question stem. Anything jump out—an ugly equation, a diagram, an indication that you might be able to estimate, etc?
• Read the question stem. Focus just on understanding what it’s telling you and what it’s asking you.
• Jot down what it’s asking, along with any other useful info (equations, etc.). Don’t solve! Just jot (write or sketch).
• Reflect on what you know so far. Lost? Guess and move on. But if you do understand everything, then consider what your best plan is. Can you estimate anywhere? How heavily? Can you use a real number and just do arithmetic? Is there a way to draw or logic it out? What are they really asking you? This reflection is how I realized I just needed a relative value on the Oil Cylinders problem.
• Organize your thoughts and your scratch work to get set up for the Solve stage. Maybe you need to redraw or add something to your diagram, as I did for Yolanda and Bob. Maybe you need to group the data or equations a little differently, as I did on the Membership problem.
• Don’t have a plan you feel pretty good* about? Forget it—guess and move on. (*You don’t have to feel 100% confident. But you want to feel like it’s a decent plan. If you don’t, let it go.)
• Be systematic. You’re almost there. Write your work down. Don’t try to compress steps or work more quickly than is comfortable for you. Keep your scratch paper organized.
• Don’t do more work than you have to. Estimate when you can. Keep an eye on the answers as you work. Eliminate impossible answers as you go. Stop as soon as only one answer letter is left.
• Be willing to bail. Even if you understand and have a decent plan, you still might get stuck. Don’t start trying some other plan at that point. Something’s not working with this one; guess and go spend your time on a better opportunity later in the test.

Finally, remember your overall goal here: You want to go to business school. The point is not to show how much of a mathematics scholar you are. The point is to learn how to think logically about quant topics—with, yes, some amount of actual textbook math tossed in there. 

Actively look for the Logic It Out / Draw It Out / Quick and Dirty approaches. They’ll not only save you time and stress on GMAT Problem Solving and Data Sufficiency, but they’ll also help train your brain for quant discussions in business school and in the boardroom.

Want more strategies to improve your GMAT Problem Solving skills? Sign up for Manhattan Prep’s free GMAT Starter Kit  and check out the section on Foundations of Math.

Happy studying!

She’s been teaching people to take standardized tests for more than 20 years and the GMAT is her favorite (shh, don’t tell the other tests). Her favorite teaching moment is when she sees her students’ eyes light up because they suddenly thoroughly get how to approach a particular problem.

GMAT Prep Online Guides and Tips

10 top tips for gmat problem solving questions.

For many test takers, the quantitative section of the GMAT is particularly daunting. The challenging section includes two types of questions: data sufficiency and problem solving. While data sufficiency questions are undoubtedly the more notorious question type, GMAT problem solving questions can also be quite tricky.

In this guide, I’ll give you an in-depth look at GMAT problem solving questions. First, I’ll cover what they are and what types of math they cover. Then, I’ll give you the top 10 tips for acing GMAT problem solving questions. Finally, I’ll walk you through solving five sample problem solving questions spanning a variety of topics.

What Are GMAT Problem Solving Questions?

Problem solving GMAT questions assess how well you can solve numerical problems, interpret graphs and tables, and evaluate information. In plainer language, problem solving GMAT questions are the “traditional” math question type that you’ll see on the GMAT quant section.

While there isn’t a set number of problem solving questions that you’ll see on the GMAT, you can bet that the quant section will be divided just about 50/50 between problem solving and data sufficiency questions. There are 31 total questions on the GMAT quant section, so you there will be either 15 or 16 problem solving questions on the GMAT quant section.

Problem solving questions look a lot like the math questions you’ve seen on other tests. GMAT problem solving questions are all multiple choice questions, with five different answers. Depending on the content tested, problem solving questions may be presented as an equation, a word problem, a diagram, a table, or a graph.

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Contrary to popular belief, the GMAT quant section doesn’t test on advanced math concepts. The quant section tests your content and analytical knowledge of basic math concepts, such as arithmetic, algebra, and geometry. The same holds true for GMAT problem solving questions – you’ll be asked to apply your knowledge of high school math concepts to questions that are presented in a more challenging and analytical way. For more information about the concepts covered on the GMAT quant section, check out our guide to GMAT quant .

10 Tips for Mastering GMAT Problem Solving Questions

Here are the top tips that you can use to master GMAT problem solving questions.

#1: Master the Fundamentals

GMAT problem solving questions only test high school math concepts. In many ways, this is good news. You’ll have likely encountered every type of math you’ll see on the GMAT before you start studying. Just because the math on the GMAT is relatively basic, however, doesn’t mean that it’s not tricky.

