GRE Workshop Quantitative Reasoning. February 13 and 20, 2018

Transcription

1 GRE Workshop Quantitative Reasoning February 13 and 20, 2018

2 Overview Welcome and introduction Tonight: arithmetic and algebra 6-7:15 arithmetic 7:15 break 7:30-8:45 algebra Time permitting, we ll start some geometry tonight Next Tuesday, 2/20: (more) geometry and data analysis Throughout: think-pair-share as we do problems. Get to know a couple of your neighbors. I ll have Brian Eikenhout post my PPT slides for you on BB.

3 General Test Resources Test Taker Prepare for the Test PowerPrep II Program (Two full-length practice tests) Math Review (Broken down by topic with practice problems) Instructional Videos on Khan Academy Math Conventions GVSU.edu/library Databases Learning Express Library Featured Resources GRE Preparation Two practice tests number2.com Practice problems based on category majortests.com/gre Practice problems based on type It is essential that you DO PROBLEMS. You can use the Math Center for support, too.

4 Quantitative Reasoning Includes: Arithmetic, algebra, geometry, and data analysis Excludes: Trigonometry, calculus, and other college-level math Question formats: Multiple choice one answer Multiple choice one or more answers Numeric entry type correct answer in box Quantitative Comparisons compare two quantities. Data Interpretation set 2 or more sets of questions concerning a display of data. You should read and study the GRE Math Conventions it s 11 pages, and the notation and ideas there are important to refresh be sure you spend time exploring and look at various question forms

5 Quantitative Comparisons - style In quantitative comparison questions you will be asked to compare two quantities and determine the relationship between them. You should familiarize yourself with the answer choices. A B C D Quantity A is greater. Quantity B is greater. The two quantities are equal. The relationship cannot be determined from the information given.

6 Four main topic areas Arithmetic Algebra Geometry Data Analysis

7 1. Arithmetic working with numbers Integers (whole numbers, positive and negative) Fractions, ratios, and percentages Basic exponents and Roots Decimals Real Numbers

8 Column A Column B A B C D Quantity A is greater. Quantity B is greater. The two quantities are equal. The relationship cannot be determined from the information given.

9 Quantitative Comparisons - suggestions Make one quantity look like the other. Compare quantities piece by piece. Do the same thing to both quantities. Don t forget to consider other possibilities for variables (Think negative) Think about simpler examples that are comparable Try to demonstrate two different relationships between the quantities (D).

10 Column A 13 ( 14) Column B 14 ( 13) Hint: If the two quantities are completely numeric the answer cannot be D. (That is, there s no way the answer is can t determine in an all-number problem like this one.)

11 Column A Column B ( 61)(112) (62)(111) Hint: Try to make the two expression look alike.

12 x 1 Column A Column B 3x 5 2x 8 Hints: try a few small values of x; think about the graphs of the linear functions, too Can also explore making the two expressions have overlapping parts

13 Order of Operations Parentheses Exponentiation (includes square roots) Multiplication and Division (from left to right) Addition and Subtraction (from left to right) Order of operations are recognized by the built-in calculator. or/

14 Column A Column B

15 Column A 50 Column B

16 Column A Column B

17 Common Fractions 1 2 = = = = = = = = 0.01

18 Using the Calculator Consider options for mental arithmetic or estimation first as most questions do not require difficult computations. Use the calculator for more complicated computations for example products of larger numbers, long division or evaluating square roots Entry may be done with the keyboard or mouse Has four arithmetic functions, square root, and parentheses The Transfer Display button will transfer a number to a Numeric Entry question with a single answer box Follows order of operations You can download a sample version to experiment with at

19 Numeric Entry Make sure you are answering the question that is being asked. Enter your answer as an integer or a decimal if there is a single answer box Enter it as a fraction if there are two separate boxes one for the numerator and one for the denominator. For a single answer box, a number can be transferred to the box from the on-screen calculator Enter the exact answer unless the question requires you to round your answer and be sure to round to the required degree of accuracy. Determine if your answer is reasonable using estimation.

20 Numeric Entry (single box) Directions: Enter your answer as an integer or a decimal if there is a single answer box To enter an integer or a decimal, either type the number in the answer box using the keyboard or use the Transfer Display button on the calculator. First, click on the answer box a cursor will appear in the box and then type the number. To erase a number, use the Backspace key. For a negative sign, type a hyphen. For a decimal point, type a period. To remove a negative sign, type the hyphen again and it will disappear; the number will remain. The Transfer Display button on the calculator will transfer the calculator display to the answer box. Equivalent forms of the correct answer, such as 2.5 and 2.50, are all correct. Enter the exact answer unless the question asks you to round your answer.

21 Numeric Entry (two boxes) Directions: Enter your answer as a fraction if there is are two answer boxes To enter a fraction, type the numerator and the denominator in the respective boxes using the keyboard. For a negative sign, type a hyphen; to remove it, type the hyphen again. A decimal point cannot be used in a fraction. The Transfer Display button on the calculator cannot be used for a fraction. Fractions do not need to be reduced to lowest terms, though you may need to reduce your fraction to fit in the boxes.

22 A retailer marks up the purchase price of an item 20% to determine the selling price. After a while the item is discounted 10% of the selling price. Determine the profit as percentage of purchase price. %

23 A committee of 20 people originally consisted of three times as many men as women. If 20% of the men are replaced by women what is the ratio of women to men on the new committee expressed as a fraction?

