The post What Kind of Math is on the GRE? Breakdown of Quant Concepts by Frequency appeared first on Magoosh GRE Blog.

]]>What GRE math question types should you prepare for? And which are truly the most important GRE math concepts? In this post, we’ll take an in-depth look at GRE Quant concepts: how they are categorized and how frequently each major category appears on the test. Get ready to read about each type of math on GRE that you should study.

The categorization and frequencies of the most important GRE math concepts in this article are based on the 180 recently-released GRE Quant practice questions from ETS: the 80 Quant questions from the two PowerPrep Plus practice tests, and 100 Quant practice questions from the practice tests in the 3rd edition of the Official Guide to the GRE General Test.

These are among the newest and freshest GRE materials, so they really do give the most up-to-date picture of Quant GRE concepts. By the time you’re done reading about this range of math problems, you’ll truly understand what the word “quant” means when it comes to GRE prep.

And for a look at the “big picture” of how these math topics fit into GRE concepts and GRE pre as a whole, see Magoosh’s free guide to the GRE test.)

Here, in a nutshell, are the 14 GRE math question types:

**Word problems:**interpreting the math in stories and descriptions**Data interpretation:**interpreting the math in charts and tables**Algebra:**includes both “pure algebra,” and algebra as applied to other GRE Quant concepts**Percents/ratios/fractions****Coordinate geometry:**shapes, lines, and angles on the coordinate plane**Two-dimensional geometry:**shapes, lines, and angles*not*on the coordinate plane**Three-dimensional geometry: volume, surface area, etc…****Statistics:**mean, median, standard deviation, etc…**Powers and roots****Probability/combinatronics****Integer properties and aritmetic****Inequalities****Functions****Sequences**

The GRE often tests multiple Quant concepts in the same problem. While there are certainly many problems that focus on just one concept, it’s not uncommon to see a problem involving two or even three quant concepts. You’re unlikely to have to deal with more than three major concepts in a given GRE Quant question, although this may happen on exceptionally advanced math prompts.

Still, certain concepts can overlap in surprising ways. Data interpretation problems, for instance, can double as geometry *and* statistics problems, when you are asked about the appropriate angles to put on statistical pie charts. Expect other interesting-but-challenging math concept mashups on the GRE as well.

Algebra in particular is one concept that overlaps *massively* with the other concepts listed in this article. You may encounter an algebra component in problems that deal with any other concept. This is why I’ve noted that the algebra category includes both problems that focus primarily on algebra and problems where algebra is a major secondary component. (In other words, there are regular algebra problems, but there are also algebra probability problems, algebra geometry problems, etc….)

Word problems of course also overlap with many concepts. Conceivably, any GRE Quant concept can be tested in a word problem. The word problem category also has an almost 100% overlap with data interpretation. Virtually any GRE Quant section question that involves chart or table reading also involves interpreting short associated word problems.

Common word problems involve mixtures, work rates, interest, and a number of other “staple” topics you’ve probably seen in other coursework and exams. But again, word problems really are pervasive on the test, and deal with math from many different angles; there are statistics word problems, arithmetic word problems, geometry word problems, and so much more. Arguably, word problems are in fact the most common GRE math problems.

In short, be especially prepared to use algebra or interpret word problems throughout the Quant section on test day! **Word problems and algebra problems should be considered the most common GRE math question types. **

With all that in mind, let’s now look at the whole table for GRE concept distribution: math problems!

In the table below, the different types of math problems are listed in order of most frequent to least frequent. And each concept is hyperlinked to an article that discusses the content in greater detail. Click the algebra link for an algebra breakdown, the statistics link for a stats breakdown, the geometry links for Magoosh geometry breakdowns, and so on. The link for Functions actually leads to a Magoosh GMAT Blog post, but that post is also relevant to the GRE.

GRE Quant Concept | Percentage frequency on the test |
---|---|

Word problems | 35.5% |

Ratios, percents, and fractions | 23.9% |

Algebra | 22.8% |

Data interpretation | 20% |

Integer properties and arithmetic | 16.7% |

Two-dimensional geometry | 15% |

Statistics | 14.4% |

Powers and roots | 7.8% |

Inequalities | 6.1% |

Probability and combinatronics | 5.6% |

Coordinate geometry | 4.4% |

Three dimensional geometry | 2.2% |

Sequences | 1.7% |

Functions | 1.1% |

**Note:** Some questions tested multiple concepts and were thus counted more than one time in more than one category. As a result, the percentages in the chart above add up to more than 100%.

The table above, grouping math exam questions by topic, provides a good road map for your GRE Quant studies. It’s worth noting, for instance, that **knowledge of percents, ratios and fractions is crucial**; these are by far the most important GRE math concepts. In fact, percents/ratios/fractions are about as important as the ever-present algebra concepts that appear in so many questions.

**Reading comprehension** is obviously crucial as well: the ability to properly interpret story problems, charts and tables has clear importance.

**Concepts related to geometry, arithmetic, and the properties of numbers can be seen as “runners up” in importance**. These GRE math question types are of intermediate frequency. As such, these concepts aren’t quite as vital to your success compared to algebra, percents/ratios/fractions, and reading comprehension. But these types of math still show up often on the test.

On the other end of the table, **sequences and functions should take the lowest priority in your studies**. If sequence or function problems appear on your exam at all, you’ll likely have one or two questions at most in these two areas. The really are not the most important GRE math concepts.

And again, remember that these concepts overlap in many typical GRE math questions. **Be prepared to deal with at least two of the concepts at the same time in the same problem**, and seek out practice questions that allow you to multitask.

Last but certainly not least, make a good study plan to ensure that you’ll cover all the important concepts during your GRE prep. For example study plans, see Magoosh’s GRE study guides and plans.

The post What Kind of Math is on the GRE? Breakdown of Quant Concepts by Frequency appeared first on Magoosh GRE Blog.

]]>The post Positive and Negative Square Roots on the GRE appeared first on Magoosh GRE Blog.

]]>Some questions to start: *Can a square root be negative?* Yes, there is such a thing as a negative square root. *How would you deal with negative roots on the GRE?* This question is actually fairly complicated, because it depends on how the GRE words the question. Let’s start with some practice problems before I explain.

First of all, consider these two similar, but not identical, Quantitative Comparison questions. These are a bit easier than you would see on the GRE, but they illustrate an important distinction about possible negative square roots.

1)

2)

All I will say right now is: despite apparent similarities, those two questions about whether roots are positive or negative, have ** two completely different answers**. The distinction between them is the subject of this article.

Explanations to these practice problems will appear at the end of this blog article. Jump ahead by clicking here.

Often, students are confused about this question. For example, in the questions above, we know +4 is a square root of 16, but can’t -4 be one as well? Or can it? Can a square root be negative? Do we include the negative square root as part of Column B or not? Does it matter how the question is framed? All of these questions about possible negative square roots are resolved by understanding the following two cases.

In case one, the test-maker, in writing the question, uses this symbol. This symbol appears printed on the page *in the question itself.*

What is this symbol? Well, the most folks call this simply a “square-root” symbol, but the proper name is the “principal square root” symbol. Here, “principal” (in the sense of “main” or “most important”) means: you take one and only one root, the most important, or principal, one — the positive root only. That is the deep meaning of this symbol.

Thus, in all cases in which this symbol appears as part of the question itself, you NEVER consider the negative square root, and ONLY take the positive square root.

In this case, that special symbol does **not** appear as *part* of the problem. What does appear is, for instance, a variable squared, or some other combination of algebra that leads to a variable squared, and you yourself, in your *process* of solving the problem, have to take the square root of something in order to solve it. The act of “square rooting” is not initiated by the test maker in the act of writing the question; rather, it is you who initiated the square-rooting.

In this case, 100% of the time, you ALWAYS have to consider **both** the positive and negative square roots.

Can a square root be negative? Well, the answer is: it depends on what was printed in the problem. The principal square root symbol never has a negative output, so if the test maker printed that symbol, it’s restrictions have to be respected: all square roots then are positive. On the other hand, if the problem contains a variable squared, or some other algebra that leads to a variable squared, and you yourself take a square root as part of the act of solving, then you always have to consider all possible solutions, both the positive square roots and the negative square roots.

If you master that distinction, you will always understand when to consider both positive and negative roots vs. when to consider only the positive root. You may want to go back to those two QCs at the beginning and think them through again before reading the solutions below.

If you’re having trouble with Quantitative Comparison questions on the GRE, I would recommend taking a look at some of these additional resources. They may help clear up some of the concepts that you’re struggling with, and can provide some extra practice.

- QC Tip #1: Dealing with Variables
- QC Tip #2: Striving for Equality
- QC Tip #3: Logic Over Algebra
- QC Tip #4: Comparing in Parts
- QC Tip #5: Estimation With a Twist

- QC: The Devil Is in the Details
- QC: “The relationship cannot be determined from the information given” Answer Choice
- QC and Manipulation
- GRE Math: Solving Quantitative Comparisons

1) Here, the principal square root symbol appears as part of the problem itself. We are in Case I. Of course, that symbol means: take the positive square root only. So Column B can only equal +4. Of course, that’s always bigger than 3. Answer = **(B)**.

2) Here, there’s no square root symbol printed as part of the problem itself. We are in Case II. For any square roots we take as part of our solution, we are liable to account for both the positive and negative roots.

Sure enough, the very first thing we encounter in the prompt is a variable squared, and when we solve for x, we have to account for both roots: x = ±4. The variable x could equal well have either one of those values.

