One of the best ways to prepare for the GRE is by studying with official practice questions published by ETS (Educational Testing Service). However, even if you *do* correctly identify the right answer on an official GRE Quant problem, you still might not understand the underlying Math principles that were used to create that particular GRE Math question, leaving yourself open to various common traps and other related pitfalls created by the ETS test question writers.

In the solutions provided below, we will use some of the core tenets of the Menlo Coaching GRE tutoring curriculum to break down three official GRE problem-solving questions and provide important principles for correctly attacking this important GRE question type.

Multiple choice “problem-solving” questions are the first of four main Math question-type variations tested on the new, shorter GRE exam. Although they are likely the most familiar GRE Math question type, many students still do not approach the majority of these questions the right way.

To succeed on GRE problem-solving questions, test takers need the requisite knowledge related to the content area being tested—math skills related to arithmetic, algebra, geometry, statistics, etc. However, it is just as important to read the question carefully, leverage every hint, and choose the right strategy (backsolving, number picking, conceptual thinking, etc.) Many people think of GRE problem-solving questions as just plain math questions, but the following sample GRE math practice questions show that, in many cases, they are much more than that. Please take a look at the following three GRE quantitative practice questions listed below.

The average (arithmetic mean) age of the people in group G is 41 years, and the average age of the people in Group H is 36 years. The average age of the people in the two groups combined is 38. If no person is in both groups, what fraction of the people in the two groups combined are in group H?

A. 2/5

B. 1/2

C. 3/5

D. 2/3

E. 3/4

Answer & Explanation

Correct Answer is (C).

At its heart, this GRE problem-solving question is just a weighted average problem comparing the ages of two different groups. However, the more fundamental error students often make here, even if they ultimately get the question right, is one of approach (i.e. selecting the optimal solution methodology).

This problem is much more time-consuming and significantly more complex to solve if you attempt to tackle it algebraically. Luckily though, this problem is significantly easier to work through if you use a Mapping technique for weighted averages instead.

The traditional way to solve this problem is to create an algebraic equation, but it is both time consuming and unnecessary. This problem, however, can be solved almost immediately by using a “mapping” technique instead. As an example, if the average age was 38.5 years old, that would mean there are an equal number of people in both Group G and Group H because 38.5 is the midpoint between the two individual averages. This scenario is shown below:

Group H | Distance | Group H & G Combined | Distance | Group G |
---|---|---|---|---|

36 | 2.5 | 38.5 | 2.5 | 41 |

The distances between the individual groups and the combined average form the ratio of each group (here the relative distances are 2.5 : 2.5 which is a ratio of 1:1).

In this problem though, you actually have the scenario mapped out below:

Group H | Distance | Group H & G Combined | Distance | Group G |
---|---|---|---|---|

36 | 2.5 | 38.5 | 2.5 | 41 |

It should be clear logically that there must be more people in Group H than in Group G because the average is closer to 36 than it is to 41. To create the ratio of Group H to Group G you must invert the distances between the individual groups and the overall average to get the ratio of `H/G : 3/2`. However, in this case, the question is asking for the ratio of Group H to the total. Since the individual group distances add up to 5, this ratio would be `H/text(Total) : 3/5`, which is answer choice (C).

**What are some final lessons to be
learned here?**

Before jumping headfirst into any GRE problem-solving question, take a moment to actively consider what topic is being tested and, ultimately, what solution methodology would work best (in this case, mapping instead of algebra).

By using a mapping technique here, this is a clear spot where
you can gain at least a minute or more over other test-takers who might be resorting to more traditional
solution
methodologies (i.e. algebra). You may not always make the right choice initially, but as you complete more and
more GRE official practice problems**, **your GRE decision-making skills will continue to become
that
much sharper.

**Remember: When using this strategy don’t accidentally forget to invert the
distances and give the initial ratio in reverse. Once you get comfortable with this approach though, you can
often answer these questions very quickly with little to no written work.**

If r ≤ s ≤ t ≤ u ≤ v ≤ 110 and the average, (arithmetic mean) of r, s, t, u, and v is 100, what is the least possible value of r?

A. 0

B. 20

C. 40

D. 60

E. 80

Answer & Explanation

Correct Answer is (D).

