## What to Expect from GRE Quant – Arithmetic

Reading Time: 8 minutesGRE quant is often referred to as GRE math as well, which could be a good indication of why so many people are so intimidated by it. Fortunately, though, the truth is that GRE math is only a very small part of the mammoth subject of mathematics.

If you’re wondering what GRE quant tests, it’s just stuff we’ve learned in school.

Honestly! We’re not kidding.

However, we understand if you still have reservations in your mind about how easy GRE math really is. So, we’ve made a series of articles on the different parts of GRE quant with one article each on arithmetic, algebra, data interpretation, and geometry.

In this article, as the title suggests, we will focus on Arithmetic. Here’s what this article will talk about at length:

The word ‘arithmetic’ is derived from the Greek word, ‘arithmos’ which means number. Arithmetic is the branch of mathematics which deals with the study of numbers and their properties.

For a fairly long time in the history of mathematics, arithmetic has been synonymous with number theory. However, when we approach Arithmetic from a Test Prep perspective, we can see it in a much broader sense.

**Arithmetic on GRE Quant**

Generally, arithmetic refers not only to Number Theory but also to its various applications including ratios and percentages and their respective applications, too.

As such, arithmetic is indeed the Queen of Math, especially when it comes to the GRE, because it constitutes almost 50% of the questions in the Quant section.

Questions from GRE Quant Arithmetic basically boil down to Number Theory, Ratios, Percentages, or Application-Based Topics.

Therefore, preparing for arithmetic entails preparation on these topics.

Let’s now look at each of these topics in detail, beginning with Number Theory.

**1. Number Theory**

One of the greatest mathematicians of all time, Carl Friedrich Gauss, once said,

“Mathematics is the queen of the sciences and number theory is the queen of mathematics.”

Since Number Theory is the root of all numbers and mathematics is nothing but numbers, it’s only right that it should be considered the most important part of mathematics. As such, Number Theory is the study of different types of numbers and their properties.

Additionally, Number Theory is also a study of the behavior of various types of numbers when mathematical operations like addition, subtraction, multiplication, division, and exponentiation are performed on them.

For the sake of GRE Quant, topics under Number Theory can be broadly divided into the following categories:

1. Properties of Integers

2. Properties of Fractions

3. Exponents & Roots

Let’s dive into what questions each of these categories could throw up.

**a. Properties of Integers**

This is probably the broadest of all categories under Number Theory. Here are some concepts you might be tested on:

– Basic properties of different types of numbers (For example, Natural Numbers, Whole Numbers, Prime Numbers)

– Division Algorithm and Divisibility Rules

– Factors & Multiples with special emphasis on HCF and LCM

– Composite Number Concepts

– Remainder Concepts

– Factorial Notation

**Sample Question:**

If a number has a remainder of 4 upon division by 5 and a remainder of 2 upon division by 3, what remainder must it have upon division by 15?

i. 5

ii. 7

iii. 9

iv. 11

v. 14

**Solution:**

The basic equation for division is Dividend = Divisor x Quotient + Remainder.

Accordingly, a number which has a remainder of 4 when divided by 5 can be written in the form of 5k + 4, where k is the quotient and can take values of 0,1,2,3 and so on.

Hence, the possible values for the number are 4,9,14,19 and so on.

Further, a number which has a remainder of 2 when divided by 3 can be written in the form of 3p + 2, where p is the quotient and can take values of 0,1,2,3, and so on.

Hence, the possible values for the number here are 2, 5, 8, 11, 14 and so on.

So, when we look at both the lists of possible answers, we find that the common number in both sequences is 14. Thus, we can conclude that this is the number we are looking for.

As a rule, if a smaller number is divided by a larger number, the remainder is the smaller number itself. So, when 14 is divided by 15, the remainder is 14 itself.

Hence, the correct answer is option E.

**b. Properties of Fractions**

This part of Number Theory tests your knowledge of fractions and decimals, which are also referred to as Real Numbers.

The inherent nature of questions on fractions and decimals is that they consume a lot of time for even basic calculations. You have to be super sorted with the basic concepts here because if you’re not, you risk losing out on time.

Under this topic, you can expect questions from:

– Comparison of fractions

– Conversion of decimals to fractions

– Identifying whether a fraction represents a terminating decimal or a recurring decimal

**Sample Question: **

Which is greater, 17/21 or 21/25?

**Solution:**

To compare fractions, cross multiply the numerator of the first fraction with the denominator of the second fraction and vice versa. The fraction on the side of the bigger product is the bigger fraction.

**c. Exponents & Roots**

This part of number theory tests your knowledge of exponents/indices and their rules. Questions from this area can include:

– Questions based on laws of indices

– Questions on cyclicity of units digit

– Maximum power of an integer in a factorial

**Sample Question:**

Find the units digit of 9^8235!

Ans: _____

**Solution:**

Any power of 9 always ends with a 9 or 1. Whenever 9 is raised to an odd power, the units digit of the resultant number is 9 and whenever it is raised to an even power, the units digit of the resultant number is 1.

Any factorial greater than 2 will always be an even number.

Hence, 8235! will be an even number.

Therefore, the question given to us can be written as 9even which will have a units digit of 1.

**2. Ratios**

Ratios are important not only because they can give direct problems but also because they are applied in other areas as well. You are expected to know the basic concepts of ratios so that you can apply them.

In this topic, you will be expected to solve problems on:

a. Interpreting ratios

b. Bridging ratios

c. Conducting mathematical operations on ratios

d. Word problems based on ratios

**Sample Problem:**

In a class of 35 students, if boys and girls are in the ratio of 2:3, how many girls are there?

Ans: _____

**Solution:**

Given that the ratio of boys and girls is 2:3. This means that for every 2 boys there are 3 girls.

Let the number of boys = 2x and the number of girls = 3x.

Then, 2x + 3x = 35

i.e. 5x = 35

Therefore, x = 7

Therefore, the number of girls which is 3x, will be 21.

**3. Percentages**

This is a very important topic because you’ll get direct questions related to it and you’ll probably also get questions related to its applications.

You can expect one or two direct questions on percentage concepts. However, the applications of percentages like Profit and Loss will also give you two to three questions.

Hence, the topic of percentages accounts for almost 40% of the Arithmetic questions. So, make sure that you are well prepared on this topic.

Direct questions on percentages could be based on

a. Basic percentage calculations

b. Percentage Change concepts

c. Successive percentage change concept

We will have a look at the application areas of percentages in the next section.

**Sample question:**

The price of a ticket increased from $80 to $84. Find the percentage increase in the price of the ticket?

**Solution:**

The percentage increase in a quantity is given by the expression:

In this case, the initial value is 80 and the final value is 84. Substituting these values in the expression given above, we get 5% as the answer.

The value 5 has to be filled in the answer box provided.

**4. Application-based topics**

Application-based topics cover a large variety of sub-categories. The ones you could be tested on are:

a. Profit & Loss and Interest which are applications of Percentages

b. Word problems based on ratio concepts

c. Word problems based on number theory

d. Time & Distance and Time & Work which are applications of Ratios

**a. Profit & Loss and Interest**

Questions on Profit & Loss and Interest are usually perceived by students to be formula oriented topics. This means that you may have a stereotype in your head that you need to memorize a number of equations to be able to solve questions from these areas.

However, this is farthest from the truth. And the truth is that if you are well versed with the basic percentage calculations and percentage change concepts, Profit & Loss and Interest problems are mere applications of these concepts.

Hence, our advice to you would be to practice as many questions as possible from the Percentages topic so that you have built up sufficient muscle memory to tackle questions from Profit & Loss and Interest.

**Sample Question: **

A shopkeeper bought a pound of almonds at $4. He made a profit of 33.33% after giving a discount of 33.33%. Find the marked price of the pound of almonds.

i. $5

ii. $5.66

iii. $7

iv. $8

v. $9

**Solution:**

The cost incurred by the shopkeeper to purchase the almonds is $4. Hence, CP = $4.

If he made a profit of 33.33%, it means he made a profit of 4/3 (because 33.33% = ⅓ and remember, profit percentage is always calculated with reference to CP).

But, Profit = SP – CP

Hence, SP = CP + Profit

Therefore, SP = 4 + (4/3)

SP = 16/3

We know that he gave a 33.33% discount. Percentage of discount is always calculated with reference to Marked Price (MP). Let MP = x.

Then, Discount = x/3 (remember 33.33% = ⅓)

Discount = MP – SP

(x/3) = x – (16/3)

To simplify, we have x = 8.

Hence, the correct answer option is D.

**b.Word problems based on ratio concepts**

#### Word problems based on ratio concepts usually require you to apply the concept of interpreting a ratio and then build a mathematical version of the statements to solve the question.