The GMAT tests basic math concepts in complicated ways. Problem solving GMAT questions often ask you to use more than one skill at one time, so you need to have strong mastery of many different concepts.

The key to GMAT problem solving mastery, then, lies in mastering the fundamentals.  Memorize the exponent rules. Memorize common roots and higher powers. Memorize the formulas for finding area of different shapes. Know how to find mean, median, mode, and standard deviation without blinking an eye. Thoroughly understanding the material covered on the GMAT will save you time and boost your score on test day.

#2: Practice Doing Calculations Without a Calculator

As I mentioned, you won’t be able to use a calculator on the GMAT. As such, y ou should prepare for problem solving GMAT questions without using a calculator to ensure that you’re used to making basic calculations by hand.

Get used to using scratch paper for calculations and double-checking your work to make sure there are no errors. In particular, make sure that you spend time practicing multiplying and dividing fractions and decimals without a calculator, as you’ll have to do both on the GMAT. The more non-calculator practice you get in before test day, the better prepared and more comfortable you’ll be.

#3: Use High-Quality Practice Materials

The best way to prepare for the GMAT is by using real GMAT problem solving questions to practice, since they’re the only questions that simulate the GMAT’s style and content with 100% accuracy.  The problem-solving questions have a unique style and logic that many unofficial resources struggle to replicate. Fortunately, there are a ton of real GMAT questions available , and some are even free !

You’ll likely want to supplement each of these resources with other third party tools to help you study. Make sure that any books or online materials you’re using are accurate, useful, and well-respected. A good way to check about the reliability of a book or resource is to read reviews of the resource on Amazon or forums like Beat the GMAT or GMAT Club. We’ve also reviewed the best GMAT books and the best GMAT online resources (coming soon) for you.

#4: Plug In Numbers

You can solve many GMAT problem solving questions by plugging in real numbers for the variables in equations.  Look for questions that have algebraic answers, or questions that ask for the values of algebraic expressions instead of just the values of variables when plugging in numbers. For instance, consider the following question.

If x < y < 0, which of the following is greatest in value?

e. 2y – x

For this question, you can pick real numbers that fit the parameters of the question (such as y = -2 and x = -3), and then plug them into each answer to see which answer has the greatest value.

Here are a few tips for plugging in numbers. First, try to use easy, whole integers that fit the constraints of the question. Second, if the question is asking you to determine a value, you can use the numbers yo’ve plugged in to find the matching answer. If the answers are also algebraic terms, keep plugging in your numbers until you get a match. Third, be careful when you’re plugging in numbers! Make sure you go through every one of the answers, as you may find two answers that match with the numbers you’ve chosen. If that’s the case, try plugging in new numbers or solving the problem in a different way, until you’ve only gotten one correct answer.

The writers of the GMAT know that people generally pick positive, whole numbers to plug into their equations. Don’t forget about negative integers, positive and negative fractions, positive and negative decimals, etc., when plugging in numbers to solve a question.

#5: Work Backwards Using Answer Choices

The GMAT normally arranges answer choices in the ascending numerical value on the quant section. Consider the following example, which we’ll go into more depth on in the next section.

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When Leo imported a certain item, he paid a 7 percent import tax on the portion of the total value of the item in excess of $1,000 dollars. If the amount of the import tax that Leo paid was $87.50, what was the total value of the item?

Notice how the answer choices are written in ascending numerical value. This arrangement means that you can try to plug in an answer and work backwards if you’ve got no idea where to start on a particular question. I’d suggest plugging in the middle answer, so that way you’ll know whether you need to go higher or lower with your answer. You can also use this method to decide which answers to try to plug in next, as well as automatically eliminate the other answers.

#6: Don’t Rely on Your Eyes

When tackling geometry questions, don’t rely on your eyes to estimate angle sizes, lengths, or areas of figures. Instead, use the numbers provided and your own mastery of geometry concepts. Geometry figures aren’t always drawn to scale, and assuming they are can get you into trouble.

You’ll never encounter a GMAT quant question that you can answer simply by visual estimation. GMAT problem solving questions are designed so that you have to use the information in the question, as well as any information in the diagrams, graphs, charts, or tables, to help you solve the question. That means that you won’t be able to see a triangle and estimate the length of one of its sides just by looking at it. You’ll need to use the information in the question to help you

#7: Remember That the Numbers Will Work Out

The writers of the GMAT know that you’re not allowed to use a calculator on the quant section. That means that you’ll be able to solve every question using your mastery of fundamental math concepts, a pencil, and scratch paper. If you’re working yourself into a quagmire of exceedingly complicated calculations, stop, take a breath, and reassess the question. You’re likely over-thinking something.

