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All You Wanted to Know About GRE Quant

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Are you looking for techniques and material to crack the GRE with a 160+ GRE Quant score?

Are looking for a no-nonsense approach to get your dream GRE Quant score?

Are you getting overwhelmed with all the advice and looking for simple GRE Quant strategies?

If your answer was a “yes” to any of the above questions, you have come to the right page!

In this article, we will look into the GRE Quant questions and syllabus in detail.

First, let’s understand what GRE Quant is all about.

Many students misunderstand the term and think that “Quant” is synonymous with “Math”.

Mathematics is different from Quantitative analysis. Educational Testing Service (ETS), which administers the GRE exam, could have easily called it “mathematical assessment” but didn’t, and there is a reason for that.

What is the difference between ‘math’ and ‘quant’?

Mathematics includes derivations, theorems, construction, mensuration, etc.

Quantitative Reasoning has more to do with reasoning than actual mathematics. It has very little to do with derivations and high level formulae. If that’s been worrying you, rest assured. Even if you have not done well in Math in your school and college, you can do well in GRE Quant.

Having said that, there are some basic mathematical concepts you need to know to fare well in the GRE, and in this article, we will look at what they are.

Understanding the GRE Score

The Quant section of the GRE forms an integral part of a student’s score. Similar to the Verbal section, the maximum score for the Quant section too is 170, out of the GRE 340 score.

The fact is, a score of 165 in GRE Quant falls only in the 89th percentile, which means that more than 11% of the students score more than 165.
Considering that the difference between 165 and 170 – the maximum possible score, is just 5 points, the fact that 165 falls on the 89th percentile makes GRE Quant much more competitive, when compared to GRE Verbal.

Percentage of Test Takers Scoring lower than Selected Scaled Scores

Topics Covered in the GRE

GRE Quant is made up of four major buckets:

• Arithmetic

• Algebra

• Geometry

• Data Interpretation

The GRE Quantitative Reasoning section tests your ability to interpret given data correctly rather than just your knowledge of formulae and concepts. Out of the four topics, Arithmetic is what is going to be tested pre-dominantly, accounting for approximately 40 to 50 percent of your questions. Arithmetic tests your skills in numbers, ratios, percentages and exponents, etc.

Hence, you should be very good at your basics, which you would have typically studied up to the Eighth or Ninth grade.

• For information about Arithmetic questions in GRE Quant, see All You Wanted to Know About GRE Quant Arithmetic

• For information about Algebra questions in GRE Quant, see All You Wanted to Know About GRE Quant Algebra

• For information about Geometry questions in GRE Quant, see All You Wanted to Know About GRE Quant Geometry

• For information about Data Interpretation questions in GRE Quant, see All You Wanted to Know About GRE Quant Data Interpretation

Question Types

The GRE Quantitative Reasoning section consists of the following question types:

• Quantity Comparisons

• Multiple choice questions (MCQ) with single answer

• Multiple choice questions (MCQ) with more than one answer

• Numerical entry question

I) Quantity Comparisons

Sample Question

X > 0

Quantity A Quantity B

X |X|

You will be provided with four default answer choices for any Quantity comparison question:

1. Quantity A is greater

2. Quantity B is greater

3. Both the Quantities are equal

4. The relationship cannot be determined with the given information.

Explanation

Understanding the Quantity comparison answer choices is very important.

In the first answer choice when they say “Quantity A is greater”, they mean that “Quantity A is always greater”, that is, for any given value, according to the condition of the question, Quantity A is always greater.

Likewise, the second and third option, “Quantity B is always greater” and “Both the quantities are always equal”.

The last answer choice, “The relationship cannot be determined”, indicates that you cannot really say which quantity is greater, or whether they are equal, because it keeps changing for different values.

The answer for the question is C (Both are equal), because given that X is positive, both the quantities will be equal for any value greater than zero.

II) Multiple choice questions (MCQ) with single answer

In this type of question, a statement is given, followed by five answer options. You have to choose the right answer to the question.

Sample Question

If x and y are real numbers, what is the value of x, given that: x + y = 3, and x + 2y = 6?