Now, when we proceed to the QC, we see that the different values of x would give different answers. If x = +4, then column B is greater, but if x = -4, then column A is greater. Different values lead to different conclusions, and this situation means we don’t have enough information to establish a definitive relationship. Answer = **(D)**

To find out where roots sit in the “big picture” of GRE Quant, and what other Quant concepts you should study, check out our post entitled:

What Kind of Math is on the GRE? Breakdown of Quant Concepts by Frequency

*Editor’s Note: This post was originally published in September, 2012, and has been updated for freshness, accuracy, and comprehensiveness.*

The post Positive and Negative Square Roots on the GRE appeared first on Magoosh GRE Blog.

]]>The post Diagonals of a Polygon in GRE Geometry appeared first on Magoosh GRE Blog.

]]>1) Regular pentagon P has all five diagonals drawn. What is the angle between two of these diagonals where they meet at a vertex of the pentagon?

(A) 12°

(B) 36°

(C) 54°

(D) 60°

(E) 72°

2) How many diagonals does a regular 20-sided polygon have?

(A) 60

(B) 120

(C) 170

(D) 240

(E) 400

Explanations to these practice problems will appear at the end of this blog article. Jump ahead by clicking here.

First, some basic terminology to begin this discussion. A **polygon** is any geometric shape all of whose sides are straight line-segments. Any **triangle** is a polygon. Any **quadrilateral** (including trapezoids, parallelograms, rhombuses, rectangles and squares) are polygons. A **pentagon** is a 5-sided polygon. A **hexagon** is a 6-sided polygon. An **octagon** is an eight-sided polygon. A circle or parabola or anything with a curved side is __not__ a polygon.

A point where two of the sides of a polygon meet is called a **vertex**. The number of vertices a polygon has is always equal to the number of sides it has.

Another important polygon fact concerns the sum of angles. You may know that the sum of the three angles in any triangle is 180°. You may even know that the sum of the four angles in any quadrilateral is 360°. This pattern generalizes. The sum of all n angle in any n-sided polygon is:

**sum of angles = (n – 2)*180°**

Thus, any pentagon (n = 5) would have angles that add up to 3*180 = 540°. Any hexagon (n = 6) would have angles that add up to 4*180 = 720°. Any octagon (n = 8) would have angles that add up to 6*180 = 1080°. (See the blog on GRE Geometric Formulas)

Finally, there is this paradoxical word “**regular**.” In everyday language, “regular” means “ordinary, unexceptional, commonplace.” In geometry, it connotes the exact opposite! A shape is regular if and only if it is both equilateral and equiangular—that is, if and only if all the sides have the same length and all the angles are equal. The “regular” version of any polygon is the most elite, most symmetrical version possible of that polygon. The “regular triangle” would be what we known as an equilateral triangle. The “regular quadrilateral” is the square. For higher polygons, you are most likely to see the regular version on the GRE, because the test (like all mathematicians) loves symmetry.

Now, we can talk about diagonals. A **diagonal** is any line through the interior of a polygon that connects two non-adjacent vertices. What does this mean? Well, first of all, starting from any vertex, an adjacent vertex is either vertex connected to the starting vertex by one side of the polygon.

Consider an irregular quadrilateral:

Let’s start at vertex A. Starting from vertex A, we are connected by sides of this quadrilateral to both B and D; vertices B & D are the ones that are adjacent to vertex A. The only vertex not connected to A by a side of the quadrilateral is C. C is A’s only non-adjacent vertex, and A is C’s. Thus, one diagonal goes from A to C. It’s not hard to see that the other goes from B to D. Any quadrilateral has just two diagonals.

Notice that **triangles NEVER have diagonals**: if we start from any vertex in a triangle, the other two vertices are adjacent. There simply are no non-adjacent vertices in a triangle, so diagonals are not possible. Among quadrilaterals, there are special rules for the diagonals of a parallelogram and the categories within parallelograms:

The diagonals of a parallelogram bisect each other: that is to say, the intersection point of the two diagonals is the midpoint of each one.

A rhombus is a parallelogram with four equal sides. The diagonals of a rhombus bisect each other and are perpendicular.

A rectangle is a parallelogram with four 90° angles. The rectangle of a rhombus bisect each other and have equal length. This is related to an old trick among carpenters. When a carpenter cuts two pairs of equal lengths to make the sides of a door or window frame, he knows he has a parallelogram because of the equal lengths, but how does he know whether he has a rectangle? Without precise equipment, it’s very hard to measure the difference between, say, an 89° or 90° angle. Well, all the carpenter has to do is measure the two diagonals: if these two easy-to-measure lengths are equal, then it’s guaranteed that he has four right angles!

A square is a parallelogram, a rectangle, and a rhombus. It is a regular quadrilateral with four equal sides and four 90° angles. The diagonals of a square bisect each other, have equal length, and are perpendicular.

A **pentagon** is any five-sided polygon, and the sum of its angles is 540°, as we saw above. The only pentagon you are likely to meet on the GRE is the most symmetrical, the regular pentagon. Since the angles are equal, we can divide the sum of the angles by five.

540°/5 = 108°

That’s the angle of each of the five angles in the pentagon.

Here’s a regular pentagon with the five diagonals drawn.

How many diagonals does a pentagon have? Any pentagon has exactly five diagonals. These diagonals trace out the shape of a classic five-pointed star, such as that those on the Flag of the United States of America. The lengths and divisions of this star are intimately related to that magical and mystical number, the Golden Ratio; Sacred Geometry is purported to give insight into the meaning of life, but you don’t need to know any of that for the GRE!

How would we find any angles in this shape? Well, we know each big angle of the pentagon is 108°. Look, for example, triangle ABC. This triangle is an isosceles triangle, because AB = BC, and we know that angle ABC = 108°. The other two angles must be equal: call them x.

108° + x + x = 180*

2x = 180° – 108° = 72°

x = 36°

This means that angle BAC = angle BCA = 36°, and so do many other symmetrically related angles around the shape. We could subtract (angle BAC) from (angle BAE) to get (angle CAE)

angle CAE = (angle BAE) – (angle BAC) = 108° – 36° = 72°

From that, we could find many other angles inside the shape. We could use analogous means to find angles involving the diagonals of any higher polygon.

A **hexagon** is any six-sided polygon, and the sum of its angles is 720°, as we saw above. In a regular hexagon,

each angle = 720°/6 = 120°

How many diagonals does a hexagon have? Starting from one vertex, two other vertices are adjacent, so 3 vertices are non-adjacent, making possible three diagonals from one vertex. From A, we can draw diagonals to C, D, and E.

From each vertex, there are three diagonals. Since there are six vertices, you might think there would be a total of 3*6 = 18 diagonals, but that counting method double-counts everything. You see, the diagonal from A to C would get counted once as a diagonal from A and again as a diagonal from C to A. Thus, the number of diagonals in a hexagon is 18/2 = 9. These can be groups into two kinds. The six shorter diagonals together make a six-sided star, the Magen David. The three longer diagonals form just three symmetrically criss-crossing segments, what is called in mathematics a “degenerate six-pointed star.”

These two diagrams show the nine diagonals of a regular hexagon. Of course, the six-pointed star simply consists of two overlapping equilateral triangles, pointing in opposite directions. (That geometric fact led to extensive mystical speculation in about the Star of David in the Kabbalah, but again, you don’t need to understand any mysticism for the GRE!)

A **heptagon** is any seven-sided polygon (n = 7). Sometimes it is called a “septagon,” but “heptagon” is the preferred mathematical name. The sum of its angles would be

(n – 2)*180° = 5*180° = 900°

This means that each of the seven angles in a regular heptagon would have a measure of

each angle = 900°/7 = 128.5714286…°

The angle measures are not integers! This is why the GRE is exceptionally unlikely to ask you anything about the regular heptagon, and it’s also why you probably never talked much about regular heptagons in high school geometry. It’s why most people aren’t even clear on the proper name for this beast! Their non-integer angle measure makes them the first “black sheep” in the regular polygon family! I won’t say anything else about them, because they almost never make an appearance on the GRE, but I will show you the two possible seven pointed stars from their diagonals: these stars are hauntingly beautiful, because of their idiosyncratic symmetry.

An **octagon** is any eight-sided polygon, and the sum of its angles is 1080°, as we saw above. In a regular octagon,

each angle = 1080°/8 = 135°

That angle is the supplement of a 45° angle. The regular octagon is the typical stop sign shape in many parts of the world.

Starting from one vertex, two other vertices are adjacent, so five vertices are non-adjacent, making possible five diagonals from one vertex. From A, B & H are the symmetrical vertices, so we can draw diagonals to C, D, E, F, and G.

Similar logic as with the hexagon: five at each vertex, eight vertices, but that counts each diagonal twice, so the total number is 5*8/2 = 20. AC and AG are what we might call “3 vertex diagonals”: there are eight of these, which form a star. AF and AD, each parallel to two sides, are what we might call “4 vertex diagonals”: eight of these form another star. Finally, AE is like a diameter of the whole octagon, cutting across its center: there are four such lines, and they form a degenerate eight-pointed star.

Just as the six-pointed star consisted of two overlapping equilateral triangles, the first eight-pointed star on the left consists of two separate overlapping squares: square ACEG and square BDFG. (The names of these squares are reminiscent of the lines at W. 4th Street!) Between these three stars (counting the degenerate thing on the right as a “star”) we have all 20 of the regular octagon’s diagonals.