#8: Use What You Know

No matter how difficult the question may look, remember that you’ll only need to use high school level math to answer it. Start small on questions by using what you know. If you break the problem down to small steps, beginning with what you know, you’ll be able to work towards an answer.

Consider the following sample diagram, which I’ll go into more depth about in the next section.

When you’re approaching GMAT problem solving questions, make sure you’re using all the information in the question and any corresponding charts, tables, or diagrams to find your answer.

#9: Practice Your Timing

One of the keys to success on the GMAT quant section is being able to quickly solve complex math problems.  If you can solve most problem solving questions in a minute or less, you’ll have plenty of time leftover to spend on more difficult questions.

To improve your timing, practice with a timer when you’re working on practice sets. Give yourself two minutes to solve every question in your practice set, and see how that feels. Slowly decrease the amount of time you’re giving yourself, until you’re averaging one minute on most questions.

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#10: Use Flashcards to Help You Memorize Important Formulas

As I mentioned in previous tips, you won’t get to use a formula cheat sheet on the GMAT. You’ll have memorize all the formulas you expect to need on test day. You should spend time before test day memorizing the formulas that you’ll see on the GMAT.

Using flashcards is a great way to build your knowledge so that you can quickly recall and use important formulas on test day. Flashcards help you learn to quickly and accurately remember information by forcing you to focus on one small piece of information at a time. Flashcards are also highly portable, and easy to carry with you so that you can practice when you have downtime, such as on your commute to and from work or school. There are many free GMAT flashcard resources out there, but it’s always best to make your own flashcards. In our guide to the best GMAT flashcards , we review the best GMAT flashcard resources out there, as well as tell you the best way to study using flashcards.

5 Sample Problem Solving GMAT Questions

These five samples questions will help you see the types of concepts covered in GMAT problem solving questions. Please note: there’s a lot of content covered in GMAT problem solving questions. While I picked sample questions that represent a wide range of topics covered by GMAT problem solving questions, there are many more content areas that you’ll see on the test.

Problem Solving Sample Question #1

We know that Leo paid $87.50 of import tax on the total value of an item in excess of $1,000.

Let’s start by saying that x stands for the total value of the item. We also know that x ≥ 1000, because Leo had to pay import tax on the part that was in excess of $1,000.

So, (x – 1000) can represent the part that Leo had to pay the 0.7% import tax on.

We can therefore write the equation:

0.7(x – 1000) = 87.50

Multiply both sides by 0.7 to isolate x, which yields us:

X – 1000 = 1250

Add 1000 to both sides to isolate x which yields us:

X = 1250 + 1000

The correct answer is C. You could also get the right answer by plugging in the different values. As I mentioned in the tips section, start with the middle number. In this case, plugging in C would yield you the correct answer. However, if it didn’t, you’d be able to use that information to eliminate other answers and decide what to plug in next, as discussed in the earlier tip.

Problem Solving Sample Question #2

If the average (arithmetic mean) of the four numbers 3, 15, 23, and (N+1) is 18, then N =

This question requires us to understand how to find the arithmetic mean. You find the arithmetic mean of a set of values by dividing the sum of all the values by the total number of values. So, in this case, that yields us the following equation:

3 + 15 + 32 + (n + 1)/4 = 18

3 + 15 + 32 + (n+1) represents the sum of all the values.

4 represents the total number of values.

Now, let’s simplify this equation. In order to isolate n, let’s first multiply each side by 4, which yields us the new equation:

3 + 15 + 32 + (n +1) = 72

We can simplify that equation to get:

51 + N = 72.

Then we can solve for n by subtracting 51 from both sides.

N = 72 – 51

The correct answer is C.

Problem Solving Sample Question #3

This question is all about interpreting graphs. The question asks us to determine the difference between the highest and lowest tides.

First, let’s start off by determining the highest tide. The highest tide seems to be at 11:30 a.m., which is 2.2 ft.

The lowest tide is 0.5 feet below the baseline, which occurs at 6 pm.

Therefore, the equation to express the difference between the heights is [2.2 – (-0.5)] = 2.7 ft.

The correct answer is E.

Problem Solving Sample Question #4

A flat patio was built alongside a house as shown in the figure above. If all angles are right angles, what is the area of the patio in square feet?

You calculate the area of a rectangle by multiplying length x width. 35 x 40 = 1400 ft.