#### Since a word problem is usually built around lengthy and complex statements, word problems also test your reading and comprehension skills.

#### They also test your ability to integrate different bits of data into a whole similar to solving a jigsaw puzzle.

#### Word problems based on ratios can be:

a. Word problems on interpreting ratios

b. Problems on ages

c. Problems on numbers and digits

**Sample Problem:**

Six years ago, the ratio of ages of Bob and Joe is 2 : 5. Four years from now, the ratio of their ages will be 4 : 5. Find the sum of their present ages.

Ans: _____

**Solution:**

Let the ages of Bob and Joe, six years ago, be 2x and 5x respectively.

Then, their ages, four years from now, will be 2x + 10 and 5x + 10 respectively.

It is given that the ratio of their ages, four years from now is 4:5.

Therefore, 2x + 105x + 10 = 45

Solving this, x = 1.

Therefore, the present ages of Bob and Joe are 2 and 5 respectively which means the sum of their present ages is 7.

**c. Word problems based on Number Theory**

Word problems based on number theory also test your comprehension and interpretation skills. Along with these, they also test your in-depth knowledge of number theory concepts.

Word problems on number theory are usually based on properties of numbers.

**Sample Question: **

Sum of the LCM and HCF of two numbers is 760, and LCM is 18 times their HCF. If one number is 360, then the other number is:

i. 20

ii. 48

iii. 64

iv. 80

v. 96

**Solution:**

Let the two numbers be x and y; let their HCF be H and their LCM be L.

Then, L + H = 760

It’s also given that L = 18H.

Substituting this in the equation above, we have 18H + H = 760

Therefore, 19 H = 760 or H = 40.

Hence, L = 18 x 40 = 720.

Let x = 360.

Product of two numbers = Product of their HCF and LCM.

Therefore, 360 x y = 40 x 720

Therefore, y = 80.

So, the correct answer is Option D.

**d. Time & Distance and Time & Work**

Both the topics mentioned above are the most significant applications of all topics that you learn in Arithmetic namely Numbers, Ratios and Percentages.

This is because questions from these two areas usually involve multiple concepts drawn from multiple topics, intertwined in such a way that a person without a firm grasp of all the concepts will find the going tough.

This is also the reason why these two topics contribute at least 3 to 4 questions to the Quant section.

**Sample Question:**

Stan drives at an average speed of 60 miles per hour from Town A to Town B, a distance of 150 miles. Ollie drives at an average speed of 50 miles per hour from Town C to Town B, a distance of 120 miles.

Quantity A | Quantity B |

Amount of time Stan spends driving | Amount of time Ollie spends driving |

Solution:

Speed = Distance/Time

Hence, Time = Distance/Speed

Stan’s Speed is 60 mph and he has covered a distance of 150 miles. Hence, the time taken is (5/2) i.e. 2.5 hours.

Similarly, the time taken by Ollie is (12/5) i.e. 2.4 hours.

Hence, Quantity A is definitely greater than Quantity B.

The correct answer option is Option A.

**Conclusion**

So, that is all for now, folks! This is everything we thought you might need some clarity on for now.

With this blog, we hope we have given you enough ammunition to ruminate on and plan your prep for Arithmetic.

Go ahead, pick up that study material and get cracking with your GRE Quant prep!

Feel free to reach out to us through the comments section below by posting any questions/suggestions you may have. We’ll get back to you ASAP!

## Everything You Need to Know About GRE Quant

Reading Time: 12 minutesDoes the very idea of GRE Quant give you the jitters?

Are multiple sources of information confusing you further rather than helping you understand GRE quant?

Would you like to take advantage of the GRE exam pattern to target a 165+ score on quant?

If you answered with a ‘No’, we’re happy to see that you have your GRE math act together! Congratulations, well done! We do hope that this article will help you anyway.

If you answered these questions with a “YES!”, look no further. You have come to the perfect place!

In this blog, we’ll be giving you a detailed insight into the Quantitative Reasoning (QR) section of the GRE. Here’s a quick roadmap to help you navigate this article.

- GRE Test Pattern
- Scoring Pattern for GRE Quant
- Quant vs Math
- Subsections in GRE Quant
- Using the Calculator

Let’s begin by refreshing our knowledge of the GRE test pattern.

**GRE Test Pattern**

Here’s what the GRE pattern looks like:

The test starts off with the mandatory AWA section which spans the first 60 minutes.

Next, you’ll either get a Quant section or a Verbal section to solve, but which one comes first is decided at random. Based on which section comes first, you can figure out how many of which section you should expect.

If your AWA is followed immediately by a VR section, it means that you will see 3 VR sections interspersed by 2 QR sections. However, if you see a QR section following your AWA section, it means you will be required to answer 3 QR sections and 2 VR sections.

Whichever way it is, you are required to answer 100 questions (20 questions per section x 5 sections) in 160 minutes to 165 minutes.

This is because one of the five sections is an unidentified experimental section, which will not be scored.

However, since it is an unidentified section, we advise you to go ahead and answer the test as though it actually contained 100 questions.

**Scoring Pattern for GRE Quant**

Your performance on the two scored sections of GRE math is first converted to a raw score. This raw score is based on how many questions you answered right and how many you answered wrong.

This raw score is then converted to a scaled score which can range from 130 to 170, in single-point increments. So, even if you get all the Quant questions wrong, you will still get a 130. But that’s the equivalent to getting a zero on GRE quant.

Statistically, scoring 165 on GRE math represents the 89th percentile.

This means that if you get 165, you have scored as much as or more than 89% of all the GRE test takers. In other words, you are in the top 11 percent of all students who took the GRE.

**Food for thought – ** The highest score one can get in Quant is 170, which represents the 100th percentile. This is a difference of just five points in terms of marks scored, but when it comes to percentiles, the same distance represents an 11-point difference. What does this tell you?

The only reason this can happen is that a substantial number of people get scores between 165 and 170. What does THIS tell you?

In our opinion, this suggests that it’s not *all that* difficult to get that perfect GRE quant score.

So! Let’s now get into dissecting GRE quant so that you have a very clear understanding of the whole thing by the time you finish reading this blog.

**Quant vs. Math**

You may notice that the words ‘quant’ and ‘math’ are used interchangeably, even in this blog.

However, here’s the thing:

Quant ≠ Math. At least not on the GRE.

Mathematics is akin to outer space, in that it is MASSIVE. You could go from school level mathematics to post-doctoral degree-level math and still not cover everything there is to know about mathematics. There’s literally nothing in the universe that is unrelated to math.

Seriously, if you have doubts or think this is crazy-talk, Google ‘golden ratio‘. Oh and also, just for fun (yes, *fun*) try this website out.

Mathematics is not only about formulae; it is also about theories, theorems, propositions, proofs, and a whole bunch of other things that are difficult to comprehend, even for the best of us.

The point is, it’s a capital mistake to mix up quant and math in your head. You’ll just end up making a mountain out of a molehill because the GRE quant is only a very small subset of the gigantic subject called Mathematics.

Quantitative Reasoning on the GRE is exactly like it sounds. It looks to gauge your reasoning skills – both analytical and logical – when it comes to numbers, along with your basic mathematical skills.

When we say ‘basic mathematical skills’, we mean the fundamental level of mathematics we all learned in high school, regardless of how our academic paths digressed from there on.

Didn’t we all learn that the sum of the three angles in a triangle is 180 degrees?

This is an example of a basic mathematical skill we all acquired when throughout our school lives. In GRE math, this is the level that you’re expected to be well-versed with.

As a rule, the GRE doesn’t look to find mathematicians. The objective is to test intelligence, which mostly means the ability to apply whatever you do know. GRE quant measures your ability to estimate, use logic, use the given answer options and eliminate them to solve the questions you face – in short, your ability to use ‘reasoning’ with numbers.

We hope that this has assuaged at least some of the fears and mental blocks you may have had with respect to GRE math/quant. Please note that we only mean a small subset of mathematics when we say ‘GRE math’ or ‘GRE quant’, and not the entire mammoth that is math.

Next, we talk about what actually constitutes ‘GRE quant’.

**Subsections in GRE Quant**

There are three ways in which you can divide GRE Quant to help you study it properly.

Let’s get into the details of each of these, one at a time.

**I. Based on Areas of Math**

#### The questions in GRE quant comprise of questions drawn from the following areas of Math:

1. Arithmetic

2. Algebra

3. Geometry

4. Data Analysis

Of these, Arithmetic takes the lion’s share of importance – almost 8 to 10 out of 20 questions come from Arithmetic. That’s almost 50%, which is why preparing well for Arithmetic questions is of paramount importance.

