Quantitative Reasoning

General Information There are 20 questions per section You can have either 2 or 3 sections You have 35 minutes to complete each section

What They Look For… basic mathematical skills understanding of elementary mathematical concepts ability to reason quantitatively and to model and solve problems with quantitative methods

What kind of math should I study??

Arithmetic topics include properties and types of integers, such as divisibility, factorization, prime numbers, remainders and odd and even integers; arithmetic operations, exponents and roots; and concepts such as estimation, percent, ratio, rate, absolute value, the number line, decimal representation and sequences of numbers.

Algebra topics include operations with exponents; factoring and simplifying algebraic expressions; relations, functions, equations and inequalities; solving linear and quadratic equations and inequalities; solving simultaneous equations and inequalities; setting up equations to solve word problems; and coordinate geometry, including graphs of functions, equations and inequalities, intercepts and slopes of lines.

Geometry topics include parallel and perpendicular lines, circles, triangles — including isosceles, equilateral and 30°- 60°-90° triangles — quadrilaterals, other polygons, congruent and similar figures, three-dimensional figures, area, perimeter, volume, the Pythagorean theorem and angle measurement in degrees. The ability to construct proofs is not tested.

Data analysis topics include basic descriptive statistics, such as mean, median, mode, range, standard deviation, interquartile range, quartiles and percentiles; interpretation of data in tables and graphs, such as line graphs, bar graphs, circle graphs, boxplots, scatterplots and frequency distributions; elementary probability, such as probabilities of compound events and independent events; random variables and probability distributions, including normal distributions; and counting methods, such as combinations, permutations and Venn diagrams. These topics are typically taught in high school algebra courses or introductory statistics courses. Inferential statistics is not tested.

Some things to remember… All numbers used are real numbers. Geometric figures, such as lines, circles, triangles and quadrilaterals, are not necessarily drawn to scale. That is, you should not assume that quantities such as lengths and angle measures are as they appear in a figure. You should assume, however, that lines shown as straight are actually straight, points on a line are in the order shown and, more generally, all geometric objects are in the relative positions shown. For questions with geometric figures, you should base your answers on geometric reasoning, not on estimating or comparing quantities by sight or by measurement.

Some things to remember… All figures are assumed to lie in a plane unless otherwise indicated. Coordinate systems, such as xy-planes and number lines, are drawn to scale; therefore, you can read, estimate, or compare quantities in such figures by sight or by measurement. Graphical data presentations, such as bar graphs, circle graphs, and line graphs, are drawn to scale; therefore, you can read, estimate or compare data values by sight or by measurement.

Types of Questions Quantitative Comparison Multiple Choice Quantity A is greater. Quantity B is greater. The two quantities are equal. The relationship cannot be determined from the information given. Multiple Choice Choose one correct answer Choose multiple correct answers Numeric Entry You type in your answer in one box (can be an integer or decimal) Fractions will have one box for the numerator and one for the denominator

Extra Tidbits You are not allowed a calculator There is an on-screen calculator for certain problems You are given paper and a few pencils Know your equations Area Circumference Surface Area Ect.

Time to test your knowledge!

Problem #1 What are all the possible solutions of | |x – 2| – 2| = 5? (Choose all that apply) -5 -3 -1 7 9

Problem #1 Answer & Explanation -5, 9 If we focus just on the 𝒙−𝟐 , we can see that the result must be positive. Stepping back and looking at the entire equation we substitute u for 𝒙−𝟐 , to get 𝒖−𝟐 =𝟓. Solving for absolute value, we get the following: u-2 = 5 u-2 = -5 Thus, u = 7 and u = -3. Because u must be positive, we discount the second result. Next, we have to find x in the original 𝒙−𝟐 , which we had substituted with u. Replacing u with 7 we get: 𝒙−𝟐 =𝒖 𝒙−𝟐 =𝟕 x-2 = 7 and x-2 = -7 x = 9 and x = -5 . A faster way is to plug in the answer choices to see which ones work.

Problem #2 If (a^2)(b) is an integer which of the following must be an integer? a b ab b^2 None of the above

Problem #2 Answer & Explanation E. None of the above Let’s choose numbers to disprove each case. By the way, the word disprove is very important here – the question says ‘must’ so by picking numbers that prove the case, we are not necessarily proving that an answer choice must always be an integer. For A. I can choose 𝟑 =𝒂, and b is any integer. Because a is not an integer, A. is not correct. For B. it’s a bit tricky. However, if you keep in mind that there are no constraints in the problem stating that a cannot equal b, we can make 𝒂= 𝟑 𝟏 𝟒 and 𝒃= 𝟑 . For C. we can choose the same numbers to show that ab is not an integer. For D. if 𝒃= 𝟑 𝟏 𝟒 and 𝒂= 𝟑 𝟑 𝟖 , 𝒂 𝟐 𝒃 equals an integer, but 𝒃 𝟐 does not.

Problem #3 How many positive integers less than 100 are the product of three distinct primes? [ ]

Problem #3 Answer & Explanation 5 Let’s write out some primes: 2, 3, 5, 7, 11, 13, and 17. I’m stopping at 17 because the smallest distinct primes, 2 and 3, when multiplied. by 17 give us 102. Therefore 13 is the greatest prime conforming to the question. Here is one instance. 2*5*13 is greater than 100 so we can discount it. Working in this fashion we can add the following instances: 2*3*5 2*3*7 2*3*11 2*5*7 3*5*7 is too great 2*3*13=78. Therefore, there are five instances.

Problem #4 -1<x<100 Column A Column B X^3 X^6 The quantity in column A is greater The quantity in column B in greater The two quantities are equal The relationship can not be determined from the information given

Problem #4 Answer & Explanation D. The relationship cannot be determined from the information given. If x is less than 0 the answer is B. If x is 0 <x<1, the answer is A. Therefore, the answer is D.

Problem #5 A square garden is surrounded by a path of uniform width. If the path and the garden each have an area of x, then what is the width of the path in terms of x? 𝐱 𝟐 𝟐 𝒙 − 𝟐 𝟐 𝟐 − 𝒙 𝟒 𝒙 𝟐 − 𝒙 𝟐 𝟐𝒙 𝟐 − 𝒙 𝟐

Problem #5 Answer & Explanation 𝟐𝒙 𝟐 − 𝒙 𝟐 If the area of the small square is x, then each side is √x. The area of the large square is 2x (you want to add the area of the small square to that of the path), leaving us with sides of √2x. If we subtract the length of a side of the small square from a side of the large square, that leaves us with √2x – √x. Remember that there are two parts of the path, so we have to divide by 2: √2x/2 – √x/2, which is (E).

Helpful Websites http://magoosh.com/gre/2011/revised-gre-math-practice- questions/ http://www.ets.org/gre

Questions?