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Math and the GRE Quantitative Reasoning

Math and the GRE Quantitative Reasoning. Lake Ritter October 20, 2012 Southern Polytechnic State University. Outline of this presentation. General information about the quantitative section The onscreen calculator General facts about real number and basic operations

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Math and the GRE Quantitative Reasoning

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  1. Math and the GREQuantitative Reasoning Lake Ritter October 20, 2012 Southern Polytechnic State University

  2. Outline of this presentation • General information about the quantitative section • The onscreen calculator • General facts about real number and basic operations • Sample Questions I: quantitative comparison • Sample Questions II: choices and numeric entry • Sample Question III: data analysis

  3. The quantitative section • The testing goal of this section is not to measure skills on advanced mathematical theory. • The goal is to measure your ability to use and reason with numbers or mathematical concepts. • The mathematical concepts are expected to be part of everyone’s background. • The four content areas are arithmetic, algebra, geometry and data analysis.

  4. Quantitative Ability You may encounter 4 types of questions: • Quantitative comparison questions • Multiple choice—select one answer—quantitative questions • Multiple choice—select one or more answers, • Numeric entry questions (no choices provided)

  5. Calculator Use “Sometimes the computations you need to do to answer a question in the Quantitative Reasoning measure are somewhat tedious or time-consuming, like long division or square roots. For such computations, you can use the calculator provided with your test.”** A calculator is provided for a paper based test. An onscreen calculator is provided for the electronic version. **Taken from ETS website 2011 (this quote still appears as of Oct. 2012)

  6. On screen calculator The calculator has basic arithmetic functions, a “c” key to clear all input and a “ce” key to clear the most recent input. The “Transfer Display” function moves the display value to the answer sheet. [M+], [MC], and [MR] are “memory with addition”, “clear memory” and “recall memory”. You should check that the transferred number has the correct form to answer the question. For example, if a question requires you to round your answer or convert your answer to a percent, make sure that you adjust the transferred number accordingly.

  7. On screen calculator This calculator respect standard order of operations (PEMDAS). For example, the key sequence 2 + 3 x 4 = results in the output 14. Multiplication (3 times 4) is performed first, then 2 is added. Again, double check the calculator answer prior to submitting your answer whenever possible.

  8. On screen calculator • Exercise: Find the value of the key sequences: • 2 x 5 + 9 = • (a) 28, (b) 19, or (c) 23 • 3  ( 5 + 1 ) + 4 • (a) 4.5, (b) 5.6, or (c) 7.6

  9. Real Numbers and Algebraic Operations • All numbers on the GRE are real numbers (no complex) • Every real number x is either positive (x > 0), negative (x < 0) or zero (x = 0). • Real numbers can be represented on a line read from left to right. • Adding forms a sum, subtraction a difference, multiplication a product, and division a quotient.

  10. Real Numbers and Algebraic Operations Integers are positive and negative counting numbers 0, ±1, ± 2, ± 3, …, Rational numbers have the form where a and b are integers with b ≠ 0 Irrational numbers can not be expressed as a ratio of integers or a repeating or terminating decimal. Prime Numbers are integers greater than 1 with no integer divisors other than 1 and themselves k is a divisor of n if n = km for integers n, k, and m

  11. Real Numbers and Algebraic Operations • The sum or difference of two odd or two even integers is even. • The sum or difference of an odd and an even integer is odd. • The product of two even or an even and an odd integer is even. • The product of two odd integers is odd. Example: n is even, k is odd what is the parity of (a) 3n, (b) 2k+1, (c) 3k+2n

  12. Real Number Basics Continued • The absolute value of x, denoted by |x|, is equal to x if x is positive, and –x if x is negative (so |x| is never negative). • If n is a positive integer, then n! denotes the product of all positive integers less than or equal to n. eg. • 5!=5*4*3*2*1= 120 • The radical sign √ means “the nonnegative square root of”. • You can’t take the square root of a negative number. • The quadratic formula: If ax2+bx+c=0, then

  13. Strategies for working with non-integers • To add and subtract decimals, make sure you line up the decimal points. • To multiply decimals, pretend they are integers and perform the calculation, then count the total number of digits after the decimal point before the operation, and place the decimal point that many to the left in the result. • To divide decimals, move the decimal point right in both decimals until you have a pair of integers (add zeros if needed).

  14. To multiply fractions, write them side-by-side and multiply across the top and bottom. • To divide fractions, invert the denominator fraction and multiply. • To add and subtract fractions, first find a common denominator. The common denominator is the least common multiple of the two denominators. If you feel rushed, multiply the denominators, add or subtract, then reduce the results. • To compare fractions, express with a common denominator.