Algebra and Data Analysis each account for 15% to 20% of the questions, while Geometry accounts for 10% to 20% of the questions.

As we said before, the questions you can expect from each of the four areas will be from topics that you learned in high school. But, this is not to say that you will be tested on all the topics that you learned in high school.

Topics like Logarithms, Progressions, Relations and even Trigonometry are not tested on the GRE.

Relieved to hear that? We know the feeling!

**II. Based on Question Types**

There is another way of categorizing Quant questions – it’s based on question types. There are four different types of questions in GRE Quant which you need to be aware of. They are:

1. Quantitative Comparison

2. Multiple Choice – Select One Answer Choice

3. Multiple Choice – Select One or More Answer Choices

4. Numeric Entry

Now, let’s examine each one of these in further detail.

**1. Quantitative Comparison (QC)**

Although this type of question tests your basic mathematical skills, it tests your reasoning and estimation skills to a larger extent. The basic structure of a Quantitative Comparison question is shown below:

Information / Constraints

Quantity A | Quantity B |

A. Quantity A is greater

B. Quantity B is greater

C. The two quantities are equal

D. The relationship cannot be determined from the information given.

As you can see, QC questions test your ability to compare two quantities and arrive at a conclusion about their magnitude. Generally, 6 to 8 out of 20 questions will be QC questions.

This goes to show the importance that this question type holds in GRE Quant.

A QC question may have additional information/constraints given above the two quantities or may not have. Based on this, you are required to compare the two quantities and mark the relevant options out of the four.

Accordingly, you will mark:

Option A if Quantity A is ALWAYS greater than Quantity B.

Option B if Quantity B is ALWAYS greater than Quantity A.

Option C if Quantity A is ALWAYS equal to Quantity B.

Option D if a definite relationship cannot be established between the two on the basis of the information provided.

The pertinent point to be noted here is the word **ALWAYS**.

For example, if Quantity A is sometimes greater than Quantity B but sometimes lesser than Quantity B, then Option A cannot be marked as the answer. Similar reasoning can be applied to the other options as well.

**Sample Question:**

Akshay is younger than Chitra

Twice Akshay’s Age | Chitra’s age |

A. Quantity A is greater

B. Quantity B is greater

C. The two quantities are equal

D. The relationship cannot be determined from the information given

Now, if you observe this question, there is hardly any math involved in solving this question except basic inequality concepts.

Consider Akshay’s age = 10 and Chitra’s age = 20; in this case, Quantity A will be equal to Quantity B.

On the other hand, consider Akshay’s age = 10 and Chitra’s age = 30; in this case, Quantity A will be lesser than Quantity B.

Since we are unable to say whether Quantity A is ALWAYS equal to B or ALWAYS less than B, the answer to this question is option D.

**2. Multiple Choice – Select One Answer Choice**

**This question type is something most of us will be familiar with. In this question type, a question will be followed by five answer options named A, B, C, D and E. Your job is to solve the question and pick one of the five options, which you think is the answer.Two points to note about this question type:**

**– There is one and only one correct answer to each question.**– The five options will be arranged in either ascending order or descending order of magnitude if numbers constitute the options.

While solving questions of this type, you need to make use of point number 2 above, by eliminating options and retaining those which you think could make the cut.

If you resort to the conventional methods in all problems of this type, you will end up wasting precious time whilst getting the same answer which does not sound commonsensical.

Approximately 6 to 8 questions of this type can appear in the Quant section. Therefore, this question type coupled with QC constitutes almost 75% of the entire Quant section of the GRE.

**Sample Question: **

Which of the following numbers is the farthest from the number 1 on the number line?

A. -10

B. -5

C. 0

D. 5

E. 10

From the given diagram, it is clear that -10 is the farthest number from the number ‘1’. Hence, Option A is the correct answer.

**3. Multiple Choice – Select One or More Answer Choices**

This is the more challenging variant of the Multiple Choice question type. Here, a question can be followed by any number of answer options ranging from 3 to 10 and therein lies the challenge. Your job is to select all the answer choices applicable under the conditions given.

A few points to note about this question type:

– These questions are marked with square boxes beside the options, not circles or ovals.

– Some questions of this type might ask you to mark a specific number of options as answers.

– Some other questions of this type might ask you to mark all those options as answers as are applicable.

– Some questions of this type might also have only one correct answer.

– No partial credit is awarded ( in this sense, this question type is analogous to the multiple blank questions that you encounter in SE or TC of the Verbal section of the GRE).

Eliminating as many options as possible by logic and estimation is a very good strategy to adopt in such questions. Elimination based on concepts is also a good method. Plugging in the remaining options into the question and checking if all of them apply, is the last stage of filtration before finalizing the answer options.

Around 3 to 4 questions in the Quant section belong to this question type. Hence, this question type does not pose that big a challenge anyway.

**Sample Question:**

Which of the following integers are multiples of both 2 and 3? A. 8 B. 9 C. 12 D. 18 E. 21 F. 36

This is a very simple question based on divisibility concepts.

Any number that is a multiple of 2, is an even number. So, if a number has to be a multiple of both 2 and 3, it has to be an even number first.

Based on this, we can eliminate options B and E, since they are not even.

Now, the next step is to eliminate the numbers which are not multiples of 3. Clearly, 8 is not a multiple of 3. Hence, Option A can be eliminated.

We are left with options C, D and F. Let us see if these are the final options which we can retain.

Any number which is a multiple of both 2 and 3 is a multiple of 6. All three numbers, i.e., 12, 18 and 36 are DEFINITELY multiples of 6. Hence, we can retain options C, D and F as the final answer.

There we go: the correct options to be marked are C, D, and F!

**4. Numeric Entry**

This is probably the only question type that we can all relate to from our school days. Because this is the question type where you have to work out the problem from start to end and *there are no options provided!*

Yep, you read it right!

No options are provided as part of the question. So, you have to be extra careful while reading and analyzing the question and working it out methodically.

You will be required to type in your answer in a single box if your answer is an integer or a decimal, or in two boxes if your answer is a fraction.

The good news is that this question type contributes a measly 10% of the total number of questions in Quant. So you may expect 1 or 2 questions from this type.

A few points to note about the Numeric Entry question type:

– Sometimes, there will be labels before or after the answer box to indicate the appropriate type of answer.

– If you are asked to round the answer, make sure you round it to the required degree of accuracy.

– Only mark the final answer in the box and not any of the intermediate answers that may be a part of your solution.

– Enter your answer as an integer or a decimal if there is a single answer box OR as a fraction if there are two separate boxes—one for the numerator and one for the Denominator.

– To enter an integer or a decimal, either type the number in the answer box using the keyboard or use the Transfer Display button on the calculator.

i. First, click on the answer box—a cursor will appear in the box—and then type the Number.

ii. To erase a number, use the Backspace key.

iii. For a negative sign, type a hyphen. For a decimal point, type a period.

iv. To remove a negative sign, type the hyphen again and it will disappear; the number will remain.

v. The Transfer Display button on the calculator will transfer the calculator display to the answer box.

vi. Equivalent forms of the correct answer, such as 2.5 and 2.50, are all correct.

vii. Enter the exact answer unless the question asks you to round your answer.

– To enter a fraction, type the numerator and the denominator in the respective boxes using the keyboard.

i. For a negative sign, type a hyphen. A decimal point cannot be used in the fraction.

ii. The Transfer Display button on the calculator cannot be used for a fraction.

iii. Fractions do not need to be reduced to lowest terms, though you may need to reduce your fraction to fit in the boxes.

**Sample Question:**

A rectangle R1 has a length of 25 and a width of 20, while another rectangle R2 has a length of 30 and a width of 20. What is the ratio of the perimeters of the two rectangles?

The perimeter of a rectangle is given by the formula 2 (l+b) where ‘l’ represents the length of the rectangle and ‘b’ represents the breadth/width of the rectangle.

Therefore:

Perimeter of R1 = 2(25 + 20) = 2(45) = 90

Perimeter of R2 = 2(30 + 20) = 2(50) = 100

Hence, the required ratio is 9:10. Remember that, because the question is asking you to find out a ratio, you have to simplify it to the lowest form before entering the numbers 9 and 10.

On the contrary, had the question asked you what fraction of perimeter of R2 is perimeter of R1, then you could directly plug in 90 and 100 without worrying about simplifying it to the lowest form.

**III. Based on the Relevance of the Question**

The two question types based on the relevance of a Maths question are:

**1. GRE Quant Questions Described in a Real-Life Setting**

Questions of this type are Maths questions where real-life scenarios are simulated/described in the questions.