Now, because the patio is missing a portion where it intersects with the house, we have to find the area of that missing portion. From the diagram, we can see that the part where the patio intersects with the house is a square with the dimensions 20 ft by 20 ft.

We can find the area of that square by multiplying length times width, so 20 x 20 = 400 ft.

Now to find the area of the patio, we simply subtract 1400 – 400 = 1000 ft.

The patio has an area of 1000 square feet.

Problem Solving Sample Question #5

Mark and Ann together were allocated n boxes of cookies to sell for a club project. Mark sold 10 boxes less than n and Ann sold 2 boxes less than n. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than n boxes, what is the value of n?

Let’s start off by defining what we know.

We know that Mark sold 10 less boxes than n. We can express the number of boxes that Mark sold as n – 10.

We know that Ann sold 2 less boxes than n. We can express the number of boxes that Ann sold as n – 2.

We also know that they each sold at least one box of cookies. Thus, we can say that n – 10 ≥ 1 and n – 2 ≥ 1.

Thus, we know that n ≥ 11, because we need at least 11 boxes to make Mark’s statement (n – 10 ≥ 1) true.

We also know that they sold less than n boxes. We can express this as:

(n – 10) + (n – 2) < n. If we solve through for n in this equation, we get that n < 12.

We therefore know that n ≥ 11 and n < 12, which tells us that n = 11.

The correct answer is A.

Review: How to Attack Problem Solving GMAT Questions

GMAT problem solving questions are more traditional than data sufficiency questions. You’ll see concepts presented in a straightforward way that is very similar to how you’ve seen math questions posed on other standardized tests.

But that doesn’t mean these questions are easy or simple! Problem solving questions cover a wide range of math concepts, from algebra to geometry to number properties and more. Work on mastering fundamental math concepts so that you can work quickly and successfully through problem solving questions on test day.

What’s Next?

There’s a lot of content covered on the GMAT quant section, so if you’re looking for specific tips on tackling a part content area check out some of our other guides (such as our guides to GMAT percents , probability , and geometry ). These guides will help you build up the fundamental knowledge you need to succeed on the GMAT.

Feel like you’ve gotten the hang of GMAT problem solving questions, but wondering what’s up with the other half of the GMAT quant section? Data sufficiency questions are undoubtedly a bit strange, and very different stylistically from any traditional math question you’ve encountered on other standardized tests. Check out our guide to data sufficiency questions  to learn more about this unique question type and how to master it.

Wondering how to build in practice on problem solving questions to your GMAT studying? Look no further than our comprehensive GMAT study plan article . In this guide, you’ll find four different GMAT study plans designed to maximize your time and boost your score. You’ll learn how much time you should devote to each section of the test and get recommendations on resources you can use to supplement your practice.

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Author: Hayley Milliman

Hayley Milliman is a former teacher turned writer who blogs about education, history, and technology. When she was a teacher, Hayley's students regularly scored in the 99th percentile thanks to her passion for making topics digestible and accessible. In addition to her work for PrepScholar, Hayley is the author of Museum Hack's Guide to History's Fiercest Females. View all posts by Hayley Milliman

Problem Solving

Concepts Tested on GMAT Problem Solving

The problems are based on various arithmetic and algebra math concepts, many of which are presented as word problems. There is no geometry, trigonometry, or calculus on the GMAT. All numbers used are real numbers; irrational numbers are not used.

• Arithmetic concepts on the test include number properties, fractions, percents, ratios, exponents and roots, and basic statistics. Also included are certain types of word problems such as rate and work, mixture, sets, probability, and basic combinatorics.
• Algebra concepts on the test include linear equations, basic quadratic equations, absolute values, and inequalities.

How to Approach GMAT Problem Solving

Read the question carefully and fully understand what is asked. Harder questions may be purposely worded in a confusing manner. For word problems, it is often helpful to translate the information presented into equations or in a tabular format. Make liberal use of the provided scratch board , as performing calculations in your head can lead to careless mistakes. Be systematic in your approach, organize the information logically, and clearly label everything. This becomes even more important as you tackle hard difficultly problems.

Before diving into calculations, examine the five answer choices for clues . Incorrect answers are typically not random numbers, but are instead created to ensnare test takers who make a careless mistake or fall into a common trap. Consider the format of the answers, so you know what you are working towards. Look for any similarities or differences amongst the available answers. If the answer choices are numbers that are far apart, some approximation may make for easier calculations .

Sample GMAT Problem Solving Question

Let’s try a sample problem. Attempt the problem on your own before viewing the answer and explanation.