Onward and upward! The methods discussed in this blog can be extended to apply to a nonagon (n = 9), a decagon (n = 10), or any higher polygon. Armed with this information, you should be able to answer anything the GRE asks you about a diagonal of a polygon! If you had any “aha” moments while reading this blog, you might want to give the practice problems on the top another peek before reading the explanations below.

1) Let’s take a look at the pentagon with its five diagonals.

An example of the angle between two diagonals at a vertex would be angle EBD, where diagonals BD and BE meet at vertex B.

We will follow the logic outlined above.

Triangle BCD is isosceles with BC = CD, and angle BCD = 108°. The other two angles are equal: call them each x.

108° + x + x = 180*

2x = 180° – 108° = 72°

x = 36°

So, angle CBD = 36°. Well, triangle ABE is in every way equal to triangle BCD, so angle ABE must also equal 36°. Thus, we can subtract from the big angle at vertex B.

(angle EBD) = (angle ABC) – (angle CBD) – (angle ABE)

(angle EBD) = 108° – 36° – 36° = 36°

Answer = **(B)**

2) If we start at one vertex of the 20-sided polygon, then there’s an adjacent vertex on each side. Not counting these three vertices, there would be 17 non-adjacent vertices, so 17 possible diagonals could be drawn from any vertex. Twenty vertices, 17 diagonals from each vertex, but this method double-counts the diagonals, as pointed out above.

# of diagonals = (17*20)/2 = 17*10 = 170

Answer = **(C)**

*Editor’s Note: This post was originally published in January, 2014, and has been updated for freshness, accuracy, and comprehensiveness.*

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]]>The post GRE Data Interpretation Practice Questions appeared first on Magoosh GRE Blog.

]]>On test day, when you sit down to take your GRE, you will have, among other things, two Quant sections. Each Quant section will have a Data Interpretation problem set towards the end of the section. This GRE Data Interpretation set will present data, information, in some graphical form, and it typically has three questions about the same data. So, you will see about 3 DI questions on each GRE Quant section, so about 6 DI questions on your test—or more if the experimental section is also a Quant section. As you can see, Data Interpretation practice is an important part of your GRE Quant prep.

Why is there Data Interpretation on the GRE? Why is this important? Well, do you know the old adage “*A picture is worth a thousand words*“? Well, a graph is worth even more. Geeky math & techie folks, such as I, absolutely love graphs and charts, because they present an efficient means to convey a truckload of information in way that is directly visually accessible. In our post-modern electronic world, the sheer amount of information available is simply mind-boggling. Graphs and charts are essential for keeping track of all this information.

Here’s one big suggestion for GRE Data Interpretation: if you are not a geeky math or techie person, start looking at graphs & charts. Look in the financial news, in scientific articles, and in international news in general. Most newspapers and news magazines are stuffed to the guppers with graphs & charts. Spend time studying them: each graph, each chart, has a “story” to tell. Spend enough time with each to understand its “story.”

Despite appearances, the GRE Data Interpretation questions are usually relatively easy compared to the rest of the questions on the Quant section. The whole point of graphs is to make information easy to see!

The GRE Data Interpretation will present information in any one of a number of visual formats. These include:

(a) pie charts

(b) bar charts

(c) line graphs

(d) scatterplots & best-fit lines

(e) boxplots

(f) histograms (*which are different from bar charts*!!)

(g) charts of numerical data

Of these, pie charts and line charts and charts of numerical data are probably self-explanatory, and there are examples below (pie chart & bar chart in Set #1, line chart in Set #2). Scatterplots and boxplots are somewhat less well-known, and each has its own blog giving more detail; there’s also an example of a scatterplot below, in Set #4.

I will say a few words here about bar charts vs. histograms. First of all, on a bar chart, there’s no particularly important difference whether the bars go horizontal or vertical (a.k.a. a “column” chart); this choice is more a matter of artistic preference in display, but it’s not meaningful for the data. In a bar chart, each bar represents a different item or group. The scale of some numerical variable is parallel to the bars, and the purpose of the graph is to compare the numerical values of the elements represented by different bars.

For example, in this bar chart, each bar is a particular continent, and the length of the bar, measured against the horizontal scale, indicates the area of the continents. For example, this bar chart indicates that Africa has considerably more area than North America, a conclusion not self-evident from a Mercator Projection map of the world.

If we want to show more complicated data, we might use segmented bars or side-by-side bars. You can read this GMAT post about this distinction. Set #3 below uses segmented bars.

A histogram is a chart that happens to use columns, but it is not a bar chart. On a true bar chart, each bar is a different element, and the scale is parallel to the bars. By contrast, on a histogram, the scale is perpendicular to the columns, and each column represents a particular range of the data. For an example, see the post on histograms. While the difference between a bar chart and a histogram is something that probably will be obvious from the problem set-up, this is an important distinction to understand for GRE Data Interpretation.

Try some GRE Data Interpretation practice to get an idea of what you can expect on test day.

The following pie chart shows the breakdown of revenues for a particular grocery store over the first quarter of last year. The bar chart shows the detail of breakdown for frozen foods.

1) What is the dollar amount of sales of canned goods in the first quarter of last year?

(A) $6,000

(B) $9,000

(C) $18,000

(D) $36,000

(E) $90,000

2) Frozen prepared meals constitute what percentage of the total sales for the first quarter last year?

(A) 2.4%

(B) 8.5%

(C) 20%

(D) 36%

(E) 54%

3) During the first quarter last year, this particular grocery store was finishing its construction of an expanded bakery facility, which, when opened at the beginning the second quarter, will offer dozens of new cakes and pies, a whole new line of pastries, and several flavors of gourmet coffee. In the first quarter, bakery sales represented 6% of total profits. Assume that in the second quarter, the bakery sales triple, and all other sale stay the same. Bakery would then account for what percentage of total sales in the second quarter?

(A) 12%

(B) 14.3%

(C) 16.1%

(D) 18%

(E) 25.3%

The following graph shows the total revenues and total costs of QN Corporation, a publicly traded company, over the span 2007-2015. Recall that, for any year, profit = (revenue) – (cost).

4) In 2013, the dollar amount of the profit at QN Corporation was approximately which of the following?

(A) $80 million

(B) $160 million

(C) $190 million

(D) $240 million

(E) $350 million

5) From 2010 to 2015, revenue at QN Corporation increased by approximately what percent?

(A) 18%

(B) 47%

(C) 63%

(D) 84%

(E) 126%

6) From 2009 to 2010, profit at QN Corporation increased by approximately what percent?

(A) 81%

(B) 165%

(C) 300%

(D) 433%

(E) 533%

The following chart shows the employee breakdown, number of marketer vs. number of programmers, at six similarly sized companies in a particular internet market. All six have similar sized management teams that are not displayed: all the non-management employees are displayed.

7) At Pindar, there are approximately how many more programmers than marketers?

(A) 6

(B) 22

(C) 35

(D) 57

(E) 73

8) At Jericho, approximately what percent of the non-management employees are marketers?

(A) 28%

(B) 42%

(C) 51%

(D) 66%

(E) 72%

9) If were to take the average number of marketers at five of the companies, excluding Fermion, then this average would be approximately how many employees larger than if Fermion were included in the average?

(A) 6

(B) 13

(C) 20

(D) 26

(E) 33

Fifteen college seniors, none of whom took any math classes in college, were selected for a clinical trial. An online GRE Math Course was made available to them, and these students were allowed to study for however many hours they wanted. Then, each took the GRE for their first time. The graph below records how many hours each student studied, using the online material, and that student’s GRE Quant score. The best-fit line model shows the best prediction, given such a student’s hours of study, of what her GRE Q score might be.

10) Among the students in this trial who studied less than 25 hours, what is the highest GRE Quant score achieved?

(A) 146

(B) 149

(C) 154

(D) 157

(E) 165

11) For the students in this trial with the three highest scores, approximately what is the average number of hours they spend studying?

(A) 29

(B) 32

(C) 35

(D) 39

(E) 43

12) Among the students in this trial who studied for 20 hours or more, what fraction of them performed worse than expected, according to the best-fit line model? Express your answer as a fraction.

1) This is a straightforward read-data-off-the-chart question. The pie chart tells us canned goods sales constitute 18% of $200,000. Don’t go to the calculator for such a straightforward percent question!

10% of $200,000 = $20,000

Double that.

20% of $200,000 = $40,000

We need an answer that is between those two, closer to $40,000. The only answer choice in that range at all is answer choice **(D)**, $18,000.

Another route to the answer. Obviously 18% of $100 is $18. Multiply both dollar amounts by 1000: then, 18% of $100,000 is $18,000. Now, double both dollar amounts: 18% of $200,000 is $36,000. There! An exact answer with no calculator, all mental math.

Answer = **(D)**

2) From the bar chart, prepared meals account for about $17,000 in sales. This $17,000 is what percent of $200,000? Again, please don’t jump to the calculator for this.

We know that $20,000 is 10% of $200,000. Take half of that: $10,000 is 5% of $200,000. So whatever percent $17,000, it must be between 5% and 10%. There’s only one answer choice in that range.

Answer = **(B)**

3) This is a tricky question, because there’s a tempting wrong answer. The bakery accounts for 6% of the total sales in first quarter, so if you triple that, it’s 18%, right? Wrong! The new amount would be 18% of the total sales in the first quarter, but we want to know what percent would it be of the total sales in the second quarter? That’s a new total because, even though everything else stayed the same, bakery sales increased.