Questions from Numbers, Word Problems, Time and Work, Time and Distance, Permutations and Combinations, Probability, Statistics & Data Interpretation, will all come under this category.

**2. GRE Quant Questions Described in a Purely Mathematical Setting**

Questions of this type are mostly concept-oriented and you may not be able to relate the situation described in the question to real life every time.

Questions from equations, inequalities, modulus, functions, and geometry, all come under this category.

**Using the Calculator**

The GRE proves to be a very student friendly test. This is testified by the fact that there is an onscreen calculator available for use in the Quant section.

Although you may think that using the calculator extensively will reduce the burden of calculations, it will actually do the opposite because you will end up spending a lot of time keying in the values.

Even if you make one error while keying in the values, you may end up getting a wrong answer, but you will not realize this until later, since you would not consciously notice that you made an error, because of the time constraints.

As such, our advice to you on using the calculator would be to use it:

1. When you are adding, subtracting, multiplying or dividing substantially large numbers so as to save time and improve accuracy;

2. When you are dealing with addition, subtraction, multiplication or division of decimals

3. When you have to find the square of a number which has 3 digits or more.

4. When your calculations involve finding a square root of a large perfect square or smaller imperfect squares

5. When estimation/approximation cannot be resorted to, to get to the answer.

On the other hand, avoid using the calculator for:

- Simple calculations, which you know can be done mentally.
- Questions where a fraction is required as an answer.

While using the calculator, make sure that:

- You do not mis-key the numbers or the signs.
- You know that all of the calculator’s buttons include Transfer Display
- You know that the Transfer Display function can be used only with Numeric Entry questions with a single answer box
- The calculator follows the PEMDAS sequence of operations while computing values
- The calculator gives an error for mathematical operations like Division by ZERO and the square roots of negative numbers.

We hope that we have given you all that you needed to know about the Quant section of the GRE.

We also hope that we have imbued you with a sense of confidence to take the Quant section head on and quell the challenge. So, go ahead and start preparing for the Quant section using some of the best resources you can find for GRE preparation, from CrackVerbal.

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## All You Wanted to Know About GRE Quant Data Interpretation

Reading Time: 3 minutesData Interpretation in the GRE can take up approximately 15 to 20 % of questions. That would be approximately 6 to 8 questions, not counting the experimental section.

**What is tested in Data Interpretation?**

**What is tested in Data Interpretation can be split into two broad buckets:**

• Charts

• Statistics and Counting methods

**I) Charts**

In the GRE, data interpretation questions (Charts) typically come between questions 11and 18, and the questions would contain Bar Charts, Line graphs, Pie charts, Box Plot graphs, Normal curve, etc.

A single chart may have three to four questions, where each question could be of a different question type (Numerical entry, MCQ and multiple answers type.)

Plan these questions wisely, because these questions tend to take more time and if this is the hard section, it would be tricky. Plan these questions towards the end of each math section, complete the rest of the questions and then come back to these at the end.

If you are a good test taker, you should have around ten to twelve minutes to solve chart questions.

**II) Descriptive Statistics and Counting methods**

Many GRE test takers don’t know the importance of this topic. It is actually very important and one can expect approximately three to five questions from this topic.

The questions will be based on Mean, Median and Mode, Range, Standard Deviation, Sets, Probability, Permutation and Combination.

**Descriptive Statistics questions test your skills at:**

• Basic Operations using Average

• How to calculate average for an evenly spaced set

• Comparing the Standard Deviation of two sets

• Finding the range

• Max and Min possible value in a set, given the average of the set

**Counting methods and Probability test your skills at:**

• Mutually and Non – mutually exclusive sets

• Finding the total number of arrangements (with or without restrictions)

• Finding the total number of selections (with or without restrictions)

• Arrangement of Numbers and words

• Probabilities of Complex events

**Sample question**

**A set of nine different integers has a range of 35 and a median of 25. What is the greatest possible integer that could be in the set?**

** 1. 55**

** 2. 56**

** 3. 57**

** 4. 60**

** 5. Cannot be determined**

**Explanation**

Given:

A set of nine different integers have a range of 35 and a median of 25.

Question: What is the greatest possible integer that could be in the set?

As the integers are different integers, we can say that they are tightly bound because we are trying to find the greatest possible integer. To find the greatest possible integer, we have to keep the smallest possible integer as the maximum value.

Let the smallest integer be X. The greatest integer will be X + 35. Maximizing X + 35 means maximizing X.

X, X +1, X+2, X+3, X+4, X+5, X+6, X+7, X+8

Median is 25, hence X + 4 = 25 and X = 21

So the integers are:

21, 22, 23, 24, 25, 26, 27, 28, 29

As range is 35, the set will become:

21, 22, 23, 24, 25, 26, 27, 28, 56.

Hence the largest integer is 56. So the answer is B.

## 5 Must-Know Tips for GRE Quant Algebra

Reading Time: 8 minutes“I love algebra! It’s so plain and simple!”

Said nobody, ever.

The mere idea of Algebra gave us nightmares when we were in school. Algebra was like a giant ‘x’ – full of random variables and symbols that we couldn’t make head or tail of!

However, you cannot hide behind excuses like how completely terrible you think Algebra is – if you want to get a 165+ in GRE Quant, that is.

Here’s the thing:

If you genuinely wish to up the ante, you cannot “not do well” in Algebra. You cannot say that you will try to cover it up in the other three sections.

So buckle up and let’s do this thing!

We’ll give you the lowdown on five basic tips to improve your performance in Algebra. By the time you finish reading this blog, you should have a fair idea of what you can do to score well in this area of GRE Quant.

**Why is Algebra Important?**

Before we answer that question, here’s an interesting fact about the origin of the word ‘algebra’. It comes from the Arabic term, ‘al-jabr’ which means the reunion of broken parts. How interesting, right?!

Questions based on Algebra account for around 15% to 20% of the questions in a GRE Quant section. So, this means you should expect 3 to 4 questions per section to be based on Algebra.

Hence, considering that there are either two or three Quant sections on the GRE, you should expect to see anywhere between 6 and 12 Algebra questions on one entire GRE test. That is a BIG chunk of the questions!

Further, GRE Quant Algebra covers 5 basic sub-categories:

- Linear & Simultaneous equations
- Quadratic equations
- Inequalities (Linear & Quadratic)
- Absolute values
- Functions

Of these, equations and inequalities get the highest weightage in terms of importance. Questions from functions and absolute values are relatively easier and fewer in number.

With this approximate picture in mind, let’s move on to exploring 5 super-useful tips to help you boost your GRE Quant score.

**1. Start Out Positive**

We cannot stress enough on the importance of starting out with a positive mindset about GRE Quant or even just algebra.

We know it is easier said than done, especially considering how much everyone seems to hate algebra.

And we know you’re probably thinking, “Dude, ‘positive’ is the last thing I can bring myself to feel about Algebra.” Trust us, we know the feeling. If you simply cannot feel positive about Algebra, try to at least develop a neutral perspective towards it.

Starting off with a negative attitude and thinking about all the horrible experiences you’ve shared with Algebra is the worst thing you could do for yourself right now. It just takes a few wrong answers for this feeling to spiral out of control and you will be back in familiar territory again cursing Algebra and saying “Algebra sucks, big time!”

And you’re not going to score a 165+ with that attitude.

Instead, if you start off by telling yourself, “Okay, let’s just give this a shot,” a few wrong answers will only end up fortifying your determination to get the next few questions right.

Every time you pick up the book to solve Algebra questions, remember that the only score you have to beat is your own previous score.

If you got 5 right answers out of 10 yesterday, all you have to do today is get 6 answers right instead.

This will help you stay calm and collected.

With a calm and collected mindset, you’ll be able to perform better on every consecutive question, too.

**2. Get Equations and Inequalities Sorted**

Having a good grasp of the basic concepts is fundamental to doing well in any area of study. It’s no different with Algebra.

If you want to be good at solving equations, you need to know some standard equations which are universally true.

Similarly, if you want to solve more inequality problems correctly, you need to be good at reproducing the basic concepts of inequalities.

Simple things are not so simple, because we usually undermine their importance.

For example, look at algebraic identities.

How many algebraic identities can you rattle out in under a minute?

We’re not kidding, this is dead serious! Try it! Here’s what we came up with:

1. (a+b)^2 = a^2 + b^2 + 2ab

2. (a-b)^2 = a^2 + b^2 – 2ab

3. a^2 – b^2 = (a+b) (a-b)

4. (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac

5. (a+b)^3 = a^3 + b^3 + 3ab (a+b)

6. (a-b)^3 = a^3 – b^3 – 3ab (a-b)

7. a^3 – b^3 = (a-b) (a^2 + ab + b^2)

8. a^3 + b^3 = (a+b) (a^2 – ab + b^2)

Wondering how knowing a few equations will help you? Let’s take a sample question to help you figure that out. 🙂

**Sample Question:**

**3. Don’t Solve Everything**

No, we’re not out of our minds – hear us out!