  15. Powers, Roots, and Inequalities

  16. Inequalities • Be careful when multiplying or dividing an inequality: operations with negative numbers can change the direction of an inequality. If a < b and c > 0, then ac < bc and a/c < b/c. If a < b and c < 0, then ac > bc and a/c > b/c. If you don’t know the sign of a value, avoid multiplication and division involving inequalities!

  17. Cautions: • Not all numbers are integers! Even if a question contains integers, the answer may be rational or decimal. • Division by zero is not defined! • Provided figures are for illustration and do not imply scale! • Avoid common error. For example 1/a + 1/b ≠ 1/(a + b), and aⁿ + bⁿ ≠ (a + b)ⁿ

  18. Sample Questions I • Directions: In the following type of question, two quantities appear, one in Column A and one in Column B. You must compare them. • Notes: Sometimes information about one or both of the quantities is centered above the two columns. If the same symbol appears in both columns, it represents the same thing each time.

  19. Column AColumn Bab = 0 • The quantity in Column A is greater • The quantity in Column B is greater • The two quantities are equal • The relationship cannot be determined from the information given

  20. Perform the algebraic operations indicated. A: a2 + 2ab + b2. B: a2 - 2ab + b2. If ab = 0, then these quantities are both = a2 + b2. The two quantities are equal Make a couple of quick guesses. 3*0 = 0 (3 + 0)2 = (3)2 = 9 (3 - 0)2 = (3)2 = 9 0*2 = 0 (0 + 2)2 = (2)2 = 4 (0 - 2)2 = (-2)2 = 4 The two quantities are equal Strategy choices

  21. Column AColumn Ba + b = 24a - b = 25 • The quantity in Column A is greater • The quantity in Column B is greater • The two quantities are equal • The relationship cannot be determined from the information given

  22. How could the difference of two numbers be greater than the sum? • Don’t forget that sometimes numbers are not positive integers. • b must be a negative number if a + b < a – b. • The quantity in Column B is greater

  23. Column AColumn B • The quantity in Column A is greater • The quantity in Column B is greater • The two quantities are equal • The relationship cannot be determined from the information given

  24. To compare two fractions, we look for a common denominator. • The two quantities are equal

  25. Column AColumn Bx5 = (13/17) • The quantity in Column A is greater • The quantity in Column B is greater • The two quantities are equal • The relationship cannot be determined from the information given

  26. Remember: if 0 < a < 1, then aⁿ < a for n ≥ 1 • x5 = (13/17) • We are comparing x and (13/17)5 • Substituting, we are comparing x and (x5)5 • 0 < x < 1 because 0 < (13/17) < 1 • So, x25 < x • The quantity in Column A is greater

  27. Column AColumn B • The quantity in Column A is greater • The quantity in Column B is greater • The two quantities are equal • The relationship cannot be determined from the information given

  28. (sidebar) Working with percents & ratios • To convert decimals to percents, multiply by 100 and add a % sign. • To convert percents to decimals, divide by 100 and remove the % sign. • a% of b = b% of a = (ab)/100 • Percent increase is (actual increase) divided by (original amount) * 100% • Percent decrease is (actual decrease) divided by (original amount) * 100% • To increase by k%, multiply by (1 + k%) • To decrease by k%, multiply by (1 - k%)

  29. If you use careful estimations, you can avoid computation. Note 53% is bigger than 50%--i.e. ½. Since ½ of 360 is 180, the answer is The quantity in column A is bigger. Alternatively: 53% = 0.53 and 0.53x360 = 190.8 The quantity in column A is bigger.

  30. Column AColumn Ba + 2b = 6dc - b = 5d • The quantity in Column A is greater • The quantity in Column B is greater • The two quantities are equal • The relationship cannot be determined from the information given

  31. (sidebar) Mean, median, & mode • The average (is the arithmetic mean) is calculated by adding up all the data points and dividing by the number of data points. • The median is the middle number in a list, or the average of the two middle numbers. • The mode is the number in a list that occurs most often.