1. -3

2. -1

3. 0

4. 3

5. Cannot be determined

Explanation

This a typical school math question. Two equations with two variables, x and y

x + y = 3………………. (1)

x + 2y = 6………….…. (2)

Multiply equation (1) by 2

2x + 2y = 6………………. (1)

x + 2y = 6…………. (2)

Subtracting both the equations, we get x=0

The beauty of the GRE lies in the fact that it is not necessary to always solve a quant question mathematically. You can use different techniques to get to an answer.

This equation can also solved by substituting the answer choices back into the equation and checking whether it gives the same “y” value.

You might feel that this a long method to solve this question. However, for some questions in the GRE, that is the way to go.

III) Multiple choice questions (MCQ) with more than one answer

This is similar to MCQ question type, with a few changes:

• You will be provided with three or more answer choices.

• Most of the times, the question expects you to select more than one answer choice, or sometimes, it could be more specific, prompting you to select two answers from the given answer options.

• Answer choices are provided with check boxes, and students need to select all the answer choices that apply to the question.

• You won’t be provided any partial credit if you have selected one of the answer choices correctly, or if you selected only one correct answer. You have to choose all the options that apply to the question, correctly.

Sample Question

If w is a non-positive integer, which of the following must be positive? Indicate all possible values.

1. −3w

2. 2w + 10

3. w^4

4. w^0

5. −w + 0.5

Explanation

Non-positive integer means negative or zero.

So, if we substitute w= -5, answer choice ‘B’ is eliminated.

And, if we substitute w= 0, answer choices ‘A and C’ are eliminated.

Therefore, the correct answer choices are D and E.

IV) Numerical entry question

This type of question is not very frequent in the GRE. It comprises less than 20% of the entire test. Questions of this type ask you either to enter your answer as an integer or a decimal in a single answer box, or to enter it as a fraction in two separate boxes, one for the numerator, and one for the denominator. In the computer-based test, use the computer mouse and keyboard to enter your answer.

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.

• First, click on the answer box. A cursor will appear in the box. Type the number.

• To erase a number, use the <Backspace> key.

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

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

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

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

• Enter the exact answer unless the question instructs you to round off your answer.

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

• For a negative sign, type a hyphen; to remove it, type the hyphen again. A decimal point cannot be used in a fraction.

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

• 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 dress that is originally sold for $150 is now selling for $120. What is the percent decrease in the cost of the dress that is originally sold?

Explanation

The original price is $150.

The new price is $120. The difference is $30.

Hence, the percent decrease is, (30/150)*100 = 20%

Almost 70% to 80% of the questions would be of the first two types.

The first six or seven questions in every Quant section would be of the Quantity Comparison question type.

What else do you need to know about GRE Quant?

• There are a minimum of 40 questions (a maximum of 60 questions if you get the experimental section as Quant). Each section is made up of 20 questions, with a time limit of 35 minutes.

• The syllabus covers portions that were taught until high school; NO advanced topics such as Binomial theorem are asked.

• Within the broad buckets of Arithmetic, Algebra, Data Interpretation and Geometry, many topics such as Arithmetic, Geometric and Harmonic progressions, trigonometry, etc. are excluded.

• You can mark and move any question within a section. You will be provided with a Mark and Review button, which you can use to review questions you mark.

• There is an online calculator provided for GRE Quant. Students are not allowed to take any calculating devices such as calculators or phones.

We hope you found this blog useful.

Please spread its value by sharing the blog on your social media channels, and letting your friends know about it.

If you have any questions about the GRE Quant section, go ahead and let us know in the Comments section.

October, 19th, 2017
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- GRE Quant
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All You Wanted to Know About GRE Quant Data Interpretation

Reading Time: 3 minutes

Data 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.

October, 19th, 2017
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- GRE Quant
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All You Wanted to Know About GRE Quant Algebra

Reading Time: 4 minutes

In GRE Quant, Algebra can take up approximately 15to 20 % of the questions. That would be approximately six to eight questions, not including the experimental section.

Those who had problems with Algebra in school need not worry when it comes to the GRE because you can use alternatives like Plugging In and Back Solving to solve an Algebra question.

What is tested in Algebra?

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

• Equations

• Inequalities and Absolute Values

• Functions

I) Equations

When it comes to Algebra in the GRE, what is tested is not more than what you learned at high school.

Algebra questions test your skills at:

• Simplifying linear equation.

• Quadratic equations – finding roots.

• Factoring equation.

• Polynomials – (very rarely tested. If tested, the questions will be more to do with logic than finding the x value).

