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.
Problem-solving questions are the first of four main Math question-type variations tested on the new, shorter GRE exam
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.
GRE Problem Solving Questions
Sample Question #1
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 right solution methodology).
This problem is much more time-consuming and infinitely more complex to solve if you attempt to tackle it algebraically. 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 in the middle with answer choice (C).
Using this value, if 3/5 of the people are in group H, this means 2/5 are in group G. Said another way, this means that for every 3 people in Group H (average age = 36), there are 2 people in Group G (average age = 41).
Since you have access to an on-screen calculator on the GRE exam, you can literally solve this problem with the calculator in less than 30 seconds by adding 36 + 36 + 36 + 41 + 41 (sum = 190) and then seeing if this value matches the given average of 38 for both groups overall.
Since 190 / 5 = 38, we know we are correct and can simply just pick (C) as the correct answer and move on to the next question.
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 most likely been trained/conditioned to solve 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 (C), you simply did not use the most efficient solution methodology available to you. This consequently 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’s one of the 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.
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: 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!
A Word to the Wise: Backsolving in the GRE
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!
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.
Sample Question #3
Ben has 30 pencils in a box. Each of the pencils is one of 5 different colors, and there are 6 pencils of each color. If Ben selects pencils one at a time from the box without being able to see the pencils, what is the minimum number of pencils that he must select in order to ensure that he selects at least 2 pencils of each color?
A. 24 B. 25 C. 26 D. 27 E. 28
Answer & Explanation
Correct Answer is (C).
Once we see that this question is asking us to calculate the minimum number of pencils, we know we have another ‘Limit’ problem. Since Ben cannot see the pencils he is selecting (meaning his selections are essentially random), he could either have very good luck (i.e. picking all the pencils he wants first) or he could have very bad luck (i.e. picking all the pencils he wants last).
Consequently, even though this question is asking us to calculate the minimum number of pencils needed to guarantee Ben will have at least two of each color, the quickest way to find the correct answer is to consider the maximum number of pencils that he would need to select (under the worst case scenario).
Let’s say the five pencil colors are Red, Blue, Green, Orange, and Yellow. Assuming Ben never selects what he wants first, he would end up picking all the Red (6), Blue (6), Green (6), and Orange (6) pencils before he picks any of the Yellow pencils. So far, this is a total of 24 pencils. Then, since he has to select at least 2 of each color to fulfill the requirements of the question, he will also need to select 2 Yellow pencils. 6 Red + 6 Blue + 6 Green + 6 Orange + 2 Yellow = 26. Therefore, (C) is the right answer. What’s a final important lesson we can learn here?
Remember: Look how quickly relatively complex and highly abstract GRE problem-solving questions can be solved when we
A) recognize upfront what type of specific GRE question type we are faced with; and
B) apply the most efficient best-practice solution methodology to solve each problem
Conclusion
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.
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