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2y y y22 + 1 Factor and reduce: Factor the numerator of 1 . We get 2 ( 5)( 2) 5 2 2 2 2 y y y y= 2 Substitute 8.000001 in 2 . We have 8.000001 2 5 5 3.000001 \u2248 3 (Answer) GRE2015_P04.indd 122 4/18/14 11:29 AM STRATEGY SECTION \u2022 123 Introduction In the quantitative comparison question, you are presented with quantities under two columns. You have to determine whether the Quantity A is less than, greater than, or equal to the Quantity B. There is also a possibility that a definite comparison cannot be made. In very, very few cases will you ever have to calculate quantities. Since you just have to compare the relative values, you can usually manipulate the columns to get a simple comparison, as you will see. 6 Quantitative Comparison Strategies Things You Must Know First 1. Memorize directions for quantitative comparison questions. The following reword ing of the standard directions given on the GRE tells you precisely how to choose an answer: \u2022 If Quantity A is always greater than Quantity B (or if Quantity B is always less than Quantity A), select Choice A. \u2022 If Quantity A is always less than Quantity B (or if Quantity B is always greater than Quantity A), select Choice B. \u2022 If Quantity A is always equal to Quantity B, select Choice C. \u2022 If there is no way to make any of the above definite comparisons, select Choice D. (In other words, if a definite comparison or relationship cannot be made, select Choice D.) Note: If there is a quantity centered between the columns, you may use that information in the columns. A symbol such as x, a, etc., means the same thing in both columns. All variables like x, n, a, represent real numbers. Make sure that you memorize these direc tions. example Quantity A Quantity B 1 2 1 In this example you would select Choice A, since Quantity A . Quantity B. 2. When there\u2019s a quantity between the columns, use the middle quantity in both columns. example Quantity A Quantity B x 5 2 x x 1 1 In this example, substitute 2 for x: 2 (2 1 1) Quantity A , Quantity B, so Choice B is correct. 3. You can add or subtract the same quantity to (or from) both columns and still get the same comparison. example Quantity A Quantity B 8 2 3 7 (Quantity A , Quantity B) Add 3 to both columns: 8 7 1 3 (Quantity A , Quantity B) GRE2015_P04.indd 123 4/18/14 11:29 AM 124 \u2022 GRUBER\u2019S COMPLETE GRE GUIDE 2015 4. You can multiply or divide both col umns by the same positive number and still get the same comparison. example Quantity A Quantity B 4 7 4 9 (Quantity A , Quantity B) Multiply by 4: 7 9 (Quantity A , Quantity B) 5. REMEMBER THE FOLLOWING: \u2022 Any variable like x, a, etc., can be negative, positive, 0, or a fraction unless otherwise specified. \u2022 If you get different comparisons when you try different num bers (for example, if you were to find that in one case Quantity A . Quantity B and in another Quantity A , Quantity B), then Choice D is correct. \u2022 Never divide or multiply columns by a negative num ber or by 0. Choose A if Quantity A is greater; Choose B if Quantity B is greater; Choose C if the two quantities are equal; Choose D if the relationship cannot be determined from the information given. SUMMARY DIRECTIONS FOR COMPARISON QUESTIONS Cancel Numbers or Expressions Common to Both Columns by addition or Subtraction If the same expression or number appears in both columns, we can then subtract it from both columns. Examples 1, 4, and 5 can also be solved with the aid of a calculator. However, to illustrate the effectiveness of Math Strategy A, we did not use the calculator method of solution in these examples. Math StratEg y a example 1 Quantity A Quantity B 2 1 1 6 1 1 17 1 17 1 1 2 1 1 7 1 Choice A is correct. Don\u2019t add fractions in columns! Cancel common quantities: Quantity A Quantity B 2 1 1 6 1 1 17 1 17 1 1 2 1 1 7 1 6 1 7 1 Quantity A . Quantity B. example 2 Quantity A Quantity B y . 