• Constructing word problems into mathematical equations.

Sample Question:

Sam is now 14 years older than Pam. If in 10 years Sam will be twice as old as Pam, how old will Sam be in 5 years?

• 9

• 18

• 21

• 23

• 33

Explanation:

	Now	In 5 years	In 10 years
Sam	X+14	X+19	X+24
Pam	X	X+5	X+10

Given X+24 = 2(X+10).

X+24 = 2X +20.

X =4.

In 5 years, Sam will be X+19 = 23.

So the answer is D.

We can also do back Solving to solve this question.

II) Inequalities and Absolute Values

This is where most of the GRE Students struggle. Students have to be very clear on the basics of Inequalities and Absolute values.

Inequalities and Absolute values questions test your skills at:

• Basic rules of Inequalities and Absolute values (This also helps in solving the Quantity Comparison questions).

• Properties of Inequalities and Absolute values

• How to find Max – Min values in Inequalities.

We can make use of a few properties of inequalities while solving a Quantity comparison question.

• 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

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

Knowing the above basic properties of inequalities helps to solve Quantity comparison questions.

Sample question

Quantity A Quantity B

b – 3a2 – 7 6 + b – 3a2

Explanation

	Quantity A	Quantity B
	b – 3a2 – 7	6 + b – 3a2
Adding 3a2	b – 3a2 – 7 + 3a2 = b – 7	6 + b – 3a2 + 3a2 =6 + b
Subtracting b	b – 7 – b = – 7	6 + b – b = 6

Answer: B (Quantity B is greater)

III) Functions:

When you represent something in the form of f(x), you are saying that the value of function f is dependent on the value of the variable “x”. Whenever f(x)”and “g(y)” are used in the GRE, they do not represent products, but functions. Sometimes, functions can also be represented by a special character like #, $,*, etc.

Functions questions test your skills at:

• Finding the input and output values when a function is given.

• Comparing two functions.

• Functions representing series and sequence.

Sample question

For which of the following functions g is g(x) = g(1-x) for all x?

1. g(x) = 1-x

2. g(x) = 1- (x^2)

3. g(x) = (x^3) – (1-x)^3

4. g(x) = (x^2)/(1-x)^2

5. g(x) = x/(1-x) + (1-x)/x

Explanation

Instead of “X” substitute the answer option with (1-x) and check with option g(x) = g(1-x).

1. g(x) = (1-x) → g(1-x) = x , so not equal.

2. g(x) = 1-(x^2) → g(1-x) = 1 – (1-x)^2 = 1-1-x^2+2x = 2x-x^2, So not equal.

3. g(x) = (x^3) – (1-x)^3 → g(1-x) =(1-x)^3 – (x^3), So not equal.

4. g(x) = (x^2)/(1-x)^2 → g(1-x) = (1-x)^2/ x^2, So not equal.

5. g(x) = x1–x + 1–xx → g(1-x) = ((1-x) / x )+ (x / (1-x)) , So both are equal.

So the answer is E.

October, 19th, 2017
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All You Wanted to Know About GRE Quant Geometry

Reading Time: 4 minutes

In 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?

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.

October, 19th, 2017
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- GRE Quant
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All You Wanted to Know About GRE Quant Arithmetic

Reading Time: 4 minutes

There is a saying in Mathematics that, “Arithmetic is the Queen of Mathematics”, because of its high importance. Similarly, in GRE Quant, Arithmetic is very important; one can easily say that in GRE, arithmetic comprises approximately 40 to 50 percent of the Quant questions.

What is tested in Arithmetic?

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

• Numbers and Operations

• Ratio and Percentages

• Rates

I) Numbers and Operations

A student who is writing GRE must be very clear about Real numbers as it is a basic concept one will need to solve any GRE question.

Questions will be based on properties of integers, fractions, Powers and Roots of a number, and Number line (Real Line).

Properties of integers questions test your skills at:

• Division algorithm

• Divisibility Rules

• Remainder Rules

• Prime factorization

• Factors and Multiples

Fractions questions test your skills at:

• How to compare fractions

• Terminating fraction and Non-terminating fraction.

Exponents and roots questions test your skills in:

• Basic exponent rules

• Cyclicity of powers (unit’s place of number when raised to an integer power)

• Maximum power of an integer in a factorial

Number line questions mostly test the decimals values, distance between two numbers, and total number of elements between two integers.