A hospital purchased 50 stethoscopes and 270 boxes of tongue depressors from a medical supply company. If the price of each stethoscope was nine times the price of each box of tongue depressors, what percent of the total bill was the price of one stethoscope?

        (A) 0.8%         (B) 1.0%         (C) 1.25%         (D) 1.45%         (E) 2.0%

Explanation to Problem

There are three general approaches to this word problem: conceptual, algebraic, and plugging-in numbers. Let’s discuss each in turn.

Conceptual approach:

The conceptual approach, likely to be taken by advanced students, is the fastest. Since the question focuses on the price of a stethoscope, we can convert the total tongue depressor cost into an equivalent stethoscope cost. The price of each stethoscope is nine times the price of each box of tongue depressors (side note – we can safely assume that all stethoscopes are equally-priced and that all tongue depressor boxes are equally-priced). Thus nine tongue depressor boxes cost the same as one stethoscope.

Divide 270 (the number of tongue depressor boxes) by nine to calculate that the cost of these 270 boxes is equivalent to the price of 30 (270 ÷ 9) stethoscopes. Therefore, the total bill is equivalent to the cost of 80 stethoscopes: the 50 stethoscopes bought plus 30 more (representing the 270 tongue depressor boxes). As a result, one stethoscope is \(\frac{1}{80}\) of the total bill.

We now need to convert this into a percent. But first let’s review the other two approaches, to make the conceptual approach more understandable.

Algebraic approach:

This approach, likely to be taken by intermediate students, puts the conceptual approach into algebraic form. For this algebra word problem, let’s assign variables to the unknowns:

         S = price of one stethoscope          B = price of one box of tongue depressors

To calculate the total bill, multiply Price × Quantity for each item and then add the results. We are given the quantities, and can use our variables for the prices:

        Total bill = 50 S + 270 B

We are told that the price of each stethoscope is nine times the price of each box of tongue depressors. Using our variables, we can write an equation to express this relationship:

         S = 9 B

A common mistake is to write this equation backwards, as B = 9 S . Since stethoscopes are more expensive, however, we need to make the bigger value S equal to nine times the smaller value B . A small number B cannot equal a big number S times 9.

Since the question asks about stethoscopes and not tongue depressors, we want to get rid of B , the variable that we don’t care about. To do this, isolate B and substitute it away. Divide both sides of our equation by 9 (same as multiplying both sides by \(\frac{1}{9}\)).         \(\frac{1}{9}\) S = \(\frac{1}{9}\)(9 B ) = B

Now let’s do an algebraic substitution into the total bill equation:         Total bill = 50 S + 270(\(\frac{1}{9}\) S ) = 50 S + 30 S = 80 S

The question asks: the price of one stethoscope is what percent of the total bill ? We are looking for a percent, as further verified by the format of the answers. When calculating a percent, a good approach is to form a fraction with the “is” number on top and the “of” number on the bottom:         Percent = \(\frac{\text{is}}{\text{of}}\) = \(\frac{\text{stethoscope price}}{\text{total bill}}\) = \(\frac{S}{80S}\) = \(\frac{1}{80}\) (since the S variable cancels out)

We’ll convert this into a percent after reviewing the plugging-in numbers approach.

Plugging-in numbers approach:

This approach is likely to be taken by less-advanced students, but is actually a great approach for this problem. When the answers represent a ratio or percent, and we don’t have specific numbers provided within the problem, then a very good technique is to pick numbers to work through the math. The price of each stethoscope is nine times the price of each box of tongue depressors, so let’s pick easy numbers. We do not need to worry about whether the numbers are accurate in the real world, just whether the numbers meet the relationship described in the problem.

        Tongue depressor box = $1         Stethoscope = $9

Now calculate the total bill, using the price numbers we made up and the quantities provided in the problem.

        Total bill = 50 × $9 + 270 × $1 = $450 + $270 = $720

The question asks: the price of one stethoscope is what percent of the total bill? As mentioned in the algebraic approach, we can calculate the percent by forming a fraction with the “is” number on top and the “of” number on the bottom:         Percent = \(\frac{\text{is}}{\text{of}}\) = \(\frac{\text{stethoscope price}}{\text{total bill}}\) = \(\frac{$9}{$720}\) = \(\frac{1}{80}\) (since 9 goes into 72 eight times)

Converting the fraction into a percent:

There are several approaches to convert \(\frac{1}{80}\) into a percent. Worst case, we could do long-hand division. But there are a couple faster approaches.