We know it makes us sound cuckoo, advising you not to try and solve everything when solving stuff is literally all that’s expected of you.

But here’s the thing:

Solving is NOT what’s expected.

Think about it – GRE Quant is about finding the answer to every math question thrown your way, sure, but solving it is not the only way to find the answer!

Most of us work on a more or less ‘automatic’ mode when it comes to dealing with math. We just get into calculations without over-analyzing the given question. And that works out well most times.

However, that will work against you when it comes to GRE Quant.

While Math requires solving, the GRE test doesn’t provide enough time for you to actually do that every time. Remember, the GRE doesn’t care *how* you arrive at a solution, they only care about whether you get to the right one in time.

As we mentioned in our blog on GRE Quant, ‘Quant’ means much more than Math. Don’t approach it like Math.

Here’s what you can do instead of solving, from time to time:

Eliminate wrong answer options till you’re left with just one possible answer. Let’s consider an example to demonstrate this.

**Sample Question:**

Solve for the range of x : 16 + x > 8x – 12

A. x > 5

B. x > 10

C. x = 4

D. x > 4

E. x < 4

**Solution:**

You know the usual way of solving this inequality-based question. Let’s try it our way:

To begin eliminating the available answer choices, we always start by plugging Option C into the question. Contact us to find out why.

In this case, Option C is x = 4. Plugging that into the inequality, we get:

LHS: 16 + 4 = 20

RHS: 32 – 12 = 20

In short, LHS = RHS. Given that this is supposed to be an inequality-based problem, Option C is clearly not the right answer.

As the next logical step, let’s try option D, which says x > 4. What’s the first number that comes to mind when you consider x > 4?

For us, it is 5. So, we’ll plug that in to see if it works, but you can choose whatever number you want.

With x > 4 (or x =5 in this case),

LHS: 16 + 5 = 21

RHS: 40 – 12 = 28

So, LHS < RHS. Meaning Option D is not the answer, either.

Consider this: if x > 4 gave us LHS < RHS, surely the values given in options A and B will also give us the same result. Thanks to this, we can not only eliminate option D, but also options A and B.

With A, B, C, and D, all eliminated, we’re only left with Option E, which HAS to be the answer.

Hope our little demo here has given you enough reason to believe that we’re not crazy and that solving everything is *actually *not necessary.

By the way, a word of caution here:

Don’t think that we are advocating the method of ‘trial and error’ as a substitute to conceptual depth. This trickery will only work with MCQs, and inequalities will appear on the test in all forms including quantitative comparison and numeric entry.

Sure, you should play smart wherever you can and reduce your workload, but rest assured, this cannot be done unless your basic concepts are clear.

**4. Don’t Get Overwhelmed by Word Problems**

By ‘word problems’, we mean those questions which have statements that look more like an AWA essay.

Sometimes, you wonder whether the problem is testing you on your knowledge of Math or Reading Comprehension!

Honestly, though, your RC skills are also being tested here. If you don’t manage to make sense of the question statements, you will end up messing up the equation and hence, the answer.

So, the best thing to do when you encounter highly verbose word problems is to keep calm and not get overwhelmed by the situation. Remind yourself that you have tackled RC passages in Verbal which had far greater verbiage.

Sounds logical, right?

The thing is this:

If you start freaking out because you can’t make sense of the question right away, you’ll lower your chances of figuring out what it means within the time limit you have. Besides, none of us is Shakespeare, we all have trouble with complicated language.

It’s okay if you don’t immediately understand. Take a moment, sip on some water and take a deep breath.

Remember this:

A lengthy word problem is like a Paper Masala Dosa. If you try eating it in one bite, you’re going to choke on it. So, take your time and eat your question masala dosa one bite at a time – interpret the word problem in parts, develop variables and then integrate the parts into a whole.

That actually brings us quite conveniently to the next part of this article!

**5. How to Develop the Right Variables**

In word problems on equations/inequations, you get the right answer only if you have framed the right equation/inequation.

Most of the time, people don’t have issues with solving equations – that’s the easy part. What’s difficult is developing equations/inequations from word problems. For those of you who just went, “YEAH DUDE!” in your heads, here are five simple steps to help you decode word problems into the appropriate math sums!

**Step 1 **

Scan the entire question by quickly going through it to get a gist of what the question demands as an answer. Your sole aim here is to figure out what the question is asking, forget all the data it gives to help you do so.

**Step 2 **

Map the important pieces of information from your first reading. This is the stage where you pay attention to the data provided in the question. When you do this, you’ll have a rough idea of what the variables could be and also what the required mathematical operations could be.

**Step 3**

Develop the variable/variables based on the mapping.

**Step 4**

Using the mathematical operations described in the statements, connect the variables and form an equation.

**Step 5**

Solve the equation.

The number of unknown factors generally represents the number of variables.

Words like more than, less than etc., represent specific mathematical operations which form the connections between the variables, represented using the symbols =, >, < , ≤, and ≥.

The following table is a ready reckoner to convert certain phrases into their mathematical counterparts:

Let’s look at a sample question to understand how this can be done.

**Sample Question:**

At a fruit stand, bananas can be purchased for $0.15 each and oranges for $0.20 each. At these rates, a bag of bananas and oranges were purchased for $3.80. If the bag contained 21 pieces of fruit, how many of the pieces were oranges?

A. 8

B. 10

C. 13

D. 15

E. 16

**Solution:**

The unknown values here are the number of bananas and the number of oranges. Hence, these are the variables which have to be assumed.

Let the number of bananas be ‘x’ and the number of oranges be ‘y’.

Here’s what we know:

Cost of each banana = $ 0.15 and the cost of each orange = $ 0.20.

If 1 banana costs $0.15, 2 bananas cost $0.30 i.e. 2 x $ 0.15. Right?

Applying a similar analogy, we can say that x bananas cost $0.15x and y oranges cost $0.20y. When we add these two individual costs, we arrive at the total cost. Combine this with what’s already given and this is what you get:

$0.15x + $0.20 y = $3.80

Since we have two variables, we need two independent equations in order to find unique values for each. It’s quite easy to obtain the second equation because we already know that there were 21 pieces of fruit in the bag. Mathematically, this is written as:

x + y = 21

What do we do next? If you answered “Solve both the equations”, then you have forgotten point #3 of this blog.

As mentioned before, we start by substituting Option C in both the equations and check if it works out. If it does, then that’s our answer.

Remember that we’re trying to calculate the number of oranges. So, the given answer options are talking about the variable ‘y’. As per Option C, y = 13; therefore, x = 8. Hence,

$0.15 x 8 = $1.20 and $0.20 x 13 = $2.60

$1.20 + $2.60 =$3.80

Well, what do you know! Both equations are satisfied if y = 13. Hence, option C is the right answer.

### Conclusion

In conclusion, the idea is to keep your mind relaxed and be aware of everything you see on the question paper. This will help you with GRE Quant Algebra, Geometry, Arithmetic, as well as Data Interpretation.

So, that is about it, folks! We hope that you found this blog useful in your preparation for Quant on the GRE. We look forward to hearing from you about how you incorporated these techniques in your prep and how they helped you.

Feel free to leave any comments and feedback in the comments section below.

## All You Need to Know About GRE Quant (Geometry)

Reading Time: 4 minutesIn GRE, Geometry takes up approximately 15% to 20% of the Quant section. That would be approximately 6 to 8 questions, not including the experimental questions.

Geometry is a vast topic in Math including all kinds of shapes, measurements, theorems, etc., but in GRE Quant, it is restricted to basic shapes (triangles, quadrilateral, etc.) and straight lines in geometry. They rarely test curves (parabola) in co-ordinate geometry. GRE Geometry mostly tests the visual skills and basic measurements (area, angles, perimeter, etc.)

Most of the Indian students do well in GRE Geometry because they are good in basic shapes and formulae.

**What is tested in Geometry?**

**What is tested in Geometry can be split into three broad buckets:**

• Lines and Angles

• Polygons (triangles, Quadrilaterals)

• Circles and 3D

• Co-ordinate geometry

**I) Lines and Angles**

Lines and Angles test your skills at:

• Parallel and perpendicular lines

• Angles of two or more parallel lines

• Properties of parallel and perpendicular lines

**Sample question**

**In the figure above, what is the value of r + s?**

** 1. 112 **

** 2. 118 **

** 3. 122 **

** 4. 128 **

** 5. 142**

**Explanation**

The sum of the interior angles of a triangle is 180 degrees

So, in triangle ACD, angle CAD = 52 degrees

Since /CAB =/CAD + /DAB,

90 = 52+/DAB

So /DAB = 38

Since r = 90

So r+s = 128

So the answer is D.