Sample Questions

**1. What is the unit’s digit of (16782345) * (30753847) ?**

1. 0

2. 2

3. 5

4. 6

5. 8

Explanation

Here, one must know the cyclicity patterns of each and every digit.

We only we need to find 82345 * 53847

8 – follows a 4 cycle pattern – that is the units place ends with either 8,4,2,6.

82345 = 82344+1 = units place is 8.

5any = always ends with 5. Five raised to any integer (positive) power always ends with 5.

8 * 5 = 40. So the answer is A.

II) Ratio and Percentages:

Ratios and Percentages will also be tested in Algebra and Geometry.

In Arithmetic, the questions will be based on Ratio and Proportion, Percentages, and Profit and loss.

Ratio and Proportion questions test your skills at:

• Finding the actual value when given a ratio

• Comparing two ratios and multiple ratios

• Constructing an equation using ratios

Percentages questions test your skills at:

• Converting a fraction to a percentage

• Successive percentage change

• Percent increase/ Decrease.

• Simple Interest and Compound Interest

Profit and loss questions test your skills at:

• Cost Price, Selling Price and Marked price

• Successive discounts

• Increasing/Decreasing one price with respect to another.

Sample question:

A used car dealer sold one car at a profit of 25% of the dealer’s purchase price for that car, and sold another car at a loss of 20% of the dealer’s purchase price for that car. If the dealer sold each car for $20,000, what was the dealer’s total profit or loss, in dollars, for the two transactions combined?

1. 1000 profit

2. 2000 profit

3. 1000 loss

4. 2000 loss

5. 3334 loss

Explanation:

A dealer sold one car at a profit of 25% of the dealer’s purchase price for that car,

let’s say C1, for $20,000.

C1∗1.25 = 20,000.

C1=16,000.

Profit = Selling price−purchase price

=20,000−16,000=4,000.

A dealer sold another car at a loss of 20% of the dealer’s purchase price for that car,

let’s say C2, again for $20,000.

C2∗0.8=20,000

C2=25,000

Loss = purchase price−selling price

=25,000−20,000=5,000;

Overall loss is 5,000-4,000=1,000,

Hence, the answer is C.

III) Rates:

Rates will be among those topics where you can expect hard questions.

Rates questions test your skills at:

• Time and Work

• Time, Speed and Distance

• Average Speed

• Group and Independent work.

• Relative Speed (speed of one object when compared to that of another object).

Rates questions, can also be solved using Back Solving (using the answer choices) and by Plugging in the values.

Sample questions:

Ben and Sam set out together on bicycles traveling at 15 and 12 miles per hour, respectively. After 40 minutes, Ben stops to fix a flat tire. If it takes Ben one hour to fix the flat tire and Sam continues to ride during this time, how many hours will it take Ben to catch up with Sam, assuming he resumes his ride at 15 miles per hour?

1. 3

2. 3.33

3. 3.5

4. 4

5. 4.5

Explanation:

The best way to solve a Time, Speed and Distance question is by picturing it.

The distance between Ben and Sam after 40 minutes (2/3 hours) is (distance) = (time)(speed) = 2/3*(15-12) = 2 miles (Ben will be 2 miles ahead of Sam).

In one hour, Sam covers 12 miles, so after an hour, Sam will be 12 – 2 = 10 miles ahead of Ben.

To catch up, Ben will need (time) = (Relative distance)/(Relative speed) = (Distance between them)/( Difference of their speeds-same direction)= 10/(15-12) = 10/3 hours.

So, the answer is B.

October, 19th, 2017
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- GRE Quant
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The Ultimate Guide to GRE Inequalities

Reading Time: 13 minutes

Are 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.

1. What are inequalities?

2. Basic properties

3. Advanced Properties

4. Quantitative Comparisons on the GRE

5. Points to remember

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

B) Squaring Inequalities

C) Square Root Inequalities

D) Reciprocal of Inequalities

E) Like Inequalities

F) Max Min Concept of Inequalities

G) Quadratic 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!

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May, 23rd, 2016
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- GRE Quant
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How to Manage Stress on GRE Quant

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Usually, 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!

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February, 4th, 2014
Posted in
- GRE Quant
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How to Reduce Silly Errors on GRE Quant

Reading Time: 3 minutes

Whenever 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.