  32. Comparing the average of a, b, c, and d with 3d • a + 2b = 6d c - b = 5d added together • a + b + c = 11d • The mean is (a + b + c + d)/4 • Substituting (11d + d)/4 = (12d)/4 = 3d • The two quantities are equal

  33. Sample Questions II • Multiple choice questions may appear in two formats: choose 1 from 5, choose 1 or more from several • The latter type should be easily identified by language such as “which two of…” or “select all such…” • Free answer questions will provide a space for a numerical answer

  34. Example: If a positive integer n, divided by 5 has a remainder 2, which of the following must be true?Select all such statements. □ A. n is odd□ B. n + 1 cannot be a prime number□ C. (n + 2) divided by 7 has remainder 2□ D. n + 3 is divisible by 5

  35. (Note the word all ) We can write n as n = 5k + 2 for some integer k. This helps us consider each option. x A. n is odd n could be 7 or 12 (k = 1 or 2) x B. n + 1 cannot be a prime number n+1 could be 13 (if k = 2) x C. (n + 2) divided by 7 has remainder 2 n+2 could be 14 (if k = 2) □ D. n + 3 is divisible by 5 n+3 = 5k + 5 = 5(k+1) So D must be true.

  36. In 1980, the cost of p pounds of potatoes was d dollars. In 1990, the cost of 2p pounds of potatoes was ½d dollars. By what percent did the price of potatoes decrease from 1980 to 1990? Example: O A. 25% O B. 50% O C. 75% O D. 100% O E. 400%

  37. The question indicates there is only one answer. Recall: percent decrease (increase) = (old – new)/old times 100% • 2p costs ½ d • p costs ¼ d • Price has gone from d for p pounds to ¼ d for p pounds. • The amount of decrease is (d - ¼ d)/d = ¾ = 0.75, 0.75*100% = 75% C.75% is the correct answer

  38. Example: Let n be a positive integer and x = 53ⁿ. Which of the following could be the units digit of x. select all such digits • 0 • 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9

  39. (Note the word all ) The key to answering this is to recognize that only the units digits affect the units digit when multiplying integers. e.g. 53² = (50 + 3)(50 + 3) = 50² + 2(50)(3) + 3² The units digit is only influenced by 3*3 = 9. 30 = 1, 31 = 3, 32=9, 33 = 27, 34 = 81 Note that the units digits cycle 1, 3, 9, 7, 1, 3, 9, 7, … The answers are (B), (C), (H) and (J).

  40. What is the average (arithmetic mean) of 330, 360, and 390? Example: • 360 • 3177 • 310 + 320 + 330 • 327 + 357 + 387 • 329 + 359 + 389

  41. The question refers to “the” arithmetic mean, so it’s clear there is one solution. • Sum the three and divide by three • (330 + 360 + 390)/3 • Distribute (330 )/3 + (360 )/3 + (390)/3 • Exponent rules 330 /31 + 360 /31 + 390/31 • 329 + 359 + 389

  42. Writing and solving equations • When adding and subtracting polynomials, make sure to keep track of parentheses and signs. Combine only like terms. • When multiplying polynomials, make sure to distribute fully. (Use FOIL) • When dividing polynomials by monomials, distribute and divide one term at a time. • Keep equations balanced. • If you have multiple equations, try adding them together. • Read questions carefully and only look for the answer you need, not all the possible information.

  43. You may have to create expressions and equations to solve The square of y is subtracted from 5 and the result is divided by 12: For a given two-digit positive integer, the tens digit is 5 more than the units digit. The sum of the digits is 11. Find the integer. If the number is ab. Then you get a pair of equations: a + b = 11 and a = b + 5.

  44. Jordan has taken 5 math tests so far this semester. If he gets a 70 on his next test, it will lower the average (arithmetic mean) of his test scores by 4 points. What is his average now? Example: a) 74 b) 85 c) 86 d) 90 e) 94

  45. Create an equation and solve it: Let his current average be x. His new average will be His new average will also be e) 94 is the correct answer.

  46. Working alone at its constant rate, machine A produces k car parts in 10 minutes. Working alone at its constant rate, machine B produces k car parts in 15 minutes. How many minutes does it take machines A and B, working simultaneously at their respective constant rates, to produce k car parts?

  47. Machine A produces k/10 parts per minute and machine B produces k/15 parts per minute. Together, they produce k/10+k/15 = 15k/150+10k/150 = 25k/150 = k/6 parts per minute. Therefore k parts are produced in (k parts)/(k/6 parts/minute) = 6 minutes Enter 6 in the box provided. 6

  48. Geometry basics • Acute angles measure less than 90º • A right angle measures 90º • Obtuse angles measure between 90º and 180º • Straight angles measure 180º • The sum of angle measures lying along a straight line is 180º • The sum of angle measures around a circle is 360º • If two lines are perpendicular, they meet at 90º angles.

  49. Vertical angles have the same measure. • The sum of measures in a triangle is 180º • The longest side of any triangle is opposite the largest angle, the smallest side is opposite the smallest angle. • If two sides of a triangle have the same length, then their opposite angles have the same measure. • a2 + b2 = c2 where c is the hypotenuse (opposite the right angle) and a and b are the two legs of the triangle.

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