Gruber's Complete GRE Guide 2015   Gruber, Gary R
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Gruber's Complete GRE Guide 2015 Gruber, Gary R


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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.
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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.
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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.
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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