We don’t need to consider the actual numbers: we can just work with the percents. Bakery sales triple from 6% to 18% — that’s the new “part.” Since the bakery goes up 12% from 6% to 18%, and all other sales stay the same, the new total is 112% — that’s the new “whole.”

Because the answers are relatively close, this is a rare problem on which it’s perfectly fine to use the calculator.

Answer = **(C)**

4) This is just a matter of reading numbers directly off the graph. In 2013, it appears that revenue = $350M and cost is about $160M. Subtract those and we get $190M.

Answer = **(C)**

5) In 2010, revenue was $250M, and in 2015, it was about $460M. The change is $460M – $250M = $210M, and we want this as a percent of the starting amount, $250. We don’t need a calculator. Notice that $210M is less than $250M, so the percent is less than 100% (choice (E) is wrong). Two thirds of $240M is $160M, so two-thirds of $10M more, $250, would be less than $170M; thus, $210 is much more than two-thirds of $250. We can eliminate (C) and anything less than that, which leaves only one answer.

Answer = **(D)**

6) We have a few different calculations to do here, but fortunately, the answers are spread out, so we can estimate. In 2009, revenue was about $110M and cost was about $80M, so profit was about $30M. In 2010, revenue was $250M and cost was about $90M, so profit was about $160M. That’s more than 5 times.

This is tricky. If something doubles, that’s a 100% increase. If it triples, that’s a 200% increase. Following this pattern, if something is multiplied by five, that’s a 400% increase, because four identical parts were added to the original part.

In this problem, $160M is more than five times, but less than six times, $30M. Thus, the percent increase has to be between 400% and 500%. The only possibility is **433%**.

Answer = **(D)**

7) At Pindar, the separation of marketers and programmers appears to be at 35, so there are 35 marketers. The top line appears to be at 92, so that’s 92 employees in total. If 35 are marketers, then 92 – 35 = 57 are programmers. How many more programmers than marketers? That’s 57 – 35 = **22**.

Answer = **(B)**

8) At Jericho, the green bar is what percent of the total green/purple bar? Well, if the green part were exactly 2/3, then two of the purple lengths would exactly equal the green length. Clearly, two of the purple lengths are shorter than the full green length, so the green is more than 2/3 of the whole bar, more than 66%. The only option this leaves is 72%.

Answer = **(E)**

9) First, take the average of the five excluding Fermion. Terrene has about 57 marketers; Pindar, about 35; Jericho, about 66; Endor, about 45; and Diomedes, about 52. The average is

average = (57 + 35 + 66 + 45 + 52)/ 5 = 255/5 = 51

Fermion has only 18 marketers. That’s a gap of 33 below the average of the other five. If we included Fermion, this gap of 33 would be divided six ways among the companies, so the new average would be 33/6 less than the original average. That’s a drop of 33/6 = 5.5, which is close to **6**.

Answer = **(A)**

__Trivia about the names of the companies in this set__ (you don’t need to know this for the GRE!) **Terrene** is an archaic English word, long out of use, meaning “*of the earth, worldly*.” Yes, Chris Lele would know this word! **Endor** & **Jericho** are from the Bible. In the first book of Samuel, Saul visits the witch of Endor and it doesn’t turn out well for him; this name, “Endor,” was borrowed by George Lucas for the forest moon in *Return of the Jedi*. Jericho, one of the oldest cities in the worlds, was a mentioned in the both the Hebrew Bible and the New Testament: the walls of Jericho fell to Joshua, and much later, a man traveling to Jericho was beset by thieves and then cared for by the Good Samaritan. **Pindar** (522 – 433 BCE) was a real person, an Ancient Greek poet. **Diomedes** was a hero in Greek mythology who was said to have fought in the Trojan War. A **fermion**, named for physicist Enrico Fermi (1901 – 1954) is a subatomic particle with half-integer spin: the three constituents of the atom (protons, neutrons, electrons) are all fermions; quarks are fermions as well.

10) For this problem, we just have to look in the correct region of the graph.

We ignore all the points that are at 25 hours studied and more: that’s the point on the vertical 25-hour line and all the points to the right—all of those are ignored. The only ones we care about are the ones less than 25-hours. Of these, the highest is the one furthest to the right, at a height of 154. This person studied less than 25 hours and got a GRE Q score of **154**.

Answer = **(C)**

11) The three highest scores, the three highest dots on the graph, are

- a) person with Q = 165 who studied 45 hours
- b) person with Q = 170 who studied 29 hours
- c) person with Q = 169 who studied 44 hours

We don’t need a calculator. Subtract 30 from each hour amount for simplicity. Then we have hours of {15, –1, 14}. The sum of those is 28, and 28/3 = 9.33. Add back 30 now: that’s an average of 39.33. The closest answer choice is **39**.

Answer = **(D)**

12) Again, we have to look in the right part of the graph. We have to look at all the dots to the right of the 20-hour line: these are the only dots that count. Dots above the line are students who exceeded the prediction. Dots below the line are students who fell short of the prediction.

The gray dots on the left are out, because those are times less than 20 hours. Of the remaining dots, the 7 green dots are above the line and 3 red dots are below the line. Ten dots altogether are above the 20-hour line, and 3/10 performed worse than expected.

To find out where algebra sits in the “big picture” of GRE Quant, and what other Quant concepts you should study, check out our post entitled:

What Kind of Math is on the GRE? Breakdown of Quant Concepts by Frequency

*Editor’s Note: This post was originally published in April, 2012, and has been updated for freshness, accuracy, and comprehensiveness.*

The post GRE Data Interpretation Practice Questions appeared first on Magoosh GRE Blog.

]]>The post GRE Geometry Formulas appeared first on Magoosh GRE Blog.

]]>Perhaps you recall with more than a little dread your high school geometry class: fifteen-step proofs and the law of cosines were two things you were certain you’d never see again. Well, if you are taking the GRE, you were by no means mistaken—proofs and indeed all of trig are nowhere to be found on the GRE.

That doesn’t mean that the GRE is void of geometry. Below are the GRE geometry rules and geometry formulas you have to know. This list is by no means exhaustive, but for must-know concepts you’ve come to the right place.

- Angles and Lines
- All Triangles
- Isosceles Triangles
- Right Triangles
- Circles
- Quadrilaterals
- Parallelograms
- Special Parallelograms
- Higher Polygons
- Three-Dimensional Shapes
- Coordinate Geometry

If you’re interested in more than geometry, try our more general GRE Math Formula eBook!

A right angle is made up of 90 degrees. A straight line is made up of 180 degrees.

In this diagram, ADC is a straight line. BTW, make sure you don’t confuse “straight” lines with “horizontal” lines. Any line that doesn’t bend or curve is “straight,” regardless of direction. By contrast, a “horizontal” line is parallel to the top or bottom of the page, parallel to the distant horizon. Line ADC is “straight” but not “horizontal.”

Notice, also, that “straightness” is one of the very few things we can assume on the GRE. If it looks like there’s no bend in the line, then there’s no bend: i.e. if it looks straight, it is straight. By contrast, we cannot assume perpendicularity: if we didn’t have that special little blue square guaranteeing that the lines were perpendicular, we could not assume that either angle ADB or angle BCD were right angles. That is something we must be told, in one form or another.

If two lines intersect, the sum of the resulting four angles equals 360°. Furthermore, the angles opposite each other have to be equal. These angles meet at just a vertex, so they are called “vertical” angles (a term you may remember but don’t need to know).

If we were told, say, that angle AED equals 40°, then immediately we would know the measures of the other three angles: angle BEC would have to equal AED, so that would also be 40°, and the other two angles would have to be the **supplements** of 40°. The supplement of an angle is 180° minus the angle. Thus, angle AEB = angle DEC = 140°; two angles that add up to 180° are **supplementary**. This way, all four angles add up to 360°.

When a line intersects a set of parallel lines, we get many more angles. We get eight angles: four “big” angles and four “small” angles. There are a number of technical names (“alternate interior angles”) that we don’t have to know for GRE geometry. We just have to know that every big angle is equal, every small angle is equal, and any big angle and any small angle are supplementary.

In the diagram, AC is parallel to DF: we could write that fact as AC // DF. That we cannot assume: we would have to be told that two angles are parallel. Once we are told that, and a third line intersects them, we get all the equal angles. Here, all the small angles are called x, and all the big angles are called y, and of course, x + y = 180°.

A triangle is a shape with three line segment sides.

Facts about all triangles:

**In any triangle, the sum of the three angles is 180°**.

That’s a big idea that Mr. Euclid (c. 300 BCE), the great geometer, first proved. Here’s another important geometry equation:

There’s more to that simple formula than means the eye. The letter b stands for the **base**: what’s the base of a triangle? Naïvely, folks will say the “bottom,” the horizontal side, is the base, but that’s not the whole story. In fact, any of the three sides can be the base. Similarly, the letter h is **height**: by “height,” we mean the length of a segment known as an altitude. An altitude is a line that passes through a vertex and is perpendicular to the side we are considering the base. Thus, for any triangle, we potentially have three different b & h combinations that would allow us to calculate the area.

Now, of course, the GRE is unlikely to give you all that information, three side lengths and three altitudes, in the course of a geometry problem. If a GRE geometry question ask for the area of the triangle, it will provide a way to find at least one base and the corresponding height. Keep in mind that the altitude divides the triangle into two little right triangles, so the Pythagorean Theorem (below) may be involved in finding some of the necessary lengths.