**II) Polygons (Convex)**

A GRE student should be aware of all the basic rules of triangles and quadrilaterals (basic polygons).

**Polygons test your skills at:**

• Sum of the interior and exterior angles of a polygon

• Area and perimeter of a triangle.

• Different types of triangles (with both sides and angle wise)

• Special triangles 30 – 60 – 90 and isosceles right angle triangle.

• Similar triangles.

• Third Side Rule of a triangle

• Area and perimeter of a Quadrilateral

• Different types of Quadrilaterals (special quadrilaterals).

• Angles and Diagonals properties in special quadrilaterals

**Sample question**

**If 3 and 8 are the lengths of two sides of a triangular region, which of the following can be the length of the third side?**

** I. 5 II. 8 III. 11**

** 1. II only **

** 2. III only **

** 3. I and II only**

** 4. II and III only **

** 5. I, II and III**

**Explanation**

Remember the third side rule of a triangle.

The difference of the other two sides < Third Side of any triangle < Sum of other two sides

So here, the third side has to be between.

5 < Third Side < 11

Only statement II is true.

So the answer is A.

**III) Circles and 3D**

**Circles in the GRE test your skills at:**

• Finding the area and circumference of the circle

• Finding the arc length and area covered by an arc (sector area)

• Central angle theorem

• Tangents

• Chords

• Inscribing polygons (circle inside a square or rectangle, or vice versa)

GRE Three dimensional geometry, tests only the basic shapes like rectangular solids, cylinder and sphere.

**3D in the GRE tests your skills at:**

• Volume and surface area of the cube and cuboids

• Volume and surface area of the cylinder and sphere

• Diagonal and center of rectangular solids, cylinder and sphere

**Sample Question**

**In the above circle, with center O if 0 < x < 40, what are all possible values of y?**

** • 40 < y < 60 **

** • 50 < y < 70**

** • 60 < y < 90 **

** • 70 < y < 90**

** • y > 70**

**Explanation**

From the diagram:

OPQ is an isosceles triangle because OP = OQ as they are the radius of the given circle.

So, /OPQ = /PQO = y

We know that the sum of the interior angles of a triangle is 180.

Given,

2y + x = 180

0 < x < 40

2y > 140

y > 70

Also, 2y < 180

So y < 90

Therefore, 70<y<90

So the answer is D.

**IV) Co – ordinate Geometry**

**Co – ordinate Geometry in GRE tests your skills at:**

• Equation of a line

• Distance between two points

• Slope of a line (also slope of parallel and perpendicular lines)

• Finding x and y intercepts

• Special lines passing through the origin(y = x and y = -x)

• Reflection of a point across x and y axes

• Midpoint of a line

• Intersection of two lines

**Sample Question**

**The vertices of a square ***s *have coordinates (–1, –2), (–1, 1), (2, 1) and (2, –2), respectively. What are the coordinates of the points where the diagonals of s intersect?

*s*have coordinates (–1, –2), (–1, 1), (2, 1) and (2, –2), respectively. What are the coordinates of the points where the diagonals of s intersect?

** 1. (1/2, 1/2) **

** 2. (1/2, –1/2) **

** 3. (3/2, 1/2)**

** 4. (3/2, –1/2) **

** 5. (√3/2, 1/2)**

**Explanation**

As the vertices points given are of a square, all sides should be equal.

Let’s plot these points in the co-ordinate plane:

From this diagram, we can see that the diagonals intersect each other in the fourth quadrant at (1/2, -1/2).

So the answer is B.

## The Ultimate Guide to GRE Inequalities

Reading Time: 13 minutesAre you wondering, why an entire blog post on Inequalities?

Well, as you may have already found out, compared to other question types on the GRE, inequality questions are an especially slippery slope! They have sent many a test-taker tumbling down on the path to not-so-great Quant scores.

By the time you finish reading this post, you will know all that you need to make sure that this does not happen to you!

So, without further ado, let us examine some must-know inequality concepts and strategies that will help us navigate these tricky questions with limited information .

We’ll first start with the fundamental concept of inequalities, followed by basic properties and then move on to explore the complexities involved with some additional properties. Finally we will summarize the key takeaways with a list of points to keep in mind while using inequalities in problem-solving and data sufficiency questions.

**4. Quantitative Comparisons on the GRE**

**1. What are Inequalities?**

Equations and inequalities are both mathematical sentences formed by relating two expressions to each other.

In an equation, the two expressions are deemed equal which is shown by the symbol =.

Where as in an inequality, the two expressions are not necessarily equal – this is indicated by the symbols: >, <, ≤ or ≥.

x > y —-> x is greater than y

x ≥ y —-> x is greater than or equal to y

x < y —-> x is less than y

x ≤ y —-> x is less than or equal to y

**Inequalities on a Number line**

Number lines, such as those shown below, are an excellent way to visualize exactly what a given inequality means. A closed (shaded) circle at the endpoint of the shaded portion of the number line indicates that the graph is inclusive of that endpoint, as in the case of ≤ or ≥.

An open (unshaded) circle at the endpoint of the shaded portion of the number line indicates that the graph is not inclusive of that endpoint, as in the case of < or >

**2. Basic Properties**

There are 2 basic properties of inequalities which we can quickly prove using the example below.

**Property 1:**

If we consider the true inequality

4 < 8

Adding 2 to both sides 6 < 10 (the inequality sign holds true)

Subtracting 2 from both sides 2 < 6 (the inequality sign holds true)

Multiplying both sides by +2 8 < 16 (the inequality sign holds true)

Dividing both sides by +2 2 < 4 (the inequality sign holds true)

Adding or subtracting the same expression to both sides of an inequality does not change the inequality.

Multiplying or dividing the same positive number to both sides of an inequality does not change the inequality.

**Property 2:**

Again considering the true inequality

4 < 8

Multiplying both sides by -2 -8 > -16 (the inequality sign reverses)

Dividing both sides by -2 -2 > -4 (the inequality sign reverses)

Multiplying or dividing the same negative number to both sides of an inequality reverses the inequality – this is also called the flip rule of inequalities.

**A little Q & A anyone?**

Now that we are done with the basic properties of inequalities, here are a couple of questions to make you think.

**Question:** Can we add or subtract a variable on both sides of an inequality?

**Answer:** Yes, because adding or subtracting a variable is the same as adding or subtracting a number.

**Question:** Can we multiply or divide both sides of an inequality by a variable?

**Answer:** No, we cannot, if we do not know the sign of the number that the variable stands for. The reason is that you would not know whether to flip the inequality sign.

**Let us illustrate this with an example –**

If x/y > 1, most test-takers make the mistake of deducing that x>y, by multiplying both sides by y. But we haven’t been given any information about the sign of the number that the variable y stands for.

If x = 3 and y = 2 then the above relation x/y > 1 will hold true, and x will be greater than y.

However if x = -3 and y= -2 then the above relation x/y > 1 will again hold true, but x will not be greater than y.

If x/y > 1, the only fact that can definitely be deduced is that both x and y are of the same sign .

**Example 1:**

**Question:** If a, b, c are non zero integers and a > bc, then which of the following must be true :

I. a/b > c

II. a/c > b

III. a/bc > 1

A. I only

B. II only

C. III only

D. I, II and III

E. None of these

**Solution:**

Now the trap answer here will be D (I, II and III). The general tendency will be to multiply both sides of the first inequality a/b > c by b to get a > bc, both sides of the second inequality by c to get a > bc and both sides of the third inequality by bc to get a > bc.

Remember that we can never multiply or divide both sides of an inequality by a variable if the sign of the variable is not known. In this problem the signs of b and c are not known. The above statements I, II and III can be true, if b and c are both positive. But they will not be true if b and c are negative. Since the question is of a ‘must-be-true’ type, the answer here must be E.

**Example 2:**

**Solve: -6x + 4 ≤ -2 **

Solving an inequality means finding all of its solutions. A ‘solution’ of an inequality is a number which when substituted for the variable satisfies the inequality

**The steps to solve a linear inequation are as follows: **

• Isolate the variable and always keep the variable positive

• Solve using the properties of inequalities

• Represent the inequality on a number line

Isolating the variable by subtracting 4 from both sides we get -6x ≤ -6

Dividing both sides by -6 and flipping the inequality sign we get x ≥ 1

**3. Advanced Concepts**

Well, so far, we saw how the basic operations are applied to inequalities.

