## 8 Solid Ways To Get A 50-51 Raw Score In GMAT Quant

Reading Time: 12 minutesAt the outset, be warned that this article on improving GMAT Quant score may make sense only if you are already scoring at the 45 raw score level in GMAT Quant.

Many of the top MBA programs are unapologetically quant-driven. Whether you are pursuing a career in consulting, marketing, or finance, what you’re walking into is a world of big data and statistics.

A world like that requires solid quant skills, no matter what role you end up taking.

The good news is that if you are willing to properly prepare for the quant section of the GMAT, using the best materials and following a strategic plan of study, you can earn an impressive GMAT quant score.

In this article, we will show you how to improve your GMAT Quant score to a 50-51 using these methods:

- Get Your Basics Sorted
- Learn to Use Numbers Effectively
- Do Not Jump to Conclusions
- Identify Simpler Solutions
- Use Logic Over Math (Especially on Hard Questions)
- Do Not Depend Only on the OG
- Never be Rigid with Techniques
- Master Data Sufficiency

If you have not yet taken a full-length test, it might be a good idea to take the official GMAT prep test now.

Let us first try and understand how the scores on the quant section have changed over time. These GMAT score charts should give you an idea about the correlation between raw scores and percentiles on the GMAT quant section for July 2016 and December 2018. (Source: mba.com)

Sample Size: 794601

Standard Deviation: 11.00

Date Period: July 2016

Notice that a scaled score of 49 now corresponds to a 74th percentile as compared to a 79th percentile in 2016.

So, what exactly does this tell us?

The simple answer is this:

More people are getting perfect or near perfect scores.

In fact, the way that the GMAT quant percentiles are now, there is no 99th percentile. You can only get to the 96th percentile, at the most. This is because a full three percent of GMAT test takers are ringing up a perfect quant score.

**Why are people doing better?**

It has to do with the shifting demographics of GMAT test takers. People from countries that tend to provide a better math education than the US have been taking the GMAT in much greater numbers, making the GMAT percentiles (on the Quant side) much more competitive.

Typically, Indians score anywhere between 45 and 51 in Quant. This is partly because most Indians who take the GMAT have an undergraduate engineering background, and partly because the Indian education system does require better-than-global-average skills in mathematics.

**However, this scale of 45 – 51 has a huge deviation in percentiles.**

A 45 yields a measly 55 percentile while just 6 more raw points ahead, a 51 sits at a comfortable 96%ile (the maximum you can score on the quant section). Although the difference in raw scores is small, the real difference in terms of percentiles is huge.

So, if you are at a 46/47, don’t assume that scaling the 50 – 51 mountain is easy. If you are an Indian IT Engineer, there is a possibility that Quant could end up being a bigger problem for you than Verbal.

In any case, here are eight specific suggestions that you can start using to improve GMAT Quant scores.

### 1. Get Your Basics Sorted

The GMAT tests you on high-school math.

Yeah, you read that right.

The GMAT assumes that you understand certain concepts and use them as the basis of reasoning to solve the Quant questions. Since this is predominantly a reasoning-based test, a lot of emphasis needs to be laid on getting your basic understanding of concepts in place.

Way too many people fuss about learning advanced concepts without investing sufficient time into getting the basic concepts right.

Mastering math, especially the math tested on GMAT quant, requires that you take a linear, systematic approach towards developing your knowledge and skills. If you skip to the hard stuff, it will be challenging for you to develop a strong command of the material.

**Try this GMAT sample question:**

A set of 15 different integers has a median of 25 and a range of 25. What is the greatest possible integer that could be in this set?

- 32
- 37
- 40
- 43
- 50

If you caught yourself saying “Uh-oh, I forgot to brush up on my basic statistics concepts”, you are in trouble on the GMAT!

Ensure that you know the basics like Pythagorean triplets ((3,4,5); (5,12,13) ;(7,24,25)), Percentage to Proportion (1/8 = 12.5%), basics of numbers… you get the hint.

By the way, if you’re wondering what the answer is – it is 43.