0 w 1 x w 1 x 1 y Choice B is correct. Since w 1 x appears in both columns, we can subtract w 1 x from both columns to get Quantity A Quantity B 0 y and from the given information we know that y . 0. example 3 Quantity A Quantity B y is an integer y , 0 21 y 1 2 1 Choice A is correct. GRE2015_P04.indd 124 4/18/14 11:29 AM STRATEGY SECTION \u2022 125 Choose A if Quantity A is greater; Choose B if Quantity B is greater; Choose C if the two quantities are equal; Choose D if the relationship cannot be determined from the information given. SUMMARY DIRECTIONS FOR COMPARISON QUESTIONS Quantity A Quantity B 21 y 1 2 1 Cancel 21: 21 y 1 2 1 \u2193 \u2193 0 y 1 Since y , 0, y 1 , 0, and Quantity A . Quantity B. example 4 Quantity A Quantity B 5 2 0.005 5 2 0.0055 Choice A is correct. Quantity A Quantity B 5 2 0.005 5 2 0.0055 Cancel 5: 5 2 0.005 5 2 0.0055 2 0.0050 2 0.0055 2 0.0050 . 2 0.0055 example 5 Quantity A Quantity B a # 15 7 1 3 1 15 7 1 3 1 a Choice D is correct. Quantity A Quantity B a # 15 7 1 3 1 15 7 1 3 1 a Cancel 7 1 3: 7 1 31 15 7 1 3 1 a 15 a Since 15 $ a (given), you can\u2019t make a definite comparison. example 6 Quantity A Quantity B l1 l3 a° b°l2 l1 \ue069 l2; l3 intersects l1 and l2 a 1 a 1 b a 1 2b Choice C is correct. Quantity A Quantity B a 1 a 1 b a 1 2b Subtract a, b from both columns: a 1 a 1 b a 1 2b 2 a 2 b 2 a 2 b \u2193 \u2193 a b Now a 5 b because if l1 \ue069 l2, alternate interior angles a and b are equal. GRE2015_P04.indd 125 4/18/14 11:29 AM 126 \u2022 GRUBER\u2019S COMPLETE GRE GUIDE 2015 Cancel Numbers or Expressions (Positive Quantities Only!!) Common to Both Columns by Multiplication or Division If the same expression or number (positive quantities only which may be multiplied by other expressions) appears in both columns, we can then divide it from both columns. NEVER divide both columns by zero or a negative number. Examples 1 and 3 can also be solved with the aid of a calculator. However, to illustrate the effectiveness of Math Strategy B, we did not use the calculator method of solution in these examples. Math StratEg y B example 1 Quantity A Quantity B 24 3 46 3 35 46 3 24 3 36 Choice B is correct. Don\u2019t multiply out! Cancel 24 3 46 from both columns (by dividing both columns by 24 3 46). Quantity A Quantity B 24 3 46 3 35 46 3 24 3 36 Quantity A , Quantity B. example 2 Quantity A Quantity B m . 1 n . 0 mn n Choice A is correct. Since n . 0 and n appears in both columns, we can divide it from both columns to get Quantity A Quantity B m 1 and we are given that m . 1. example 3 Quantity A Quantity B 14 3 3 7 5 3 3 2 7 5 3 14 3 3 4 3 Choice B is correct. Quantity A Quantity B 14 3 3 7 5 3 3 2 7 5 3 14 3 3 4 3 Cancel common fractions: 14 3 3 7 5 3 3 2 7 5 3 14 3 3 4 3 3 2 4 3 3 2 , 4 3 example 4 Quantity A Quantity B A B D C h 60° ABCD is a parallelogram h 3 AB AD 3 CD Choice B is correct. GRE2015_P04.indd 126 4/18/14 11:29 AM STRATEGY SECTION \u2022 127 When a Comparison of the two Columns Is Difficult, Use Numbers in Place of Variables Sometimes by using numbers in place of variables, you can show that different comparisons exist, making choice D correct. Math StratEg y C example 1 Quantity A Quantity B b . 1 a . 1 a fi b b a a