It is now time to delve into more complex properties of inequalities, dealing with :

A) Inequalities in fractions

F) Max Min Concept of Inequalities

**A) Inequalities in Fractions**

All proper fractions on the number line can be represented using the range -1 < x < 1 where x represents the proper fraction

All positive proper fractions can be represented using the range 0 < x < 1 where x represents the positive proper fraction

For all proper fractions (0 < x < 1), √x > x > x2

If x = ¼ then √x = ½ and x^2 = 1/16

Clearly here ½ > ¼> 1/16

**Example:**

**If x = 0.888, y = √0.888 and z = (0.888)^2 which of the following is true **

A. x < y < z

B. x < z < y

C. y < x < z

D. z < y < x

E. z < x < y

**Solution:**

Since 0.888 is a fraction,

√0.888 0.888 > (0.888)^2

y > x > z

Reversing the inequality we get z < x < y

**Answer : E**

**B) Squaring Inequalities**

We cannot square both sides of an inequality unless we know the signs of both sides of the inequality.

**If both sides are known to be negative then flip the inequality sign when you square.**

For instance, if a < -4, then the left hand side must be negative. Since both sides are negative, you can square both sides and reverse the inequality sign : a^2 > 16. However, if a > -4, then you cannot square both sides, because it is unclear whether the left side is positive or negative. If a is negative then a^2 < 16, but if x is positive then x^2 could be either greater than 9 or less than 9.

** If both sides are known to be positive, do not flip the inequality sign when you square.**

For instance, if a > 4, then the left side must be positive; since both sides are positive you can square both sides to yield a^2 > 16. However if a < 4 then you cannot square both sides, because it is unclear whether the left side is positive or negative.

**If one side is positive and one side is negative then you cannot square.**

For instance, if you know that a < b, a is negative, and b is positive, you cannot make any determination about x^2 vs. y^2.

If for example, x = -2 and y = 2, then x^2 = y^2.

If x = -2 and y = 3, then x^2 < y^2.

If x = -2 and y = 1, then x^2 > y^2.

It should be noted that if one side of the inequality is negative and the other side is positive, then squaring is probably not warranted.

**If signs are unclear, then you cannot square.**

Put simply, we would not know whether to flip the sign of the inequality once you have squared it.

**C) Reciprocal Inequalities**

Taking the reciprocal of both a and b can change the direction of the inequality.

The general rule is that when a < b then:

• (1/a ) > (1/b). When a and b are positive , flip the inequality.

Example: If 2 < 3, then ½ > 1/3

• (1/a) > (1/b). When a and b are negative , flip the inequality.

Example: If -3 < -2, then 1/ -3 > 1/ -2

• For (1/a) < (1/b). When a is negative and b is positive , do not flip the inequality.

Example: If -3 < 2, then 1/ -3 < 1/2

• If you do not know the sign of a or b you cannot take reciprocals.

In summary, if you know the signs of the variables, you should flip the inequality unless a and b have different signs.

**Example:**

** If 3 ? 6/ (x+1) ? 6, find the range of x **

Taking the reciprocal of the above range and flipping the inequality sign since the entire inequality is positive

1/3 ≥ (x + 1)/6 ≥ 1/6

Multiplying throughout by 6

2 ≥ (x + 1) ≥ 1

Subtracting 1 from all sides

1 ≥ x ≥ 0 –> 0 ≤ x ≤ 1

**D) Like Inequalities**

The only mathematical operation you can perform between two sets of inequalities, provided the inequality sign is the same, is addition.

If the signs are not the same then use the properties to flip the inequality sign and then add the two sets of inequalities.

**Example:**

If 4a + 2b < n and 4b + 2a > m, then b – a must be

A. < (m – n)/2

B. ≤ (m – n)/2

C. > (m – n)/2

D. ≥ (m – n)/2

E. ≤ (m + n)/2

Given 4a + 2b < n and 4b + 2a > m. We can always add like inequalities.

Multiplying the second inequality

4b + 2a > m by -1 we get -4b – 2a < -m.

Now adding the two inequalities

4a + 2b < n and -4b – 2a < -m

4a + 2b < n

-4b – 2a < -m

________________

2a – 2b < n – m

Dividing both sides by 2

a – b < (n – m)/2

Multiplying both sides by -1

b – a > (m – n )/2

**Answer : C**

**E) Min and Max Inequalities**

Problems involving optimization: specifically, minimization or maximization problems are a common occurrence on the GRE .

In these problems, you need to focus on the largest and smallest possible values for each of the variables.

This is because some combination of them will usually lead to the largest or smallest possible result.

Read on to learn from an example.

**Example 1:**

If -7 ≤ x ≤ 6 and -7 ≤ y ≤ 8, what is the maximum possible value for xy?

To find the maximum and minimum possible values for xy, place the inequalities one below the other and make sure the inequality signs are the same. You need to test the extreme values for x and for y to determine which combinations of extreme values will maximize ab.

-7 ≤ x ≤ 6

-7 ≤ y ≤ 8

The four extreme values of xy are 49, 48, -56 and -42. Out of these the maximum possible value of xy is 49 and the minimum possible value is -56.

Whenever two ranges of inequalities are given in x and y and you need to evaluate the value of x + y , x * y, and x – y then use the max-min concept

1. Place the two inequality ranges one below the other

2. Make sure the inequality signs are the same in both cases

3. If the signs are not the same use the properties we have discussed before to make them the same

4. Now add/multiply/subtract both in a straight line and diagonally to get 4 values

5. The greatest value will be max and the lowest value will be min

1/2 < x < 2/3 , and y^2 < 100

Quantity A Quantity B

xy 6

Since y^2 < 100 —> -10 < y < 10

Now placing the two ranges one below the other and finding out the extreme values of xy

1/2 < x < 2/3

-10 < y < 10

The four extreme values of xy here are -5, -20/3 , 5, 20/3. Out of these the maximum value of xy is 20/3 and the minimum value of xy is -20/3. Now since Quantity A can take values from -20/3 to 20/3 a definite relationship cannot be determined with Quantity B.

Answer : D

**F) Quadratic Inequalities**

**3x^2 – 7x + 4 ≤ 0 **

Factorizing the above quadratic inequation

3x^2 – 7x + 4 ≤ 0 —> 3x^2 – 3x – 4x + 4 ≤ 0 —> 3x(x – 1) – 4(x – 1) ≤ 0 —> (3x – 4)(x – 1) ≤ 0

we get 1 and 4/3 as critical points. We place them on number line.

**Since the number line is divided into three regions, now we can get 3 ranges of x**

i) x < 1 (all values of x when substituted in (3x – 4)(x – 1) makes the product positive)

ii) 1 ≤ x ≤ 4/3 (all values of x when substituted in (3x – 4)(x – 1) makes the product negative)

iii) x > 4/3 (all values of x when substituted in (3x – 4)(x – 1) makes the product positive)

At this point we should understand that for the inequality (3x-4)(x-1) ≤ 0 to hold true, exactly one of (3x-4) and (x-1) should be negative and other one be positive. Let’s examine 3 possible ranges one by one.

i) If x > 4/3, obviously both the factors i.e. (3x-4) and (x-1) will be positive and in that case inequality would not hold true. So this cannot be the range of x.

ii) If x is between 1 and 4/3 both inclusive, (3x-4) will be negative or equal to zero and (x-1) will be positive or equal to zero. Hence with this range inequality holds true. Correct.

iii) If x < 1, both (3x-4) and (x-1) will be negative hence inequality will not hold true.

**So the range of x that satisfies the inequality 3x^2 – 7x + 4 ≤ 0 is 1 ≤ x ≤ 4/3**

**The steps to solve a quadratic inequation are as follows: **

1. Isolate the variable and always keep the variable positive.

2. Maintain the Inequation in the form ax^2 + bx + c > 0 or < 0.

3. Obtain the factors of Inequation.

4. Place them on number line. The number line will get divided into the three regions.

5. Mark the rightmost region with + sign, the next region with a – sign and the third region with a + sign (alternating + and – starting from the rightmost region).

6. If the Inequation is of the form ax^2 + bx + c < 0, the region having the – sign will be the solution of the given quadratic inequality.

7. If the Inequation is of the form ax^2 + bx + c > 0, the region having the + sign will be the solutions of the given quadratic inequality.

**Question:** Will the above procedure hold good even for a cubic or a fourth degree equation?

**Answer:** YES. For a cubic inequality we get 3 critical points which when plotted on the number line divides the number line into 4 regions. Mark the rightmost region as +ve and alternate the sign as shown below

Now based on whether the right hand side of the cubic inequality is < 0 or > 0 we get the solution to lie in 2 of the 4 regions.