### 2. Learn to Use Numbers Effectively

One thing you have to realize is this:

In GMAT Quant, the difference between the guy scoring 45 and the one scoring 51 is NOT that the latter knows more formulae.

It’s just that the 51 scorer is better at “hacking” his way through the test.

Remember, the GMAT does not worry about ‘how’ you get to the answer, but ‘if’ you get to it.

**Try this GMAT sample question:**

The number 75 can be written as the sum of the squares of 3 different positive integers. What is the sum of these 3 integers?

- 17
- 16
- 15
- 14
- 13

Is there really any formula you can apply here?

It’s about how your brain is going to pick the right values and “hack” its way to the answer. The better you can prepare your brain to do that, the better your GMAT Quant scores will be!

So here’s how you hack your way through this one:

Begin by plugging in values on the number line (especially for inequalities).

The standard values are a large negative number, a large positive number, -1, +1, a negative fraction between 0 and -1, a positive fraction between 0 and +1, and the number 0 itself.

The question here is a classic case – the square of 9 is 81, so you know the numbers should be a number between 1 and 8.

Next, quickly write down the squares 1, 4, 9, 16, 25, 36, 49, and 64.

Now, start playing with the numbers. You know it has to be one large plus one small number, so keep adding any two values (1 and 64, 4 and 49, etc.) and subtract it from 75 to see if it fits any remaining value.

Soon enough, you’ll realize that the numbers are 1, 49, and 25, i.e. 1, 7, and 5 = 13.

While practicing, try to get the answer within two minutes. Then try it without time-limits.

If you arrive at an answer, check the explanation to see if you are right. If you are wrong, then without looking at the solution, take a stab at solving the same question again – repeat the above loop.

Exercising your brain this way can be a lot of fun! No, seriously, just try it!

### 3. Do Not Jump to Conclusions

“How many questions can I afford to answer incorrectly if I am targeting a score of 50/51 in Quant?”

This has always been, and will always be, a question that students toy with at some stage of their quant prep.

Here’s why that is a bad idea altogether:

The quant section has a total of 31 questions, 3 of which are experimental and do not contribute to your overall quant score. You will not know which ones the experimental questions are. Your score depends on getting the remaining 28 right.

Check out this blog on the GMAT Enhanced Score Report to know how this scoring thing works.

If you go by the logic of allowing yourself to make mistakes, you will never know which side of the fence you’ll land up on.

If you get 3 questions wrong and none of them are experimental, you can kiss your perfect quant score goodbye.

To get a 51, a test taker can only afford a maximum of 1-2 errors and to get a 50, a maximum of 3-4 errors. This means that careless errors are going to adversely affect your score.

Remember, the test maker is always trying to get one up on you. Questions (and answer choices) are designed specifically to trip you up.

**Here is an example of a relatively simple GMAT question. Try solving it.**

The sales tax on an item is 8%. If sales tax had been only 5% Ben would have paid $12 less sales tax. What was the total amount that Ben paid for the purchase

- 368
- 380
- 400
- 420
- 432

Now if on solving you got an answer of 400 and picked C, then you fell into a very well-laid trap.

Notice that the question does not ask us for the price of the item (which is 400). It asks us for the TOTAL AMOUNT that Ben paid for the purchase. The total amount will be the price of the item plus the sales tax (8% of 400 = 32).

The correct answer here is E.

On an adaptive test like the GMAT, making silly mistakes on problems that you should get right can have a devastating effect on your score. Not only do you get that question wrong, but now you’re being served easier questions subsequent to that, with an even more heightened necessity of avoiding silly mistakes there.

So, make it a point to notice the mistakes you make on practice tests so that you’re careful not to make them again. Particularly under time pressure in a high-stress environment, we’re all susceptible to making mistakes.

### 4. Identify Simpler Solutions

Every GMAT quant question has a certain finesse to it, a finesse that is engineered by the test-maker.

So, when preparing for the GMAT quant section, you must learn to think like the test-maker. Even when you are correct, spend time trying to understand if you could have done a problem faster.

Let us start explaining this concept with a GMATPrep question.