**The sum of any two sides is greater than the third**. If one side is 3 and one side is 5, call the third side x. We know that x + 3 > 5, which means that x > 2—this means that x could be 2.01, because it doesn’t have to be an integer. We also know that 3 + 5 > x, or 8 > x; again, x could be 7.999, as long as it’s less than 8. Thus, x has to be between 2 and 8, so it could be an integer {3, 4, 5, 6, 7} or it could be any fraction or decimal in that range. Remember that numbers are not necessarily integers on the GRE.

The largest side is always opposite the largest angle, and the smallest side is always opposite the smallest angle.

In this diagram, PQ is clearly the shortest side, so angle R has to be the smallest angle. Similarly, PR is longest side, so angle Q must be the largest angle.

One geometry rule related to this last fact concerns one elite category of triangles: the **isosceles triangles**. These are triangles with two equal sides.

The angle between the two equal sides may be

(a) an acute angle (less than 90°), as in triangle ABD

(b) a right angle, as in triangle DEF, or

(c) an obtuse angle (greater than 90°) as in triangle KLM

If two sides are equal, then we know the opposite angles are also equal: in triangle ABC, angle A = angle C; in triangle DEF, angle E = angle F; and in triangle KLM, angle K = angle M. In fact, Mr. Euclid pointed out that this geometry rule works both ways: if we are told two sides are equal, then we know two angles are equal, and if we are told two angles are equal, we know two sides are equal.

The line down the middle of an isosceles triangle is special:

As long as we are told that JKL and PQR are isosceles triangles, then this midline has some special properties:

(a) it is perpendicular to the base: angle KML = angle QSR = 90°

(b) its lower point bisects the base: JM = ML and PS = SR

(c) it bisects the upper angle: angle JKM = angle MKL, and angle PQS = angle SQR

These are all good geometry equations to associate with isosceles triangles.

Super-technically, an isosceles triangle is one that has *at least *two equal sides. This means that a very special case of isosceles is **the equilateral triangle**.

All three sides have equal length and each angle equals 60°. This is the most symmetrical triangle. Be careful not to assume that a triangle is equilateral: you cannot assume one is equilateral just because it appears as one. You would have to be told about either three equal lengths or all 60° angles.

Any right triangle has two sides touching the right angle: these are called **legs**. The longest side, always opposite the right angle, is called the **hypotenuse**.

One geometry rule that applies to all right triangles is theorem named for Mr. Pythagoras (570 – 495 BCE), whom some scholars consider the first mathematician.

Notice that the side we call “c,” the side alone on one side of the equation, has to be the hypotenuse. Also, notice that we can apply this formula unless we know a triangle is a right triangle, and that is not something we can assume simply from looks. If we get the litter perpendicular square, as we have here, then we know it’s a right triangle and we can apply this geometry equation.

Of course, some or all of the sides of a right triangle can be decimals, but it is possible, in special cases, for all three sides to be integers. These are sets of three integers that satisfy the Pythagorean Theorem. The most common is {3, 4, 5} and its multiples, but other good ones to recognize are {5, 12, 13}, {8, 15, 17}, and {7, 24, 25}.

There are two very special triangles that you have to understand for GRE geometry. The first is the **30-60-90** triangle:

We can multiply these lengths by any multiple, but they are always in this ratio. The short side, opposite the 30° angle, is always half the hypotenuse. The longer leg is always the square root of 3 times longer than the shorter leg. The ratios and angles in this triangle come direction from an equilateral triangle that was cut in half: for a detailed discussion of this view, see this GMAT post.

The other triangle is the **45-45-90** triangle, also known as the **Isosceles Right Triangle**.

Again, the ratios always are the same and we can multiply by any number. The two legs are always equal because this is an isosceles triangle, and the hypotenuse is always the square-root of two times any leg. GRE loves all the geometry formulas associated with these two triangles.

pi describes the ratio of any circle’s circumference (c) to its diameter (d). This number, pi, can be approximated by the decimal 3.14 or by the fraction 22/7, but it is a really a decimal that goes on forever with no repeating pattern.

The distance from the center out to the circle in any direction is called the radius (r). Here are the first few geometry formulas associated with the circle:

The first is just the fact that any diameter can be thought of as two radii in 180° opposite directions. The second is just a restatement of the definition of pi, given above. The third results from plugging the first into the second, and it’s actually the most useful formula for circumference, as we shall see.

Then, there’s another famous geometry equation, the formula for the area of a circle:

Archimedes (287 – 212 BCE), one of the greatest mathematicians of all times, gave us this remarkable formula.

We can divide a circle or its area up into pieces. Either a piece of the circle or a piece of the area would depend on the central angle, which is a part of the 360° all the way around the circle.

The curvy line from A to B, the path along the curve of the circle, is called an **arc**: this is a “piece” of the circumference. An arc is like the crust of a slice of pizza. The shaded area, which is like the whole slice of pizza, is called a **sector**. A sector is a piece of the whole area. We calculate either by setting up a geometry formula that compares two part-to-whole ratios, one of which involves the central angle, AOB.

These are two geometry formulas that you should **not** need to memorize: you should be able to figure them out by setting up part-to-whole ratios.

A quadrilateral is a shape with four line segment sides.

In the first, an irregular quadrilateral, we are given that two angles are right angles. The last one *looks like* a square, but of course, we can’t assume either equal sides or 90° unless we are told.

The only geometry rule that applies to all quadrilateral is the fact that **the sum of the four angles in any quadrilateral is 360°**.

A parallelogram is a quadrilateral with two pairs of parallel sides.

Any parallelogram has four amazing properties that always go together

(1) opposites sides are parallel: AD // BC and AB // CD

(2) opposites sides are equal: AD = BC and AB = CD

(3) opposite angles are equal: angle BAD = angle BCD and angle ABC = angle CDA

(4) diagonals bisect each other: AE = EC and BE = ED

The **diagonals** of any quadrilaterals are the lines connecting opposite vertices: here AC and BD are the diagonals.

Those four are the “BIG FOUR” parallelogram properties. If any of them is true, all the others automatically have to be true. Once you know it’s a parallelogram, you get a boatload of geometry formulas you can use.

The area of a parallelogram can be tricky. It’s just A = bh, but we have to be careful: the base b is a side, but the height h is the distance between two parallel bases.

In the diagram, we could consider either AD or BC the base (they are equal!) and BE would be the height. What if we are not given the height? Well, notice that ABE is a right triangle, in which the Pythagorean Theorem would apply: for example, if we knew AE and AB, we could find BE with the Pythagorean Theorem.

As a general rule, whenever a diagram has some vertical lines, some horizontal lines, and some slanted lines, and you have to find a particular length, chances are very good that there’s a right triangle somewhere in the diagram you can use to find the length you need! The Pythagorean Theorem is a truly remarkable geometry formula: it shows up all over the place!

There are three categories of special parallelograms: rhombuses, rectangles, and squares. The BIG FOUR parallelogram properties above apply to all of them.

A **rhombus** is an equilateral quadrilateral, that is, a quadrilateral with four equal sides.

Most “diamond” shapes, such as those on playing cards, are simply rhombuses turned sideways, like the one on the right above. If the side is s, then the perimeter is always just 4s. The diagonals of a rhombus are always perpendicular. In fact, any rhombus can be subdivided into four congruent right triangles.

As with a general parallelogram, A = bh, where the b is any side and the h is the length of a perpendicular segment: as with a general parallelogram, the Pythagorean Theorem may play a role in finding one of the lengths that you need.

(A pedantic note: the correct plural of “*rhombus*” is “*rhombuses*.” You see, words that end in –*us* from Latin can take the Latin ending –*i* in the plural: these would be words such as “*syllabus/syllabi*,” “*focus/foci*,” etc. Words with the –*us* ending from Greek do **not** use the Latin plural: hence “*rhombus/rhombuses*,” “*octopus/octopuses*“, “*hippopotamus/hippopotamuses*,” etc. Obviously, much confusion circulates in colloquial speech about these finer points.)

A **rectangle** is an equiangular quadrilateral, that is, a quadrilateral with four 90° angles.

Each angle is 90° and the two diagonals are always equal in length: it’s an old carpenter’s trick to verify that a doorframe has four right angles simply by checking the lengths of the two diagonals. Of course, it’s usually easy to find the length of a diagonal using the Pythagorean theorem.

The area of a rectangle is simply A =bh, where the base & height are simply lengths of any two adjacent sides. The perimeter is P = 2b + 2h.

Finally, we get the **square**. The fact that the square is one of the most familiar shapes, one of the first shapes learned in one’s early years, profoundly obscures how elite and sophisticated a shape it is. A square is a quadrilateral, a parallelogram, a rhombus, and a rectangle—and it has all the properties of all those categories, including four equal sides and four 90° angles. If the side is s, then

If we *know* a shape is a square, if we are told that it is a square, then all these geometry rules apply to the shape. BUT, just because something looks close to a square doesn’t mean that it **is** a square! Remember, no GRE geometry diagram is guaranteed drawn to scale, so if we are not given some explicit guarantee, something drawn to look like a square might be *any quadrilateral*!

A polygon is any closed figure with three or more line-segment sides. We have already discussed triangles and quadrilaterals. Here is a handy chart for the names:

For any n-sided polygon, the sum of all the angles in the polygon is given by sum = (n – 2)(180°). Any triangle has a sum of 180°, any quadrilateral has a sum of 360°, any pentagon has a sum of 540°, etc.