Notice that \(\frac{1}{80}\) = \(\frac{1}{8}\) × \(\frac{1}{10}\). We recommend that students memorize the decimal equivalents of common fractions. So we should ideally know that \(\frac{1}{8}\) = 0.125. Multiplying by \(\frac{1}{10}\) is the same as moving the decimal one place to the left, resulting in 0.0125. Converting a decimal into a percent is done by moving the decimal two places to the right, so 0.0125 = 1.25%.

Another good shortcut takes advantage of the fact that a percent is equivalent to a fraction with a denominator of 100. How can we turn our denominator of 80 into 100? We can increase 80 by 25%, the same as multiplying 80 by 1.25. To leave the value of our fraction unchanged, we must multiply numerator and denominator by the same number.         \(\frac{1}{80}\) = \(\frac{1(1.25)}{80(1.25)}\) = \(\frac{1.25}{100}\) = 1.25%

Finally, if we are very short on time and don’t see an easy way to do the conversion, we could at least quickly eliminate three answers by recognizing that:         \(\frac{1}{100}\) < \(\frac{1}{80}\) < \(\frac{1}{50}\)  →  \(\frac{1}{100}\) < \(\frac{1}{80}\) < \(\frac{2}{100}\)  →  1% < \(\frac{1}{80}\) < 2%

Our correct answer has to be somewhere between 1% and 2%, leaving only answers C and D. As we have seen above, C is the correct answer.

GMAT Problem Solving—Be Flexible in Your Approach (and Know What You Need to Know)!

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Probably the most common misconception about the GMAT is that the quant section is a “math test.” Obviously, math skills are essential to success and your fluency with underlying math concepts directly affects your score. However, the problems in the quant section are testing much more than math:

• Who is good at creative problem solving?
• Who can deal with abstract presentation of simple concepts?
• Who leverages every resource and hint in a problem?
• Who reads carefully and follows instructions properly?

Of all the question types on the GMAT, test prep companies feel most comfortable creating unofficial quant problems. They hire math whizzes to crank out content questions and consumers gobble them up in their preparation for this section.

While these questions will help with content improvement, they usually lack the type of difficulty you see in a full 75% of the quant questions on the GMAT! By using mainly official Problem Solving questions, you not only improve your understanding of underlying content but also prepare yourself for the other types of difficulty that plague a majority of students on hard quant questions.

Best Practices for GMAT Problem Solving Questions

On the quant section of the GMAT, it is helpful to think of Problem Solving questions in two categories:

Type 1 : These questions are more just math questions and require you to apply conceptual knowledge and practical math approaches to solve a question.

Type 2 : These questions are made difficult by abstract presentation, complex or tricky wording, red herrings, your choice of approach—i.e. those in which just understanding the math will not get you to the correct answer efficiently (if at all!).

Type 2 questions have always been the mainstay of the quant section on the GMAT, and these questions are what make the test so hard for students. To explain this type of question, I have always used this example:  on the exam, GMAT test writers turn 1 + 1 into a 90 th percentile problem by making it exceptionally hard to sort through all the garbage and see that you just need to do a simple addition. When you miss this question, you don’t need to go do more addition drills, you need to learn how to sort through abstract presentation and deal with complex wording! People do not spend enough time improving these types of skills that are so essential on hard official quant questions.

In the shift to the GMAT Focus exam, I expected to see even fewer Type 1 questions but so far this has NOT been the case. Anecdotally, I would say that ¼ of the questions fall into category 1 and ¾ fall into category 2 on the new exam, the same proportion as on the legacy version of the GMAT.

This is still a small percentage of the quant questions overall, but you should think of these Type 1 questions as gifts on the GMAT quant section:  if you do the proper prep and understand the math, you will get these questions correct with little effort. It is also important to note that the standard for solving pure math questions around the globe is very high. If it is mostly just a math question, you really need to get it right to be competitive on the GMAT quant section.

So, when I review missed questions with students and I see that they are missing a Type 1 question, I say: “Know What you Need to Know! and this question would feel easy.”   There is no better example of this than the first question covered in this section, a question I see far too many students miss.

GMAT Problem Solving: Example Question #1

Detailed Explanation For Question 1

This process is made more difficult in this official question with two mechanisms:

• The denominator does not just contain individual terms with roots but also integers or multiple roots added together. This makes for a more difficult version of rationalizing the denominator in which you must recognize the difference of squares and” multiply by one” using the conjugate of the denominator. Even though this is harder, these are both core best practices that you learned in algebra in high school and that you must know for the GMAT. This skill has been tested so many times on the GMAT that you should recognize two things immediately: you should use the conjugate and the denominators will simply disappear with the numbers they have used.
• Three terms with roots in the denominator are being added together, so the test-maker entices students to try to find a common denominator or take some other incorrect approach.