**Take around two minutes to solve this one. **

According to the directions on a can of frozen orange juice concentrate, 1 can of concentrate is to be mixed with 3 cans of water to make an orange juice. How many 12-ounce cans of concentrate are required to prepare 200 6-ounce servings of orange juice?

- 25
- 34
- 50
- 67
- 100

Take the ratio as 1:4 (concentrate: juice) and ask yourself how many 12-ounce cans of concentrate you would need to make 100 12-ounce servings of the juice. The answer is 25!

However, the same question can be convoluted if you take a ratio of concentrate to water instead of concentrate to juice or start converting everything into a single unit (ounce). Be careful of overcomplicating solutions.

### 5. Use Logic Over Math (Especially on Hard Questions)

One of the themes we always stress is that the GMAT Quant section is not, primarily, a math test.

Though math is certainly involved – How could it not be? – logic and reasoning are far more important factors than the conventional mathematical facility.

This is particularly true on harder questions on the test. Learn to hone your reasoning skills and always be on the lookout of critical pieces of information that you can leverage to make your working simpler.

**Try this hard GMAT sample question:**

In a certain class, 1/5th of the boys are shorter than the shortest girl in the class, and 1/3rd of the girls are taller than the tallest boy in the class. If there are 16 students in the class and no two people have the same height, what percent of the students are taller than the shortest girl and shorter than the tallest boy?

- 25%
- 50%
- 62.5%
- 67.5%
- 75%

A large percentage of test-takers see this question, rub their hands together, and dive into the algebra.

But here’s the thing:

Even if you were fortunate enough to possess the algebraic virtuosity to solve this question using algebra, you’d likely chew up 3 or 4 minutes.

The best way to solve this question, instead, is to leverage the information the question stem provides.

1/5th of the boys are shorter than the shortest girl. Now, common sense tells you that the number of boys must be a multiple of 5. So, the number of boys can be 5, 10 or 15.

1/3rd of the girls are taller than the tallest boy in the class, so the number of girls needs to be a multiple of 3. By that logic, the number of girls can be 3, 6, 9, 12 and 15.

Since there are 16 students in total, we can easily conclude that there must be 10 boys and 6 girls. Beyond this, the question just deals with a basic percentage concept.

If you are wondering about the answer to the question – it is 62.5%.

**Important Note:** For those who answered 75%, you fell for the trap of not understanding what the question asks you. The question here reads, how many students are BETWEEN the shortest girl and the tallest boy. So, the number of students in between will be 10 and not 12.

### 6. Do Not Depend Only on the OG

Don’t rely only on the Official Guide.

As awesome a source as the OG is, it still caters to people in the middle of the bell curve, i.e., those who get around 40-44 as a GMAT quant raw score.

If you are gunning for a raw score of 51, you should be looking at GMATPrep questions available freely online from various sources. Try your hand at these threads on GMATClub for both PS and DS problems from the GMATPrep tests.

CrackVerbal students get a curated compilation of GMATPrep questions put together by our team (of course!)

Ensure that you take the GMATPrep tests a minimum of three to four times before you start practicing these questions; otherwise, you will get inflated and unreliable scores.

Another downside to relying on the official guide is this:

The explanations that accompany the questions tend to be biased towards algebraic solutions.

Before you begin to wonder what’s wrong with using algebra to solve an algebraic question, give us a moment to explain.

Quite often on the GMAT, using pure algebra takes longer than you have to solve a given question. In general, the GMAT is a test of your intelligence, not your knowledge. This means that the test is trying to evaluate how smart you are, not how much you know.

Thanks to this, although the given algebraic solutions are always technically correct, they present suboptimal ways to solve the question.

That’s why we offer this important piece of advice:

Never take a formal solution to a problem at face value. All you’re seeing is one way to solve a given question, but that doesn’t mean it’s the only (or the best) way there is.

The best workaround in this situation is to review the questions on GMATClub or on the CrackVerbal forum and take notes about alternate approaches used to solve the question.

### 7. Never be Rigid with Techniques

The whole point of reviewing questions using GMATClub or the CrackVerbal forum is to bring to light new techniques that can help you solve questions faster.