This next question is what’s known as a ‘limit’ problem. ‘Limit’ problems on the GRE are essentially GRE problem-solving questions that ask you to minimize or maximize data to find a particular threshold (usually the minimum or a maximum of something).

For a variety of reasons, most students find these ‘Limit’ question types relatively abstract, fairly time-consuming to solve, and overall, quite challenging. The number one reason why many students have trouble with these questions is that they can be very counterintuitive to solve, especially if you don’t approach them from the right mathematical perspective.

Here’s a hypothetical example: If you are given a problem containing two variables (x and y), and you are asked to maximize the value of x, intuitively, what will most students do? The short answer here is that almost everyone will start by trying to maximize for the value for x.

Unfortunately, when ETS creates new higher difficulty level GRE questions, they almost always structure them in a way where the question can be solved using one of two different approaches. One of these approaches (usually the more intuitive one) will either be extremely difficult or time-consuming to implement, while the other approach (generally the more counterintuitive one) will often be incredibly simple.

Going back to our hypothetical example, the two approaches here would be as follows:

1. Maximize for x (intuitive approach); or

2. Minimize for y (counterintuitive approach)

Applying this concept to Question 1, we are asked to minimize for the value of r. Therefore, instead of (intuitively) thinking about what the minimum value of r could be, let’s consider what the maximum values of s, t, u, and v could be. Since the inequalities provided contain less than or equal to signs, all of the other variables could each be equal to 110. Therefore, the sum of s + t + u + v = 110 + 110 + 110 + 110 = 440. Since average = sum divided by # and the average of the 5 variables = 100, therefore the sum = 5 x 100 = 500. Furthermore, if the sum of s, t, u, and v = 440, the sum of r would have to be 60 (500 – 440). Therefore, (D) is the correct answer.

**What’s our final lesson here? **

As you continue to work through more GRE math practice questions, keep on the lookout for this type of GRE ‘Limit’ problem and don’t fall for the ‘intuitive’ trap ETS is setting you up for.

**Remember: The quickest way to solve for the **__maximum__ of one variable may be to solve for the __minimum__ of the other variable(s) and vice versa.

By approaching these GRE ‘Limit’ questions with a counterintuitive approach, you will end up saving significant exam time that can now be more effectively spent working on other GRE Quant questions within the same exam section.

This next question is what’s known as a ‘limit’ problem. ‘Limit’ problems on the GRE are essentially GRE problem-solving questions that ask you to minimize or maximize data to find a particular threshold (usually the minimum or a maximum of something).

For a variety of reasons, most students find these ‘Limit’ question types relatively abstract, fairly time-consuming to solve, and overall, quite challenging. The number one reason why many students have trouble with these questions is that they can be very counterintuitive to solve, especially if you don’t approach them from the right mathematical perspective.

Here’s a hypothetical example: If you are given a problem containing two variables (x and y), and you are asked to maximize the value of x, intuitively, what will most students do? The short answer here is that almost everyone will start by trying to maximize for the value for x.

Unfortunately, when ETS creates new higher difficulty level GRE questions, they almost always structure them in a way where the question can be solved using one of two different approaches. One of these approaches (usually the more intuitive one) will either be extremely difficult or time-consuming to implement, while the other approach (generally the more counterintuitive one) will often be incredibly simple.

Going back to our hypothetical example, the two approaches here would be as follows:

1. Maximize for x (intuitive approach); or

2. Minimize for y (counterintuitive approach)

Applying this concept to Question 1, we are asked to minimize for the value of r. Therefore, instead of (intuitively) thinking about what the minimum value of r could be, let’s consider what the maximum values of s, t, u, and v could be. Since the inequalities provided contain less than or equal to signs, all of the other variables could each be equal to 110. Therefore, the sum of s + t + u + v = 110 + 110 + 110 + 110 = 440. Since average = sum divided by # and the average of the 5 variables = 100, therefore the sum = 5 x 100 = 500. Furthermore, if the sum of s, t, u, and v = 440, the sum of r would have to be 60 (500 – 440). Therefore, (D) is the correct answer.