The important point with a manipulation like this is that you simply must recognize what to do! We cover these types of important math skills in detail in our Refresh Modules and then you need to practice them with questions like this. Once you see what to do on the first term, then just do the same type of manipulation on each fraction individually and add the simplified terms together.

With each of the three fractions simplified and the denominators disappearing, you are simply adding together the following three terms:

The correct Answer is thus (E).

While these three steps look tedious on paper, the reality is that a lot of people taking the GMAT are going immediately to the last step shown above without any written work. You want to be one of those people!

If you don’t know what to do on this problem algebraically (and the point of this example is that you should!), it is important to note that this question can also be solved cleverly using answer choices and simply estimating the roots. Since the first four answer choices are all less than 1/2, you know the answer must be (E). Estimating the two roots in the question stem allows you to see that the sum of the three expressions will get close to 1, and none of the other answers are close. If you solved this question with this technique, good for you! However, this question could easily contain 6/7 as an answer, and then you would be in trouble.

As you prepare for the exam, pay special attention to any misses on questions like this that just require math knowledge. They are easier to prepare for and it is important that you get them right. With difficult abstract problems involving lots of red herrings or tricky wording, you simply can’t get win them all, but for these types of questions, you can develop complete mastery.

GMAT Problem Solving: Example Question #2

One year ago, a window washing service charged $100 for setup and an additional $30 per hour for on-site washing. This year the company charges $20 for setup and an additional $50 per hour for on-site washing. Which of the following is equivalent to the percentage change from last year to this year that the company charges for setup and x hours of on-site washing?

Detailed Explanation For Question 2

This example is a classic type 2 question—it feels abstract and you must read carefully. You can be comfortable with most percent questions on the GMAT and still get this wrong (or waste a lot of time) if you don’t choose the right approach.

If you search the internet for explanations on this question, you see everyone explaining one tedious algebra step after another AND you see many people who have either botched that algebra or made a mistake setting up the percent change. In 20 years of preparing people for the test, I have only seen a few variable-in-answer choice percent questions for which algebra was a better approach than number picking. Here the algebra is not as tedious as in other questions of this kind, but number picking is unquestionably easier.

As a best practice for the exam, always take a little time to decide on your approach (algebra, conceptual thinking, backsolving, or number picking) before jumping into a question. Don’t swim upstream with a long math approach when you can take advantage of answers or use your own numbers. As we teach in our curriculum, whenever you see percent change questions with variables, try number picking first and only go to algebra if that is not working. When number picking, it is important that you are careful with the number(s) that you choose for variables—that is, anticipate and use numbers that will make solving the question as easy as possible. On harder number picking questions, you may choose the wrong numbers first and only realize which ones will work better as you move into the question.

The final step after you solve with the number(s) you have chosen is to plug that number into each answer, looking for your solution, in this case 8%. By plugging 5 into each answer, it is clear that (D) and (E) are wrong as they would be negative. (A) is way too big and (C) would leave 27 in the denominator (i.e. not reduce to 8) so the correct answer must be (B).

Thinking about this question broadly, it is really quite simple with number picking as long as you pick a good number!!!! One risk in number picking is that you get buried in awkward calculations. Imagine if you picked say 3 or 7 for x. With 3, you would be starting at $190 and calculating the % change to $170. Ugly. With 7 it would be $310 to $370. Also, ugly.

Number picking is an essential strategy for GMAT word problems with variables in answers and for many other question types (percent questions or others in which the starting number can be anything). You must practice and hone this strategy in the same way that you do with certain quant skills and calculations, but most people are not doing that in their preparation. As you move through official questions, take the time to consider alternative approaches after you have solved a question, particularly if your method seemed tedious or time-consuming. As a final exercise, think about how easy you can make a problem like this compared to how it first seems:  if I asked you what the percent change was from 250 to 270, I am confident that all of you could get it correct in less than a minute!

Caution: Avoid Unofficial GMAT Problem Solving Questions (except when you need content help)

Utilizing unofficial Problem Solving questions is not as worrisome as using unofficial Verbal or Data Insights questions, which can actively hurt your score. Since most unofficial Problem Solving questions are more just about the math, they can help improve your mastery of underlying math content.

With that being said, you better move to official questions early in your preparation once your content knowledge is solid. Without using the complex and cleverly made official quant questions, you are not preparing for the more complicated problems in which you must sort through clever wording, use answer choices actively, number pick to simplify the problem, etc. People with strong quant skills (engineers, math majors, etc.) are often surprised that their quant scores are not higher, and it is often because they are not prepared for this “Type 2” difficulty that appears in a majority of quant questions on the GMAT.

