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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
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
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
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
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
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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
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
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