**Let’s take a sample question and review the various methods we can use to solve it: **

If |4x − 4| = |2x + 30|, which of the following could be a value of x?

- -35/3
- -21/2
- -13/3
- 11/5
- 47/5

As far as we can see, this question can be solved using 3 methods.

The first method is to use pure algebra: create 4 different equations and solve every one of them to get the solution.

Clearly, this method is tedious.

So we look for an easier way, which is to use the “plug-in” method. Here, you have the absolute value equation with the potential values of x given in the answer choices, but the values of x that are given are all fractional, making this method tedious as well.

A third, faster method is to be aware that |x| = √x^2.

The above equation can be written as √(4x – 4)^2 = √(2x + 30)^2. Squaring both sides gives us (4x – 4)^2 = (2x + 30)^2. If we take the term (2x + 30)^2 to the left-hand side, we have an equation of the form a^2 – b^2 which can easily be broken down to (a + b) (a – b). If we solve this, we get x = -13/3 and x = 17.

So, the answer to the question is option C, x = -13/3.

Make it a point to take notes of these techniques and try putting them to practice on different questions.

Here’s a great resource to similar techniques that can be used on Inequalities and Absolute Values.

### 8. Master Data Sufficiency

Data Sufficiency questions generally make up just under half of the questions in GMAT quant. Since there are 31 questions in the Quant section, **about 14/15 of them will be data sufficiency questions**.

Simply put, the DS section of the GMAT is different.

It’s different because you aren’t trying to solve for one answer. Instead, your goal is to test for sufficiency. That means you’re trying to determine whether you’ve been provided enough information to definitively answer the question stem.

Since you are not required to get down and dirty with the math, these questions are time savers.

But, on the flip side, since this question type relies heavily on reasoning, the chances of getting the logic wrong and falling for well-laid traps is high. It is important that you master this question type if you want to improve your GMAT quant score.

Here are a few points that will help you approach DS questions better:

- Write out all the important information in the question stem, whether it be certain constraints (integer, pos/neg etc.), equations, or the actual question. This helps avoid silly mistakes.
- Always break down and rephrase any equations, inequalities, fractions etc. For example, try to write expressions with exponents in such a way that all the bases are the same, or whenever you see a sum with a root contained in it, try to multiply by its conjugate.
- Always translate word problems on DS into an algebraic equation. There will most likely be an opportunity to break this equation down further and rephrase the question. This skill is key as it saves time in computing the two statements.
- Never try to prove a statement to be sufficient. Always try to prove insufficiency. Being a contrarian on DS questions will help you avoid well-laid traps.
- Never make assumptions. On most questions where there aren’t any constraints given, people sometimes still assume that there is some constraint and thus only test integer cases. By really paying attention to what is given and what is NOT given, you can find the correct answer on a lot of questions.
- Never eliminate a statement just because you ‘feel’ the statement is insufficient. This is a very common DS trap. Especially with quadratic expressions, people assume the data is insufficient since they can’t find a single solution. This is a big mistake. Always finish your calculation, because you don’t want to spend a lot of time on a question and end up being wrong just because you skipped the last step(s) of the calculation.
- In questions where you are forced to work with numbers, always test numbers systematically. Test for 0, positive and negative integers and positive and negative fractions. On some occasions, it’s also crucial to pick boundary values, for e.g. if a question asks whether x is less than or equal to 1 and a statement gives you x² less than or equal to 1.3, you can pick x=1.1 to prove insufficiency.

We hope you have found this blog useful to improve your GMAT quant score. Do let us know what you think in the comments section below!

## A Guide to GMAT Geometry

Reading Time: 4 minutes

*Most of the students when they hear the word “Geometry”, they think of trigonometry, they think about theorems etc. But in GMAT it’s quite different.*

**In this article, let’s discuss about what is tested in GMAT geometry and what could be the best preparation strategy for GMAT Geometry?**

Even though it carries only approximately 3 to 6 questions in GMAT, it has its own importance in GMAT.For these 3- 6 questions, you are expected to know the basic formulae and shapes. GMAT also tests your visualizing skills.