Another important idea is the idea of a **regular** polygon. This is a particularly tricky word: in ordinary speech, “regular” means ordinary, commonplace. In geometry, it has almost the opposite meaning. The regular polygons are the most elite and symmetrical shapes. A regular polygon is one that has all equal sides and all equal angles. The “regular triangle” is the equilateral triangle, and the “regular quadrilateral” is the square. In a regular polygon, because we can compute the sum of the angles, we can divide this by the number of angles to get the individual angle measure.

The regular octagon, of course, is the stop sign shape in many countries around the world.

As we saw above, quadrilaterals have two diagonals. Pentagons have five, and higher polygons have many more. See this post about the diagonals of a regular octagon. The diagonals of the regular pentagon trace out the standard five-pointed star, such as the stars on the flag of the United States of America.

A **rectangular solid** is a box-shape: it has six faces that are rectangles, and all the faces meet at right angles. Let’s say that it has a height of h, a length of l, and width of w. Then the volume and surface area would be:

The surface area is simply the sum of the areas of the six rectangular faces.

If all the lengths of a rectangle solid are equal, then it’s a **cube**. A cube has six congruent square faces that all meet at right angles to each other. If the edge length is s, then

All the 3D shapes that involve a circle were figured out by Archimedes (287 – 212 BCE), including the cylinder and the sphere. The volume of a **cylinder** can be thought of as base times height, where the “base” is the area of the circular base.

The GRE does not expect you to memorize the formulas for cones and spheres. If the GRE were to ask about these shapes at all, it would have to give you the formula and then expect you to do some calculation or algebraic manipulation with it. It’s good to be generally familiar with those formulas, enough so that they are not strange and intimidating, but you do **not** need to memorize them. If you’re curious, here they are:

A line in the coordinate plane can be represented in a few ways: the most common is **slope-intercept form**:

**y = mx + b**

The b is the y-intercept, the place where the line intersects the y-axis. The m is the **slope**. Slope is an indication of how tilted a line is. The slope of a horizontal line is zero. A vertical line has infinite slope. A 45° line has a slope of m = 1. To find the slope between two points A & B, draw a slope triangle:

If you do even a rough sketch, you should be able to read the rise and run off the graph, and then slope is just rise over run. Keep in mind, there are a few ways to think about any slope.

A slope of m = +3 means any of the following:

(a) to the right one unit, up 3 units

(b) any multiple of (a) (e.g. right 7 units, up 21 units)

(c) to the left one unit, down 3 units

(d) any multiple of (b) (e.g. left 5 units, down 15 units)

A slope of m = – 5/3 means any of the following:

(a) to the right 1 unit, down 5/3 of a unit

(b) to the right 3 units, down 5 units

(c) any multiple of (b) (e.g. right 9 units, down 15 units)

(d) to the left 1 unit, up 5/3 of a unit

(e) to the left 3 units, up 5 units

(f) any multiple of (e) (e.g. left 18 units, up 30 units)

The geometry rules concerning slope are very important to remember.

To find the y-intercept, set x equal to 0 and solve for y.

To find the x-intercept, set y equal to 0 and solve for x.

**Parallel** lines have **equal slopes**. **Perpendicular** lines have slopes that are **opposite-signed reciprocals**: for example, if the slope of one line is – 3, the slope of the perpendicular line is + 1/3; if the slope of one line is +4/7, the slope of the perpendicular line is – 7/4.

Finally, there is a thing out there called the “distance formula” for coordinate geometry, but we are not going to state it here, because it is an abomination before God and man. It makes monstrously difficult something that is quite straightforward. Here’s how to find distance between any two points. Again, draw or sketch the slope triangle:

Even if you sketch this, you should be able to read the “rise” and “run” directly from the graph. Those are the legs of a right triangle, shown in green in this diagram. The distance between A & B is simply the hypotenuse of the slope triangle, so use the Pythagorean Theorem. That’s all you have to do to find distance in the coordinate plane.

Finally, a word of caution. Knowledge of a formula, by itself, does not equal mathematical understanding. The GRE Quant doesn’t care about your memorization of formulas: instead, it designs problems specifically to probe your understanding. Yes, it’s important to be conversant in the formulas, but knowledge of these formulas is less than 10% of what you need to understand on the GRE Quant. For more on this, see this GMAT blog: even if you skip the three hard practice problems at the beginning, the rest of the discussion is quite pertinent to the GRE Quant as well as the GMAT.

To find out where geometry sits in the “big picture” of GRE Quant, and what other Quant concepts you should study, check out our post entitled:

What Kind of Math is on the GRE? Breakdown of Quant Concepts by Frequency

*Editor’s Note: This post was originally published in March, 2012, and has been updated for freshness, accuracy, and comprehensiveness.*

The post GRE Geometry Formulas appeared first on Magoosh GRE Blog.

]]>The post GRE Math Formula eBook appeared first on Magoosh GRE Blog.

]]>Based on your suggestions, we’re excited to offer you this eBook compilation of must-know GRE math formulas for you to reference and memorize as you study!

Though memorizing a math formula (or twelve) isn’t the only way to study for GRE Quant, knowing certain frequently-tested formulas by heart will help you improve your speed on test day. Once you learn these formulas, be sure to practice using them! A little timed practice can go a long way.

Download the Magoosh math formulas eBook and read on to learn a little bit more about what to expect from the math section of the GRE.

The GRE Quantitative Reasoning section tests concepts that you likely learned during sophomore and/or junior year of high school. These concepts DO NOT INCLUDE higher level math like trigonometry, calculus, or geometry proofs. They DO INCLUDE things you’ve probably long forgotten like properties of shapes, integer properties, exponent rules, and word problems.

The thing that takes these concepts to the GRE level is their complexity. Figuring out what each question is asking you to accomplish can be really tricky, whereas the math involved in solving them tends to be fairly straightforward. And you won’t get partial credit for showing your work, so picking the correct answer choice is very important!

Since it may take you longer than you’d like to read and understand what a GRE quant question is asking of you, it’s important that you won’t waste time solving the problem. Knowing which math formulas to use, and then using them quickly and correctly, can really help you do well on GRE quant. Time is of the essence!

Here is a list of the main concepts and formulas you’ll need to know for the GRE. Each of these concepts is addressed in the Magoosh’s Complete Guide to Math Formulas eBook. In this resource, you’ll also find recommended strategies for how to best use (or not use!) and remember these math formulas on the GRE.

- Arithmetic and Number Properties
- Types of Numbers
- Order of Operations: PEMDAS
- Commutative, Associative, and Distributive Properties
- Prime Numbers
- Factorization
- Divisibility
- Absolute Values
- Fast Fractions
- Percentages
- Ratios
- Powers and Roots
- Exponents
- Roots
- Algebra
- Simplifying Expressions
- Factoring
- Solving Equations
- Geometry
- Angles
- Polygons
- Triangles
- Circles
- Squares
- Rectangles
- Trapezoids
- Quadrilaterals
- 3-D Shapes
- Coordinate Geometry
- Lines
- The Distance Formula
- Quadratics
- Word Problems
- Distance, Rate, and Time
- Work Rate
- Sequences
- Interest
- Statistics
- Counting
- Probability

Don’t be intimidated! Some of these formulas are easy to remember, and some you’ll still know from your high school days. Focus your time on re-learning the formulas and concepts that are trickiest and newest to you.

Below are the main math formulas you’ll need for the GRE. I’ve just given a quick overview — a Cliff Notes version — since later on in the post you’ll find the full Math Formula eBook, which contains these formulas and *many* more. So if you want an in-depth take on this stuff, then head to the eBook at the bottom of the page. If not, look below for your math formula edification.

That said, I should offer a quick word on the temptation of thinking that knowing a formula is the key to automatically answering a question correctly. This is decidedly not the case on the GRE, on which you have to decipher, unscramble, unlock (pick your metaphor!) the question before you can answer it correctly. In other words, the formula is the final step to answering a math question right. (I expound on this in the intro of the eBook below.)

**Area:** , where is the radius of the circle

**Circumference:** , where is the diameter of the circle

**Surface area:** , where s is the length of a side of the cube

**Volume:** , where s is the length of a side of the cube

,where is the base and is the height of the triangle

=

=

=

=

=

=

**FOIL:** First Outer Inner Last

Total degrees = , where equals number of sides of polygon

We recommend that you print it out, share it, and start a pile of Magoosh eBooks (General, Vocabulary), that you can study on the go. Enjoy! Be sure to let us know you like this resource. 🙂

*Editor’s Note: This post was originally published in June 2012 and has been updated for freshness, accuracy, and comprehensiveness.*

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]]>The post GRE Math: What’s the Difference Between Combination and Permutation? appeared first on Magoosh GRE Blog.

]]>Do you know the difference between permutation and combination? No? You’re not alone. Combinations and permutations are the bane of many students. Yet, what I’ve noticed over the years is it’s not so much both of them that are the issue as it is which one to use for a particular problem: the combination vs permutation question. In other words, students have no difficult identifying whether a question is a combinations/permutations problem. The difficulty is in knowing exactly which one it is—combinations or permutations?

One way to think of it is to think of **permutations** as the number of arrangements or orderings within a fixed group. For example, if I have five students and I want to figure out how many ways they can sit in five chairs, I’m going to use the permutations formula. First off, the number in the group is fixed. Secondly, I’m looking for how many ways I can “arrange” the students in five chairs.

