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b Choice D is correct. Let us take numerical examples that satisfy a, b . 1. case 1 a 5 6, b 5 3 Then the columns become Quantity A Quantity B 3 6 5 3 6 3 5 2 1 and the quantity in Quantity A is greater. case 2 a 5 4, b 5 12 Then the columns become Quantity A Quantity B 12 4 5 3 1 4 12 5 3 and the quantity in Quantity B is greater. In one case, Quantity A . Quantity B. In the second case, Quantity B . Quantity A. Thus, a definite comparison cannot be made. example 2 Quantity A Quantity B a . 0 a 1 a Choice D is correct. 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 Label AB 5 a and AD 5 b A B D C hb b a a 60° AB 5 a AD 5 b CD 5 a Quantity A Quantity B h 3 AB AD 3 CD So h 3 a b 3 a Cancel a: h b Since b is the hypotenuse of the right triangle, it must be greater than h. So h , b. GRE2015_P04.indd 127 4/18/14 11:29 AM 128 \u2022 GRUBER\u2019S COMPLETE GRE GUIDE 2015 Often you can find a number for the variable that makes the columns equal. Then all you have to find is another number that will make them unequal. In the above exam ple, choose a 5 1. This makes the columns equal. You can see that any other value of a, like a 5 100, will make the columns unequal. Thus, a definite relation cannot be obtained. example 3 Quantity A Quantity B 30 . ab . 5 a and b are whole numbers a 1 b ab Choice D is correct. Try a 5 6, b 5 4. We get: Quantity A Quantity B 6 1 4 6 3 4 10 24 10 , 24 Try a 5 1, b 5 6. We get: Quantity A Quantity B 1 1 6 1 3 6 7 6 7 . 6 A definite comparison cannot be made. examples 4\u20136 Quantity A Quantity B 4. a . b . 1 a and b are whole numbers ab ba 5. x is an integer x x 1 1 (x 1 1)x 6. ab fi 0 2a2b ab2 explanatory answers for examples 4\u20136 Quantity A Quantity B 4. (D) a . b . 1 ab ba Let a 5 3, b 5 2: 32 23 9 . 8 Let a 5 4, b 5 2: 42 24 16 5 16 A definite comparison cannot be made. Quantity A Quantity B 5. (D) x is an integer x x 1 1 (x 1 1)x Try x 5 1: 11 1 1 (1 1 1)1 1 , 2 Try x 5 2: 22 1 1 (2 1 1)2 8 , 9 Make sure. Try x 5 3: 33 1 1 (3 1 1)3 81 . 64 In one case Quantity A , Quantity B; in another case Quantity A . Quantity B. Thus, a definite comparison cannot be made. Quantity A Quantity B 6. (D) ab fi 0 2a2b ab2 You can\u2019t cancel or divide by a or b because a or b may be negative. Try a 5 1, b 5 1. We get: 21 , 1 Try a 5 21, b 5 1. We get: 21 5 21 A definite comparison cannot be made. For examples 7\u201310, use the Gruber equal/not equal method to prove that Choice D is correct If Possible, Find a Number (or Numbers) That Makes the Columns Equal and Another Number (or Numbers) That Makes Them Unequal to Ensure That Choice D Is the Right Answer. If you feel that a definite comparison cannot be made, try to find one particular number that when substituted for the variable in the columns will make the columns equal. All you then have to do to prove that Choice D is correct is find another number that will make the columns unequal. example 7 Quantity A Quantity B x2 5 y2 x2 xy Choice D is correct. GRE2015_P04.indd 128 4/18/14 11:29 AM STRATEGY SECTION \u2022 129 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 x2 5 y2 x2 xy You can\u2019t divide by x since x may be negative. Try x 5 1, y 5 1. 12 (1)(1) 1 5 1 Now try x 5 21, and y 5 11. (21)2 (21)(11) \u2193 \u2193 11 . 