*Below mentioned three things are must for a GMAT student who wants to fair really well in Geometry:*

1. Draw for the questions even if the figures are not provided. If the figures are provided then re-draw the figure.

2. Know the basic shapes like triangle, quadrilaterals, circles, rectangular solids and basic properties of straight lines in co-ordinate geometry really well.

3. Do not make any assumption unless or otherwise it is specified in the question.

Let’s understand what GMAT has to say about the figures in Geometry.

**GMAT Directions for Geometry (GMAC™)**

Figures:

1. For problem solving questions, figures are drawn as accurately as possible. Exceptions will be clearly noted.

2. For data sufficiency questions, figures conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2).

3. Lines shown as straight are straight, and lines that appear jagged are also straight.

4. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero.

5. All figures lie in a plane unless otherwise indicated.

**So, how should a student approach a question on Geometry?**

In GMAT particularly for a DS questions, one should not solve a question based on the diagram or figure given in the question. Try to draw your own figure based out of the information provided in the question and the statements.

For a PS question, mostly the diagram will map to the question provided unless stated otherwise.

That is, they will give a note below a figure stating “Figure is not drawn to scale”, In that case one should be really careful about the diagram and try to redraw according to the information given to the question.

To just to give an example of this, take a look at the below question,

*In the figure given, if PR is a line segment, what is the sum of the lengths of the curved paths from P to Q and from Q to R?*

XQ = QY = 5 centimeters.

Every point on arc PQ is 5 centimeters from point X, and every point on arc QR is 5 centimeters from point Y.

**Explanation:**

Given is a figure that looks like two semi circles of equal areas with center X and center Y respectively. However we cannot assume any of this as we are not told this in the question.

Here anybody could easily fall for the diagram, and say X and Y are centers, which is what we have to be very careful and look at the question and the statements, and search for anything with suggests that X and Y are centers. Here statement II only suggest that X and Y are centers.

Statement I is insufficient: XQ=QY=5.

This would have been sufficient if we were given X and Y are the center of the circles. If they were centers then XQ and QY were radius of the circle and we could have calculated the length of the arc. However as we do not know if they are centers they could be any arbitrary points. Hence not sufficient.

**Statement II is sufficient: **

The meaning of the statement is that X and Y are the centers of the 2 semi circles each with radius of 5. We can calculate length of arc by 2* Pie* Radius.

Hence it is sufficient.

So the Answer is B.

Let’s look at another figure:

Here let’s say the question is ABCD is a quadrilateral and all sides are equal.

Here basic assumption students may end up doing is considering this is a square. Because all sides are equal and each angle looks like it is 90 degrees. But here one should be really careful about this.

It looks like each vertex angle is 90 degrees, does not mean that it is a square. If somewhere in the question if they have specified each vertex angle is 90 degrees then you can take the figure as a square otherwise not.

You have to seriously look for other information in the question. Because you cannot assume anything like angles and lengths based out of a diagram.

**Tips for Co-ordinate Geometry:**

Best part about GMAT co-ordinate geometry question is they test your visualizing skills in straight lines than the formulae. Very rarely they test curve like parabola. Even if they test curves you can still solve it by plugging in and drawing it out.

If you look at it, lines alone could easily have 20 odd formulae; one cannot be mugging up all these to do well in GMAT.

**Just keep the following tips in mind while solving a co-ordinate geometry questions:**

1. A student will be provided with a scratch pad during the exam which has grid lines. So make good use of this and draw x-y plane in the scratch pad and consider each small square as x and y units.

2. When finding the intercepts, slopes and distance try to use the grid lines to solve rather than formulae. Formulae would be handy only for easy questions but not for hard questions. You need to develop the skills of finding the distance, slope and intercepts using the x-y plane.

3. Use the answer choices, so that you can do some POE (Process of Elimination) when it comes to a hard question. That would save lot of time.

**If you follow the above points while practicing the questions it would definitely help you in the actual GMAT exam.**

To sum it up, GMAT geometry is the easiest topic one could expect. So don’t go about learning all the unnecessary formulae. That is definitely not going to help. Try to keep it simple. Practice as many questions keeping in mind the above points.