**Combinations**, on the other hand, are useful when figuring out how many groups I can form from a larger number of people. For instance, if I’m a basketball coach and I want to find out how many distinct teams I can form based on a group of people, I want to use combinations.

To make sure you understand this important distinction, here are three different scenarios. Your job is not to solve the question but to determine whether you use the combinations or the permutations formula to solve them.

1. Joan has five panels at home that she wants to paint. She has five different colored paints and intends to paint each panel a different color. How many different ways can she paint the five panels?

2. How many unique combinations of the word MAGOOSH can I form by scrambling the letters?

3. There are seven astronauts who are trying out to be part of three-person in-space flight team. How many different flight teams can be formed?

1. Permutations

She wants to arrange the colors. The number of panels is fixed. Had she been choosing five panels from a total of 8, let’s say, then we would need to use combinations.

2. Permutations

Okay, this was a little bit of a trick, since I used the word “combinations”. But that word I used colloquially, not mathematically. In this case, the number of letters is fixed. We are simply rearranging them.

3. Combinations

We are choosing a group from a larger group. One way to think of it is that when you use the word choose in the context of selecting from a group, you are dealing with combinations. And “choose” and “combinations” both begin with the letter ‘C’. There is an exception to this rule that I’ll talk about in the next section.

Big Idea: If you are forming a group from a larger group and the placement within the smaller group is important, then you want to use permutations.

Imagine a group of 12 sprinters is competing for the gold medal. During the award ceremony, a gold medal, silver medal and bronze medal will be awarded. How many different ways can these three medals be handed out?

Remember that with permutations ordering is key. Even though the top three spots for the sprinters forms a subgroup, it is the ordering within that subgroup that matters greatly, and is the difference between a gold, silver, and a bronze medal. An easy way to solve this question mathematically is to imagine that the dashes below are the podium upon which each sprinter will stand (albeit the dashes are at the same level):

____ ____ ____

gold silver bronze

To find out the number of different arrangement, ask yourself how many athletes can stand on the gold medal podium? Well, we have a total of 12 athletes. What about the silver medal podium? Now, we have one athlete fewer—since one is already on the gold medal podium. So that gives us a total of 11 for the silver medal spot. Finally, that leaves us with 10 athletes for the bronze medal podium.

The math looks like this:

12 x 11 x 10 = 1320

You might notice that this is the fundamental counting principle. The idea that when we are looking for total number of outcomes, we multiply numbers—or in this case, the number that goes on top of each dash—together. For instance, if I have six pairs of shorts and four pairs of t-shirts and I’m wondering how many different combinations of shorts and t-shirts I can wear, I want to multiply each, not multiply them:

___4_______ x _____6______ = 24

# of shirts # of shorts

I do not want to add them, which would give me 10, the wrong answer, in this case.

The reason I bring up the fundamental counting principle is that some questions will actually combine combinations with the fundamental counting principle (though you’ll likely use the fundamental counting principle more often with permutation questions). To give you an idea of how a combinations can show up along with the fundamental counting principle, try the following question:

Mrs. Pearson has 4 boys and 5 girls in her class. She is to choose 2 boys and 2 girls to serve on her grading committee. If one girl and one boy leave before she can make a selection, then how many unique committees can result from the information above?

(A) 9

(B) 12

(C) 18

(D) 22

(E) 120

The first step in this problem is recognizing whether we are dealing with combinations or permutations. Since, I am ‘choosing’ from a larger group, in this case two separate groups, I want to use combinations. Remember: once we’ve chosen the 2 boys or 3 girls, the position within the committee doesn’t matter. That is, either you are in the committee or out of the committee (there are no gold medalists here!)

The next thing to notice with this problem is that of the original 9 students, 2 leave, one boy and one girl. So that leaves us with 3 boys and 4 girls. We want to choose two each. Therefore, we have to set up one combination for boys and one for girls.

For boys, we have 3C2 and for girls we have 4C2. This gives us: 3C2 = 3 and 4C2 = 6

The final step is once we’ve figured out the combinations above, we have to use the fundamental counting principle and multiply the total number of possibilities in a committee, not add them: 3 x 6 = 18, answer (C).

Many students get stuck on this step and wonder, why don’t I add them. It’s a good question, so what I want you to do is imagine that you have 3 shirts and 3 pants. How many different shirt-pants getups can you wear? Well, for each shirt there are 3 options of pants. Therefore, we multiply and get 9.

My advice is to try about 5 or 6 more combinations/permutations problems so that you can get the hang of it. With a little practice, you’ll able to deal with most of the problems the GRE hands you. Even if you miss a question—likely because it is very difficult—the fundamentals in this post should be enough to help you understand the explanation to that question, so that you can get a similar question right in the future.

*Editor’s Note: This post was originally published in May 2011 and has been updated for freshness, accuracy, and comprehensiveness.*

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]]>The post How to Round to the Nearest Integer | GRE Math appeared first on Magoosh GRE Blog.

]]>Rounding means to make a number shorter or simpler, but keeping it as close in value as possible to the original number. Let’s take a closer look:

The most common type of rounding is to round to the nearest integer. The rule for rounding is simple: look at the digits in the tenth’s place (the first digit to the right of the decimal point). If the digit in the tenths place is less than 5, then **round down**, which means the units digit remains the same; if the digit in the tenths place is 5 or greater, then round up, which means you should increase the unit digit by one.

**Here are a couple other things to know:**

Observation #1: under most circumstances, rounding changes the decimal to whatever integer is closer. For example, 4.3 is rounded to 4, and 4.9 is rounded to 5. The exception is when the decimal is smack dab between two integers: 4.5 is exactly equidistant to both 4 and 5, but because of the “tie-breaker” rule of rounding, anything with a 5 in the tenths digit is rounded up. This is the only case in which the “go to the closer integer” interpretation will fail.

Observation #2: **Do NOT double-round**. Some people look at a number like, say, 7.49, and they erroneously think — well, that 9 would round the 4 up to 5, and then a 5 gets round up, to this number would round to 8. WRONG! Never round a number “in stages.” Rounding is a one-shot deal, a one-step process. When the number we need to round is 7.49, we only need notice that the tenth’s digit is a 4, which means the number is rounded down to 7. One step, case closed. In fact, all of the following numbers get rounded to 7:

**Here’s the truly mind-boggling part: **

How many numbers would there be larger than this last number, but still lower than 7.5? INFINITY! No matter how many additional 9’s we slap on to the end of that number, there’s still a continuous infinity of decimals larger than that number and below 7.5 No matter how finely we chop up the real number line, each tiny fragment of the line, no matter how small, still contains a continuous infinite of numbers.

Observation #3: the “tie-breaker” rule can be tricky with negative values. For example, +2.5 gets rounded up to 3, but –2.5 gets rounded *down* … to –3. As with positive numbers, the negative number ending in .5 is rounded to the higher absolute value integer, but with negatives, that’s rounding *down*. (This is not the only way to formulate this rule, but this is the convention that ETS follows.)

Rounding to the nearest integer is really rounding to the nearest units place. Sometimes, you will be asked to round to the nearest hundreds, or to the nearest hundredths — to some decimal place other than the units place. The rule is just a more generalized version of the previous rounding rule.

Suppose we are asked to round to some specific decimal place — call this the “target place.” You always look at only one digit, the digit immediately to the right of the target place. If this digit immediately to the right is {0, 1, 2, 3, 4}, then you “round down”, and the digit in the target place remains unchanged. If this digit immediately to the right is {5, 6, 7, 8, 9}, then you “round up”, and the digit in the target place increases by 1.

Not many GRE questions will say, “Here’s a number: round it to the nearest such-and-such.” By contrast, many questions, in the course of asking something else, could ask you to round your answer to the nearest such-and-such. In this way, rounding is one math skill you need to know for the GRE. There are a few tricky issues, which I will address here.

Very occasionally, a GRE math question may ask you not to round to a particular decimal place, but rather to the nearest multiple of something. For example, suppose you are asked to round, say, to the nearest 0.05 — how do you do that?

Well, let’s think about the results first. The result of rounding to the nearest 0.05 would be something divisible by 0.05 — that is to say, a decimal with either a 0 or a 5 in the hundredth place, no digits to the right of that, and any digits to the left of that. The following are examples of numbers which could be the result of rounding to the nearest 0.05:

- 0.35
- 1.40
- 3.15
- 5.2
- 8

Notice: the second, (b) is the square root of 2 (sqrt{2} = 1.414213562….) rounded to the nearest 0.05, and the third, (c), is pi rounded to the nearest 0.05.

Let’s demonstrate the rounding by means of an example. What numbers, when rounded to the nearest 0.05, would be rounded to 2.35? Well, for starters, 2.35 and other “tenths” around it would be rounded to 2.35

2.32 — rounded down to 2.30

2.33 — rounded up to 2.35

2.34 — rounded up to 2.35

2.35 — stays at value

2.36 — rounded down to 2.35

2.37 — rounded down to 2.35

2.38 — rounded up to 2.40

Now, the tricky regions are those between the values that are rounded in different directions. For example, 2.32 is rounded down and 2.33 is rounded up, so something fishy is happening between those two. Let’s think about the hundredths between 2.32 and 2.33 — exactly between them is 2.325, the midpoint between 2.30 and 2.35, and like all midpoints, according to the “tiebreaker” rule, it gets rounded up. Thus:

2.320 —- rounded down to 2.30

2.321 —- rounded down to 2.30

2.322 —- rounded down to 2.30

2.323 —- rounded down to 2.30

2.324 —- rounded down to 2.30

2.325 —- rounded up to 2.35 (the “tie-breaker” rule)

2.326 —- rounded up to 2.35

2.327 —- rounded up to 2.35

2.328 —- rounded up to 2.35

2.329 —- rounded up to 2.35

2.330 —- rounded up to 2.35

This is all probably far more detail than you will need to know for GRE math, but this does demonstrate the steps you would take to round any decimal to the nearest 0.05. By analogy, you could round any decimal to any specified multiple.