21 Thus a definite comparison cannot be made. example 8 Quantity A Quantity B 0 . a (a 1 4)(a 1 5) (a 1 4)2 Choice D is correct. You may be tempted to cancel (a 1 4) from both columns. Because (a 1 4) may be zero or negative, you are not allowed to do this. If you did, you\u2019d get Quantity A Quantity B a 1 5 a 1 4 5 . 4 and you would think Choice A correct, which is wrong! Here\u2019s the best way: Let a 5 24. That way the columns become equal (both equal to 0). Now let a 5 21: (21 1 4)(2l 1 5) fi (21 1 4)2 Don\u2019t bother to calculate out, because you can see that the columns aren\u2019t equal. Thus a definite comparison cannot be made. example 9 Quantity A Quantity B x . 0, y . 0, z . 0 x y z 1 + + x 1 y 1 z Choice D is correct. Let x 5 y 5 z 5 3 1 to get the columns equal. Thus: 1 3 1 3 1 3 1 1 1 1 5 1 Any other numbers, such as x 5 2, y 5 2, z 5 2, will make the columns unequal: 2 2 2 1 1 1 fi 2 1 2 1 2 Therefore, Choice D is correct. example 10 Quantity A Quantity B (a3)4 a7 Choice D is correct. Quantity A Quantity B (a3)4 a7 First calculate (a3)4 5 a12 (remember your basic math skills?). The columns become: a12 a7 Now try a 5 1 to get the columns equal. If we then try a 5 2, we can see that the columns are unequal. Thus a definite comparison cannot be made. GRE2015_P04.indd 129 4/18/14 11:29 AM 130 \u2022 GRUBER\u2019S COMPLETE GRE GUIDE 2015 to Make a Comparison Simpler\u2014Especially of Fractions\u2014 Multiply, Divide, add to, or Subtract from Both Columns by a Quantity (Never Multiply or Divide by Zero or by a Negative Number) Examples 1, 3, 4, 12, 13, 16, and 18 can also be solved with the aid of a calculator and some with the aid of a calculator allowing for exponential calculations. However, to illustrate the effective- ness of Math Strategy D, we did not use the calculator method of solution in these examples. Math StratEg y D example 1 Quantity A Quantity B 1 7 9 9 7 Choice A is correct. Don\u2019t divide 7 __ 9 by 9 __ 7 . Multiply both columns by 9 __ 7 to get rid of the complicated fraction in Quantity B: Quantity A Quantity B 1 3 9 __ 7 7 9 9 7 3 9 __ 7 9 __ 7 7 __ 9 Quantity A . Quantity B. example 2 Quantity A Quantity B a0\u20131 3 8 a 8 2 a Choice B is correct. Get rid of the minus sign by adding a to both columns: Quantity A Quantity B a 1 a 8 2 a 1 a 2a 8 Divide by 2: a 4 Now look at the diagram: a , 3, so a , 4. Quantity A , Quantity B. example 3 Quantity A Quantity B 19 2 3 16 Choice B is correct. First, get rid of the minus sign by adding 3 to both columns. Quantity A Quantity B 19 16 1 3 Now square both columns. Quantity A Quantity B 19 2^ h 16 3 2+^ h 19 16 1 3 1 2 16 3 19 19 1 2 16 3 Cancel the 19s: Quantity A Quantity B Quantity B . Quantity A. 0 2 16 3 For Examples 4\u20137, try to get rid of minus signs by adding. To Simplify a Problem, You Can Often Get Rid of Minus Signs Just by Adding the Same Quantity to Both Columns. You learned addition before you learned subtraction, and addition is more fundamental and basic than subtraction. Thus it would seem that it is easier and more natural to add whenever you can rather than to subtract. So try not to deal with minus signs or subtraction if the problem looks tedious\u2014try to use addition. GRE2015_P04.indd 130 4/18/14 11:29 AM STRATEGY SECTION \u2022 131 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