**4. Quantitative Comparisons on the GRE**

Now that we are through with the properties of inequalities, lets see how we can make use of these properties in quantitative comparisons.

A quantitative comparison question is a big inequality in itself since it asks you to compare and determine which of the two quantities is greater. So the rules of inequalities can be used here, provided the initial comparison is not tampered with.

For e.g. If we consider a basic quantitative comparison question where quantity B is clearly greater than quantity A,

Quantity A Quantity B

4 6

Adding 2, Quantity A becomes 6 and Quantity B becomes 8. Quantity B is still greater.

Subtracting 2, Quantity A becomes 2 and Quantity B becomes 4. Quantity B is still greater.

Multiplying by +2, Quantity A becomes 8 and Quantity B becomes 12. Quantity B is still greater.

Dividing by +2, Quantity A becomes 2 and Quantity B becomes 3. Quantity B is still greater.

Multiplying by -2, Quantity A becomes -8 and Quantity B becomes -12. Quantity A is greater.

Dividing by -2, Quantity A becomes -2 and Quantity B becomes -3. Quantity A is greater.

It is very evident that if we multiply or divide by a negative number the comparison will never be consistent with the initial comparison.

**Points to Remember**

Here are a few things you need to remember when you are using the properties of inequalities to simplify complex quantitative comparison questions:

1.Add or subtract any value to both quantities.

2.Multiply or divide by a positive value.

3.Square both sides only when the quantities are both positive.

4.Never multiply or divide both quantities by a negative number.

5.Never multiply or divide both quantities by a variable if the sign of the variable is unknown.

6.If the sign of the variable is always positive then it is possible to multiply or divide both quantities by the positive variable (for e.g. x2 ,since x2 is always positive).

After reading our simple guide, you should now know what strategies you must employ for inequality questions on the GRE!

We hope this guide helps you along the way to a 170 on GRE Quant!

*You can now have a copy of your own Inequalities guide here!*

## How to Manage Stress on GRE Quant

Reading Time: 2 minutesUsually, it is GRE Verbal that gets the bad rap – both with regards to prep and final scores. However, for at least some of you, studying for GRE Quant can be more stressful that studying for Verbal. Here are a few tips that will help you stay on track:

**Know your start point: **

It is very important that you start your GRE Math journey with a diagnostic test. This will help you ascertain your skill level and determine how much ground you need to cover. Understand that the first test score will always be an underestimation of your skill level, since you are completely unaware of the question types and test features.

**Determine what you are going to study:**

It is very important that you plan out the study material that you are going to cover. In case you decide to study by yourself (self study) make sure you research well about the books or the online portal that you are going to buy – choose the one that fits you best!

**Have a Study Plan:**

The next and the most important part is having a study plan. You might want to determine your strengths and weaknesses before jumping into the questions. Target your weaknesses and sharpen your strengths. Plan out how many hours you can take out from your busy schedule. A minimum of 4 to a maximum of 16 hours should be allocated every week.

**Be Healthy:**

Health is wealth at all times. You need to get good sleep and remain physically fit if you want to tame the GRE. Keep your mind off other stressors- focus on the GRE. Keep taking breaks; reward yourself when your observe progress in your study!

**Take Tests:**

You will never be able to judge your performance just by practicing questions. You need to sit and take tests which simulate the GRE test environment. Mark your improvements and make sure that you review your mistakes. Do not forget to maintain an error log.

Good luck!

*What are your areas of stress on the GRE? Leave your comments below and our GRE experts will guide you! *

*Looking for help with prepping for your GRE test?*

## How to Reduce Silly Errors on GRE Quant

Reading Time: 3 minutesWhenever you review your GRE test and categorize a mistake as “careless error”, you think that you can’t help it. Well you can! You can narrow down the cause of careless errors in GRE to a few habits. If you improve on these habits your “careless errors” will definitely go down. If you are aiming to achieve a 160 + in each section then definitely you will have to be disciplined no matter how talented you are.

Let us see the habits which will help you reduce the so called silly and unavoidable errors.

#### Scratch Paper

You won’t believe this, majority of careless errors happen because of unorganized scratch paper work. Keeping the scratch paper organized requires practice and some techniques which are part of the Crackverbal GRE course.

An ideal scratch paper should look like this:

There are lot of benefits to keeping the scratch paper organized; for example, it allows you to come back to a half attempted question and start from the point you left it at. You don’t make stupid calculation or algebra errors which you would have otherwise.

#### Read the Question Layerfully

Most of the GRE questions are framed in such a way that you miss the most important information if you read the question in a hurry. You have to identify a lot of things in a question like what is given, what is asked, etc. The information that you require to solve the question is coded with layers of extra information and you have to break it down.

#### Don’t Skip Steps

Especially in the GRE Math section students have the tendency to skip simple steps. From 4x^{2} = 32, they would straight away write the value of x in their scratch paper. While skipping steps, we make so many careless errors that hamper our score.

It is always advisable to write the step down in your scratch paper no matter how easy it is. It allows you to review quickly if you have made any mistake.

#### Eat in Pieces

Ever eaten a Subway sandwich? If yes then how do you eat it? Bite by bite right? So if you see a wordy math question then take a breath and do the question in pieces.

It is very important that you read the first sentence and convert the English to Mathish ( we just made that up 🙂 ) and then proceed further. This will again help you to not to make any careless error.

**It’s Logic, Not Math **

People who are good at Math are the ones who get most affected by standardized testing because they tend to think that GRE or any other standardized test will test math. It is always advisable to approach the question logically rather than mathematically. So graphs, logarithms won’t help much in GRE.

Always try to find simpler solutions to the question and you will be surprised to see that each question has a simple solution and does not require any advanced concept at all. After all it is STANDARDised testing.

#### Test Features

There are various test features which help you in NOT making careless errors. For example the review button for every section. Before you submit a section make sure that you have clicked the review button and checked whether you have answered all the questions or not.

Sometimes in a double click we miss a question altogether. The top bar looks like the following.

Don’t forget that you have a calculator to simplify calculations – that doesn’t mean you would start doing simple arithmetic calculations like 20 times 3 on a calculator. Don’t click the Help button it won’t give you the answer 🙂

After developing these habits you would see a great reduction in silly errors, however they won’t count to zero as we are humans and are bound to make mistakes.

Best of Luck!

*So, what trips you up on GRE Quant? Leave your comment below and our GRE experts will guide you!*

*Are you looking for expert help with your GRE preparation? Explore our GRE Course Offerings!*

## GRE Math: Easy or Difficult?

Reading Time: 2 minutes

The question of whether math is easy or difficult revolves in every student’s mind while preparing for the GRE. Going through the official guide will make you believe that quantitative reasoning in GRE requires nothing more than a mere brush up. You could be wrong!

#### Easy but Deceptive

GRE Quantitative Reasoning section is easy but deceptive.** **The questions in the section have a way of camouflaging their cunning. They look easy; however they can be very tricky. Let us discuss some of the features of GRE Quantitative Reasoning questions which make it a part of the test that cannot be ignored.

#### Content Versus Strategy

Even though the concepts tested are at high school level, such as percentages, averages and geometry, the structure of the questions is tricky. GRE Math has a weapon known as Quantitative Comparison questions. These questions ask you to compare two quantities and then identify which of the two quantities is greater. It may look like an easy thing to do; however, these questions are smartly created. Therefore, if you are planning to prepare for GRE Math, don’t forget to learn the strategies that are specific to Quantitative Comparison: approximately 40 percent of the test will be of this type!

#### Time Eaters

Another type of question that makes GRE Math a force to reckon with is the Data Analysis type of question. These questions are not difficult, but they eat up your time, which is a vital resource on the test. Our advice would be to attempt the chart questions at the end. These questions are longer and usually take up more time.

#### Scoring Blues

The last issue is the scores students receive on the GRE Math section. Students typically score higher in GRE Math compared to GRE Verbal; this makes getting even a seemingly high score of 155 pointless. If you want to get into the top universities, you must aim for at least 165 in GRE Quant.

#### Bottom-line

This section works as a hygiene factor:

**You do well in this section – not a big deal!**

** You screw up this section – and the universities notice!**

Therefore, even though the questions are easy, you have to be consistent; getting even a few questions wrong can easily take your score down to the 150s.

Thus, do not take GRE Quant too lightly – make sure you pay it the attention it deserves!

*What has your experience been with GRE Quant? Leave your comments in the comment section below.
Are you looking for guidance with your GRE preparation for Quant? *

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