As you continue to work through more GRE math practice questions, keep on the lookout for this type of GRE ‘Limit’ problem and don’t fall for the ‘intuitive’ trap ETS is setting you up for.

By approaching these GRE ‘Limit’ questions with a counterintuitive approach, you will end up saving significant exam time that can now be more effectively spent working on other GRE Quant questions within the same exam section.

An investor placed a total of $6,400 in two accounts for one year. One of the accounts earned simple annual interest at a rate of 5 percent, and the other earned simple annual interest at a rate of 3 percent. The investor made no deposits or withdrawals from the accounts. If each account earned the same amount of interest after one year, what was the total amount of interest earned from both accounts?

A. $128

B. $144

C. $240

D. $256

E. $512

Answer & Explanation

Correct Answer is (C).

At its heart, this GRE problem-solving question is just a double simple interest problem, with two different principal amounts being invested at different interest rates. However, the most fundamental error many students make here, even if they ultimately get the question right, is one of approach (i.e. applying the right solution methodology).

This problem proves to be quite tricky in that it is both time-consuming and unexpectedly complicated to solve if you attempt to solve it from a purely algebraic standpoint. Luckily though, this problem is significantly easier to work through if you leverage the five answer choices by**backsolving** it instead.

Since answer choices on the GRE are always listed in order (ascending or descending), as long as you’re confident you can infer what direction (bigger or smaller) you’d need to go in after your first attempt, you should start by plugging in the middle answer choice (C).

There is an additional twist to this question though as the answer choices contain the total amount of interest earned from both of the accounts together, not the individual amount of interest each from each account. How should we best deal with this additional subtle complexity then?

Students who read the question carefully will notice that each account apparently earned the same amount of interest after one year. If one thinks this through logically, this means that each account must earn half (50%) of the total amount of interest earned. Thus, if we take the value in answer choice (C) $240 and divide it by two ($240/2 = $120), we now have a value that we can use to backsolve in both interest account equations.

For example, in the first interest account, 5% of the principal equals $120. Therefore, the principal amount invested would equal $120 divided by 5%. Using the available online GRE calculator, you can solve for this amount in two seconds ($120/0.05 = $2,400).

Doing the same for the second investment account yields the following interest equation (3% of the principal equals $120). Using the same logic as before, the principal amount invested in this account would equal $120 divided by 3%. Once again, with the aid of the handy GRE calculator you get a value of ($120/0.03 = $4,000).

Together these two account values sum to $6,400 ($2,400 + $4,000), which matches the overall principal amount listed in the original question. Since $6,400 = $6,400, we know that the answer we plugged in is in fact correct. As such, we can now simply just pick answer choice (C) and move on to the next question.

To summarize, despite the fact there are multiple ways to solve almost all GRE problem-solving questions, the majority of untrained students generally default to using more traditional approaches, such as algebra, as they have been trained/conditioned to solve most algebra questions this way throughout high school and college.

However, if you chose to solve this problem any other way than by simply backsolving it using answer choice (C) first, you simply did not use the most efficient solution methodology available to you. Consequently, this means there’s a greater chance you either got it wrong, spent 2+ minutes on it, or both. Either way, this is going to cost you on the test.

**What are some final lessons to be learned here?**

Before jumping headfirst into any GRE problem-solving question, take a moment to actively look at the answer choices and consider what approach would work best. It may ultimately be algebra, but likely not. More often than not, backsolving the answers by plugging in the middle answer first is generally a better place to start.

You may not always use the right approach initially, but as you complete more and more GRE official practice problems**, **your ability to select the optimal solution methodology upfront for each question you work on with will continue to improve.

**Remember: Given the strict time requirements on the GRE Quant section (average of 1.75 mins per question) you can’t always solve these problems the traditional way if you want to get through all the questions in the time allotted **__AND__ obtain a super high score!

At its heart, this GRE problem-solving question is just a double simple interest problem, with two different principal amounts being invested at different interest rates. However, the most fundamental error many students make here, even if they ultimately get the question right, is one of approach (i.e. applying the right solution methodology).