I hope this article helped you in understanding – *how to take on GMAT geometry and shine through.*

If you loved the blog, please let us know in the comments!

** Pro Tip:** Curious about how to kick off your mission to your dream business school? Download this free e-book – A guide to GMAT to get started.

## The Ultimate Guide to GMAT 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 GMAT, 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?**

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)

Muhhhltiplying 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) Square Root Inequalities**

**If x^2 < a ^2, then x > -a and x < a, the range of x will be – a < x < a **

For e.g. if x^2 < 100 then the values of x that are going to satisfy the inequality are values of x < 10 and values of x > -10.

**If x^2 > a^2, then x > a and x < -a, the range of x will be from (-∞, -a) and (a, ∞)**

For e.g. if x^2 > 100 then the values of x that are going to satisfy the inequality are values of x > 10 and values of x < -10.

**Example:**

**If (y – 5)^2 < 36, find the range of y**

If x^2 < a^2 then the range of x is -a < x < a.

Now x here is replaced by y – 5 and a is replaced by 6 (since 6^2 = 36).

(y – 5)^2 < 36 —-> -6 < y – 5 < 6.

Now adding 5 throughout and isolating the variable y we get,

(y – 5)^2 < 36 —-> -6 + 5 < y – 5 + 5 < 6+ 5

(y – 5) ^2 < 36 —-> -1 < y < 11

**D) 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:

**If (1/a ) > (1/b) when a and b are positive **. That is, flip the inequality.

If 2 < 3, then ½ > 1/3

**If (1/a) > (1/b) when a and b are negative **. That is, flip the inequality.

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

**If (1/a) < (1/b) when a is negative and b is positive **. That is, do not flip the inequality.

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

**E) 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**

**F) Min and Max Inequalities**

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

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.

**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

**Is xy < 6 ?**

**I. x < 3 and -y > -2**

**II. y^2 < 100 , 1/2 < x < 2/3**

Evaluating Statement 1 individually

x < 3 and y < 2 (Multiplying both sides by -1 and flipping the inequality sign)

Now If x = 2 and y = 1 then xy = 2 which gives a YES

If x = -3 and y = -2 then xy = 6 which gives a NO

**Statement 1 is insufficient**

Evaluating Statement 2 individually

y^2 < 100 —> -10 < y < 10 ;

1/2 < x < 2/3.

Placing the ranges of x and y one below the other and using the max min concept we have the maximum value of xy to be 20/3 which gives a YES and the minimum value of xy to be -20/3 which gives a NO.

**Statement 2 is Insufficient**

Combining Statements 1 and 2,

Now the range of x is still going to be 1/2 < x < 2/3 but the range of y is going to be – 10 < y < 2 (Since y < 2 from statement 1)

Now using the max min concept for the above ranges of x and y, we have the maximum value of xy to be 4/3 and the minimum value of xy to be -20/3. All possible values of xy here are less than 6 which gives a definite YES.

**Answer : C**

**G) 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.

**Example:**

How many of the integers that satisfy the inequality (x + 2) (x + 3) (x – 2) >=0

are less than 5?

A. 1

B. 2

C. 3

D. 4

E. 5

The 3 critical points here are at -2, -3 and 2. Now using the concept of quadratic inequalities and plotting the critical points on the number line we get

Now since the right hand side is >= 0 we need to consider the positive regions of the number line. The range of x where the given inequality expression is positive is x >= 2 and -3<= x < = -2 . From the range of x the integer values less than 5 are 2, 3, 4, -3 and -2

**Answer : E**

**Points to Remember**

Here are a few things you need to remember when you are using the properties of inequalities to simplify complex PS and DS inequality problems.

1.Add or subtract any quantity on both sides of the inequality without changing the inequality sign.

2.Multiply or divide by a positive value without changing the inequality sign.

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

4.When multiplying and dividing by a negative number always flip the inequality sign.

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).

7.The only mathematical operation that you can perform between two sets of inequalities is addition. Never subtract, multiply or divide.

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

We hope this guide helps you along the way to a 51 on GMAT Quant!

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

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