*Editor’s Note: This post was originally published in November 2012 and has been updated for freshness, accuracy, and comprehensiveness.*

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]]>The post GRE Quantitative Comparison Geometry Practice Problems appeared first on Magoosh GRE Blog.

]]>1) In the diagram, AC = 6. CE = 12, DF = 4, and AB is parallel to DE.

2)

3) In the diagram, JL = 4 and JK = 6.

4) In the diagram, O is the center of the circle, and AB is a diameter. Region J is the area between chord AC and the arc of the circle.

5) In the diagram, triangle MNP is equilateral.

6) In the diagram, JKLM is a square. Point S is the midpoint of KL, and point T is the center of the square. Point O is on segment ST, and is the center of the circle, which passes through both K and L.

7)

As you may appreciate, there’s no guarantee that any geometry diagram on the entire GRE Quant section is drawn to scale. In fact, most diagrams that appear are specifically designed to hoodwink you in the most viciously deceptive way imaginable. For example, if the GRE gives this

with no further explanation or specification, the naïve gullible GRE test taker will think, “OK, an equilateral triangle,” whereas in fact, it could be any of these:

Never be duped by believing a diagram. In particular, one large group of GRE QC Geometry questions gives you diagrams that appear a certain way, but which leaves some ambiguity open, and your job is to spot the ambiguity, the different geometric possibilities, and not be sucked in by trusting the misleading diagram. Some of the questions above are this sort.

Another batch of GRE QC Geometry questions specific everything in detail, and your job is to perform some kind of calculation, a length or angle or area, and compare it to something.

At the very least, it’s important to recognize, when looking at a GRE QC Geometry question, whether it’s the ambiguous-diagram kind or the everything-specified kind. The former case has a high likelihood of having **(D)** as an answer!

If the foregoing discussion gave you some insights into the question, you may want to go back and give them a second glance. Please let us know about your experience with GRE Geometry questions in the comment sections below.

1) The triangles CBA and CDE are similar, so all corresponding lengths are proportional. Each length in CBA is 1/2 as big as the corresponding length in CDE. Think about DF, the altitude in CDE, and the length to which it would correspond in CBA: an altitude from vertex B to the base AC.

Call that **h**. Since DF = 4, h = 2.

Now, for CBA, we have an altitude h = 2 and a base AC = 6.

Answer = **(B)**

2) Everything here is fixed and specified. Triangle ABC is a right triangle, because the longest side is the diameter. See this GMAT blog post for details. In right triangle ABC, AB = 4 is the hypotenuse. Notice, this is NOT a 3-4-5 triangle, because the hypotenuse is 4.

We could take the square root to find the length of AC, but we don’t need to. AC is the side of the square, so AC squared is the area of the square. The square has an area of 7.

Answer = **(C)**

3) This is a deceptive diagram. The angle at L *appears* to be right angle, but absolutely nothing in the diagram or the text guarantees that it is a right angle. The triangle could be:

On the left, the triangle has been “crushed” close to flat, and it could be crushed even further, to an area of almost zero. Certainty the area could be less than 11. By contrast, if JK and JL are perpendicular, then the area would be A = 0.5bh = (0.5)(4)(6) = 12, which is more than 11. Depending on the diagram, it could go either way.

Answer = **(D)**

4) We know AB is a diameter, and we know that angle ACB is 90°. All of that we know. We don’t know the position of point C. In the diagram given, point C is positioned so that the two areas look approximately equal. By moving point C around, we could radically change the situation. For example, we could move C toward B, making Region J much larger.

Or we could move point C toward the other side, closer to point A, in which case the triangle would be much bigger and Region J much smaller.

Since we can vary point C to produce different relationship, we cannot produce a fixed answer.

Answer = **(D)**

5) This is a hard problem, but we can do it entirely without touching a calculator.

The first step is to draw the altitudes from each vertex of the equilateral triangle to the midpoint of the opposite side. This will subdivide the equilateral triangle into six congruent 30-60-90 triangles. For more details on these triangles, see this GMAT blog. It’s very handy to remember that any equilateral triangle can be subdivided this way.

Notice that OP, OM, and ON are all radii of the circle. Let’s just say for convenience that r = 2, so OPO = OM = ON = 2. Look at 30-60-90 triangle TOP. The hypotenuse is OP = 2, so the side opposite the 30° angle is half of this, OT = 1. The remaining side, opposite the 60°, must have a length of:

The area of triangle TOP would be:

Six of these triangles make up the entire equilateral triangle, so that’s

That’s the area of the entire equilateral triangle. Meanwhile, the circle has a radius of r = 2, so

Now the shaded area is the area of the circle minus the area of the equilateral, so the area of the shaded region is:

Now, we have numbers for the areas of both regions:

Whatever the square-root of 3 equals, it is clearly less than 2, so Quantity A is less than 2, and Quantity B is slightly more than 2.

Answer = **(B)**

6) Notice that the one thing not fixed in the problem is the position of point O on the segment ST. It could be toward the top or bottom of that segment. Think about the implications.

If we move point O toward the top of the circle, so that it is very close to S, then we get relative small circles.

In the limit in which O was at S, the circle would be one whose diameter would equal the side of the square: such a circle could fit entirely inside the square, simply touching each side at its midpoint. Such a circle clearly would have much less area than the square.

On the other hand, we could move point O down toward point T, the center of the square. This produces relatively large circles.

In the limit in which O was at T, the circle would neatly contain the square, passing through the four corners. Such a circle clearly has more area than the square.

Because we can choose a position for point O that allows for a relationship either way, we can determine no conclusive relationship between the quantities.

Answer = **(D)**

7) This is a hard question. Let’s approach it this way. Sometimes it is very helpful to introduce a variable into a geometric problem. I am going to say that AC = x and BC = 1. Notice that AB = x + 1. Then

Now, what happens if x = 3? Notice that:

Clearly, they are not equal, as they should be. What happens if we make x bigger? Well, then AC/BC = x simply becomes bigger, but AB/BC becomes smaller, so they wouldn’t get closer to being equal. In fact, they would be getting further away from each other.

What happens if x get smaller? Well, then AC/BC = x clearly gets smaller, and the fraction 1/x gets bigger when its denominator gets smaller, so AB/BC would get bigger. In that case, they would be moving toward each other: the bigger one getting smaller, and the smaller one getting bigger. That indicates they could become equal if we move x in that direction, smaller than 3. In other words, x must be smaller than 3.

Right there, that’s enough to determine that the answer to the QC question is **(B)**.

More information for the curious:

If you are interested, notice that if x = 2, then:

They’re closer, but not equal. If x = 1, then

Here, they have shot past each other. Now, the little one is too big, and the big one is too little! We overshot. This tells us the correct value of x is between 1 and 2.

In fact, a few students may recognize that the diagram & equation above is the definition of the Golden Ratio, and in fact, this is the correct value of x.

If you happened to be familiar with the Golden Ratio, that would have been a shortcut on this particular problem, but in general, you don’t need to know anything about the Golden Ratio for the GRE. The Golden Ratio plays a big role in sacred geometry and all kinds of esoteric wisdom (Egyptian temples, the Parthenon, Leonardo’s paintings, the Gothic Cathedrals, etc.), so you may find it interesting, but all this is well beyond anything relevant to success on the GRE!! 🙂

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]]>The post FAQ: Should I Study Probability? appeared first on Magoosh GRE Blog.

]]>There is perhaps no other topic on the GRE math that instills as much dread in students as does probability. Understandably, students will obsess over this topic more than they will other topics. However, such an approach won’t only lead to a poor night’s sleep, it will also be counterproductive to your GRE score. The reason is that instead of studying for topics in which you can make quick gains—topics that are likely to show up on a test—you will squander your precious time on a question that may not even show up test day. And even if it does, you’ll likely get only one probability question.

What also makes studying for probability frustrating is that it requires more intuition than most of the other question types. Studying formula after formula isn’t going to help you make progress. Doing practice questions will help, but doesn’t ensure you’ll correctly answer that one probability question that you end up getting.

Let me state clearly that I’m not implying that you should skip probability altogether. What I’m saying is that you should focus on number properties, data interpretation and quantitative comparison, the three areas that students tend to overlook—or at least under study—even though these question types, taken together, account for as much as half of the quant section.

It would be terrible for you to forget that ‘1’ is not a prime, or that the square root of a fraction between 0 and 1 increases that number, because you were burning the midnight oil on a probability question about the likelihood that three different students would sit together in a row of nine chairs. In other words, you focused not only on a very limited question type (probability) but also on a very specific aspect of probability, one that you’re very likely not to see on the actual test.

It is important to know your weaknesses; it also important to know the frequency of question types involving that weakness. So, study what is likely to show up on the test and figure out your weaknesses in that area. If you have diligently made your way through all the other concepts, and only probability is left, then study it. However, if you only have thirty days to study for the test, the probability of that happening is pretty low.

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