This problem proves to be quite tricky in that it is both time-consuming and unexpectedly complicated to solve if you attempt to solve it from a purely algebraic standpoint. Luckily though, this problem is significantly easier to work through if you leverage the five answer choices by

Since answer choices on the GRE are always listed in order (ascending or descending), as long as you’re confident you can infer what direction (bigger or smaller) you’d need to go in after your first attempt, you should start by plugging in the middle answer choice (C).

There is an additional twist to this question though as the answer choices contain the total amount of interest earned from both of the accounts together, not the individual amount of interest each from each account. How should we best deal with this additional subtle complexity then?

Students who read the question carefully will notice that each account apparently earned the same amount of interest after one year. If one thinks this through logically, this means that each account must earn half (50%) of the total amount of interest earned. Thus, if we take the value in answer choice (C) $240 and divide it by two ($240/2 = $120), we now have a value that we can use to backsolve in both interest account equations.

For example, in the first interest account, 5% of the principal equals $120. Therefore, the principal amount invested would equal $120 divided by 5%. Using the available online GRE calculator, you can solve for this amount in two seconds ($120/0.05 = $2,400).

Doing the same for the second investment account yields the following interest equation (3% of the principal equals $120). Using the same logic as before, the principal amount invested in this account would equal $120 divided by 3%. Once again, with the aid of the handy GRE calculator you get a value of ($120/0.03 = $4,000).

Together these two account values sum to $6,400 ($2,400 + $4,000), which matches the overall principal amount listed in the original question. Since $6,400 = $6,400, we know that the answer we plugged in is in fact correct. As such, we can now simply just pick answer choice (C) and move on to the next question.

To summarize, despite the fact there are multiple ways to solve almost all GRE problem-solving questions, the majority of untrained students generally default to using more traditional approaches, such as algebra, as they have been trained/conditioned to solve most algebra questions this way throughout high school and college.

However, if you chose to solve this problem any other way than by simply backsolving it using answer choice (C) first, you simply did not use the most efficient solution methodology available to you. Consequently, this means there’s a greater chance you either got it wrong, spent 2+ minutes on it, or both. Either way, this is going to cost you on the test.

Before jumping headfirst into any GRE problem-solving question, take a moment to actively look at the answer choices and consider what approach would work best. It may ultimately be algebra, but likely not. More often than not, backsolving the answers by plugging in the middle answer first is generally a better place to start.

You may not always use the right approach initially, but as you complete more and more GRE official practice problems

Answer choices on the GRE are always listed in order (ascending or descending). When backsolving on the GRE, it’s smart to start in the middle with answer choice (C). That way, you can infer which direction (bigger or smaller) you’d need to go in after your first attempt.

It’s not always feasible to solve GRE Quant questions in the traditional way, and learning these tricks can save you valuable time on test day!

GRE and GMAT Tutor

The GRE Quantitative section uses multiple-choice questions and numeric-entry questions to test various math topics, including statistics, data interpretation, and number properties, asking questions of varying difficulty levels. The above official GRE quantitative reasoning sample questions illustrate the importance of honing your skills before you are faced with the math sections on the real GRE.

GRE multiple choice questions can be solved using a variety of methodologies, some of which may diverge from typical algebraic approaches. GRE numeric entry questions require test takers to generate their own solutions to plug into an answer box, meaning backsolving will not be a viable methodology for reaching the correct answer.

In addition to problem-solving and ‘Limit’ questions, you may run into quantitative comparison questions on test day. These quantitative comparison multiple choice questions ask you to compare two quantities to figure out if quantity a is greater, if quantity b is greater, if the two quantities are equal, or if the relationship cannot be determined.

Before test day, it is important not only to ensure that you can find the correct answer to GRE practice questions but also that you understand the most efficient methodology to get to these correct answer choices when working through the real questions. Practicing with official GRE problem-solving questions shows you what to expect as you attempt to figure out the correct answers on the quant sections of the GRE.

Reading the explanations under GRE prep math problems can also help you prepare to take the real GRE, teaching you how to reach the correct answer choice in the correct way. It can also be useful to sign up for a GRE test prep course or hire a GRE tutor to learn how to work through the GRE sections proficiently and accurately.

Elevate your GRE performance with the assistance of experienced GRE tutor professionals. Discover how our online GRE tutors can provide effective guidance and support for your exam journey.

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