To strengthen your skills and tackle these “Type 2” difficulty questions with confidence, consider joining our live GMAT prep course . These sessions are designed to guide you through the complexities of official quant questions in a supportive, interactive environment.

Questions about this article? Email us or leave a comment below.

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GMAT Problem Solving Questions With Answers

Problem Solving questions constitute a major chunk of the Quant section of the GMAT. Of the 31 questions that appear in this section, you can expect close to 50% of the questions from problem-solving. In this article, we will be looking into –

What are GMAT Quant Problem Solving Questions?

• 3 examples of GMAT Problem Solving Practice Questions
• 5 tips that can help you ace these questions.

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Problem Solving or PS questions are questions that have a question followed by five options. They are very similar to any MCQ question that we come across in our school or university examinations. You need to solve the question and arrive at the answer and mark the correct choice.

There is only one correct answer in Problem Solving Questions.

Problem Solving questions are considered easier than Data Sufficiency questions because they are straightforward. Having said that, a preparation without a proper plan might prove disastrous for these questions.

Let us now take a look at a few Problem-Solving questions to get you started.

Examples – GMAT Problem Solving Questions

Question 1:.

If a regular unbiased die is rolled twice, what is the probability of getting a sum greater than 10 in the two rolls?

In probability, the probability of an event E = The number of favourable outcomes / Total number of outcomes

The first step is to calculate all favourable outcomes.

Let {x,y} represent the two rolls.

For getting a sum greater than 10, we can get a sum of 11 it a sum of 12.

Now, for a sum of 11, we can have these cases – {5,6}, {6,5}

For a sum of 12, we can have these cases – {6,6}

Hence, there are three favourable outcomes.

Total number of outcomes = 6*6 = 36

Hence, probability = 3/36 = 1/12.

Question 2:

The length of the equal sides of an isosceles triangle is 6 cm. What is the maximum possible area of the triangle(in square centimetres)?

Let us assume that one of these 6 cm sides is the base and the angle that the other 6 cm side makes with the base is k degrees.

Now, we know that the area of a triangle whose 2 adjacent sides are known and the enclosed angle is known can be calculated as:

Area = (1/2)*a*b*sin(k)

a = First side

b = Second side

k is the angle enclosed.

Now, a and b are fixed as 6 cm each

If sin k is maximum = 1, the value will be maximum.

Hence, the maximum possible area = (1/2) x 6 x 6 = 18

Question 3:

If a set A has 16 distinct elements all of which are integers and set B necessarily has the square of all elements of set A, what can be the minimum number of elements in set B?

If all 16 elements in A are distinct, say 1, 2, 3, …., 16, then B also has 16 elements 1, 4, 9, ….., 256.

However, to reduce the number of elements of B, we need to have as many elements as possible in A that also has their negative counterpart in A. For example, if 2 is an element, -2 should be an element as well. As a result, if 4 (square of both 2 and -2) is present in B, it will account for 2 elements in A.

Hence, we can have 8 positive numbers in A and their corresponding negative numbers in A. In this way, we can say that B will have a minimum of 8 elements.

Hence, the minimum possible number of elements in B is 8.

Though these examples provide a good sense of what type of GMAT Problem Solving questions you can expect, in no way do they represent the exhaustive list of concepts required for the Quantitative section of GMAT.

Tips to keep in mind:

• Try to take some time out from easy Problem Solving questions so that you can use that time to solve tricky Data Sufficiency questions. Data sufficiency questions usually take more time to solve than Problem Solving questions.
• Do not get stuck in a question for long. If you find yourself trapped in a question for long, take a guess and move on.
• Read the units carefully.
• Look out for negation words. For example: Which of the following are NOT possible values of x?
• Some questions can be solved faster by the use of options. Make sure you don’t solve these questions in a conventional way.

You can check out the Free GMAT Daily Targets on our platform .

Also, check out the Free GMAT Verbal Tests and Quant Tests .

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Hope this article was helpful. Wish you all the best for the GMAT.

Student Question Bank: Math Questions

Because each question on the Math section deals with different numbers and mathematical scenarios, it's not as simple as the Reading and Writing section to identify exactly what each question stem will look like. You can still use the descriptions in this section to determine which math domains and skills you want to focus on in the Student Question Bank.

Math Questions

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Free GMAT Practice Questions

What is the price of an orange?

• (1) The price of 3 oranges and 2 apples is $7.
• (2) The price of an orange and the price of an apple are both integers.