24 Percentages part = percent x whole What is 8 percent of 80? 32 is what percent of 40? 12 is 2 percent of what number?

25 Percentages Percent increase = amount of increase x 100% original number A company had sales in January of $3,000. Their sales in February were $3,200. If they expect sales to increase by the same percentage, what should sales be in March?

26 Mary earns 20% more than Bill. Rachel earns 10% less than Bill. If Bill earns $120, how much more does Mary earn than Rachel?

27 Multiple Choice (Multiple Answer) The number of choices will vary with the question. Pay attention to the number of answers expected. Some questions will specify the number of choices Questions that do not specify the number of solutions can be from 1 to all answers selected. Credit is given only for choosing all of the correct answer and no extras. Consider limiting values to eliminate answer choices. Look for patterns in the correct answer choices.

28 Integers whole numbers, positive & negative Factors/Divisors (include negatives) Multiples Least common multiple Greatest common divisor/factor Even Integers Odd Integers Prime Numbers Prime Factorization Composite Numbers

29 Which of the following integers are multiples of both 2 and 3? Indicate all such integers. (a) 8 (b) 9 (c) 12 (d) 18 (e) 21 (f) 36

30 Given that a is an even number and b is odd number determine which of the following values are even. Select all that are even. a) a + b b) a b c) ab d) a b e) b a f) 3a + 2b

31 Which of the following could be the units digit of 23,where n is a positive integer? Indicate all such integers. (a) 0 (b) 1 (c) 2 (d) 3 (e) 4 (f) 5 (g) 6 (h) 7 (i) 8 (j) 9 n

32 P and Q are prime numbers, with P < Q. R is the smallest prime number greater than P and S is the smallest prime number greater than Q. Column A P R Column B Q S

33 Fractions Adding & subtracting fractions Multiplying fractions Dividing fractions Mixed number fractions Exponents Positive and Negative Exponents Properties of Exponents Roots

34 Ratio The ratio of one quantity to another is a way to express their relative sizes. Notations: a to b a:b a b

35 Arithmetic Exercises p You should do all of these on your own. We ll focus on #4, 5, 14, 15

36 Break

37 2. Algebra working with unknown quantities (variables) Operations with Algebraic Expressions Rules of Exponents Solving Linear Equations Solving Quadratic Equations Solving Linear Inequalities Functions Applications Coordinate Geometry Graphs of Functions

38 Working with algebraic expressions Permitted Operations: Addition Subtraction Multiplication by positive numbers Division by positive numbers Avoid (or use carefully) the following operations: Squaring both quantities Only use when quantities are known to be positive Multiplication/division by negative numbers Switches the statement of an inequality

39 Rules of Exponents Properties of Exponents: If m and n are counting numbers, then b b m n m n b n n n bc b c b m n b mn b b m n b m n n b c b c n n Understanding negative exponents and zero as an exponent: If b 0 0 b 1 and n is a counting number, then b n 1 n b 1 n n b b

40 Solve the linear equation for x. 3x + 2 = 4 x 1 + 5

41 Solve the given system find the value of x + y 3x + 2y = 2 5x y = 25

42 Given the system, find the ratio of x to y: 3x + 2y = 19 x = 3y 12

43 Quadratic Equations Two options: 1. Factor, and use the zero product property 2. Use the quadratic formula

44 Solve the following equations: x 2 3x x x x 15 0

45 Ben can walk at a speed of 6 mph and bike at a speed of 15 mph. How many miles can he travel in four hours without biking more than three times as long as he walks? (A) 66 (B) 33 (C) 51 (D) 60 (E) 58

46 Ben can walk at a speed of 6 mph and bike at a speed of 15 mph. How many miles can he travel in four hours without biking more than three times as long as he walks? (A) 66 (B) 33 (C) 51 (D) 60 (E) 58

47 If it takes 3 days for 10 workers to finish building one house, how long will it take 15 workers to finish 4 houses? (A) 15 (B) 10 (C) 8 (D) 6 (E) 4

48 In applied problems, we often introduce a variable to represent the quantity we are seeking. How many ounces of pure hydrochloric acid should be added to 20 ounces of a 25% solution of hydrochloric acid to obtain a 40% solution of hydrochloric acid. (A) 5 (B) 10 (C) 15 (D) 18 (E) 25

49 At a concession stand candy can be purchased for $0.25 each and bags of popcorn for $0.50 each. If 18 items were purchased for $6, how many bags of popcorn were purchased. (A) 0 (B) 4 (C) 6 (D) 10 (E) 12

50 Use the definition a b ( a b)( a b) to evaluate the following expressions

51 Use the definition a b a 2 b 2 a b to evaluate the following expressions

52 Coordinate Geometry Graphs will be drawn to scale (but pay attention to scale). Terms to know: x-axis y-axis Origin Quadrants Coordinates Intercepts Reflections Reflection about x-axis (symmetric about the x-axis) Reflection about y-axis (symmetric about the y-axis) Reflection about the origin (symmetric about the origin)

53 Coordinate Geometry Equations: Lines y = mx + b Parallel Lines Perpendicular Lines Parabolas y = ax 2 + bx + c Vertex Line of Symmetry Circles (x h) 2 +(y k) 2 = r 2 Center Radius Piecewise-defined functions

54 Determine which of the following lines intersect the line y = 1 x + 2 in the first quadrant. 3 Select all that apply. (A) y = 2 3 x 1 (B) y = 1 3 x + 1 (C) y = x 1 (D) y = 1 3 x + 1 (E) y = 2 3 x + 2

55 Algebra Exercises p You should do all of these on your own. We ll focus on part (a) in each of #1-8 and try to work through several examples from