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

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```examples 4\u20137
Quantity A Quantity B
4. 12
7 2 14
1 14
6
5. 0 . a . b
23b a

a b\u20132 \u20131 +1 +206.
b 2 a a 2 b

a c
d
b
140°
40°
Note: Figure is not drawn to scale.
7. a2 2 c2 d2 2 b2
Quantity A Quantity B
4. (A) 12
7 2 14
1 14
6
1 to both columns to get rid of the
minus sign:
12
7 2 14
1 1 14
1 14
6 1 14
1
12
7 2 14
1 1 14
1
14
7
12
7 .
14
7
Quantity A Quantity B
5. (A) 0 . a . b
23b a
23b 1 3b a 1 3b
\u2193 \u2193
0 a 1 3b
Since 0 . a and 0 . b, a 1 3b is negative. So
Quantity A . Quantity B.
Quantity A Quantity B
a b\u20132 \u20131 +1 +20
b 2 a a 2 b
6. (A)
columns to get rid of minus signs:
b 2 a a 2 b
Add a: b 2 a 1 a a 2 b 1 a
b 2a 2 b
Add b: b 1 b 2a 2 b 1 b
2b 2a
Divide by 2: b a
From diagram: b . a
7. (C)
a c
d
b
140°
40°x°
Note: Figure is not drawn to scale.
The sum of the internal angles of a quadrilateral is
360 degrees. Find remaining angle:
140 1 40 1 90 1 x 5 360
x 5 90
Now draw a line and label it h.
a c
d
b
h
140°
40°
Note: Figure is not drawn to scale.
By the Pythagorean Theorem,
a2 1 b2 5 h2
d2 1 c2 5 h2
so a2 1 b2 5 d2 1 c2 1
Add c2 and b2 to both columns:
GRE2015_P04.indd 131 4/18/14 11:29 AM
132 \u2022 GRUBER\u2019S COMPLETE GRE GUIDE 2015
Quantity A Quantity B
a2 2 c2 d2 2 b2
Add c2: a2 d2 2 b2 1 c2
Add b2: a2 1 b2 d2 1 c2
The columns are therefore equal because of 1 .
For Examples 8\u201314, multiply or divide to simplify the
problem, but never multiply by 0 or by a negative
number.
Often You Can Multiply or Divide Each Quantity
by the Same Number to Simplify the Problem and
Avoid Tedious Calculations.
examples 8\u201314
Quantity A Quantity B
8. a fi 0
0 a
5
2
9. a . 0
2a
.
a
0 4
10. a . 0
a
1 a
11. a . b . 0

a b
a b2 2
-
+ a 2 b
12. 7 3 812 8 3 712
13. 35 3 65 34 3 66
14. a . b . 0

a b
ab2
+

2
a b+
Quantity A Quantity B
8. (B) a fi 0
0 a
5
2
Since a2 . 0 (even if a is negative, a2 . 0) you
can multiply by a2:
0 a
5
2
0 3 a2 a
5
2 3 a2
0 5
0 , 5
Quantity A Quantity B
9. (B) a . 0
2a .
a
0 4
Multiply by 0.4 to get rid of the denominator:
(2a)(0.4) .
a
0 4 0.4
\u2193 \u2193
0.8a a
Since a . 0, divide by a:
.a
a0 8 a
a
\u2193 \u2193
0.8 1
0.8 , 1
Quantity A Quantity B
10. (D) a . 0
a
1 a
Since a . 0, multiply both columns by a:
a . 0
a
1 3 a a 3 a
1 a2
Now you have to be careful: If a . 1, certainly a2 . 1.
But if a is a fraction such as 4
1 , then a2 5 16
1 , 1, so a
definite comparison cannot be made. This is because in
one case, a2 . 1, and in another case, a2 , 1.
GRE2015_P04.indd 132 4/18/14 11:29 AM
STRATEGY SECTION \u2022 133
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
11. (A) a . b . 0

a b
a b2 2
-
+ a 2 b
Since a . b, a 2 b . 0,
So we can multiply both columns by a 2 b:

a b
a b a b
2 2
#
-
+ - (a 2 b)(a 2 b)
a2 1 b2 a2 2 2ab 1 b2
Cancel a2 1 b2 from both columns:
0 22ab
Since a . b . 0, 2ab . 0 and 22ab , 0. Thus
Quantity A . Quantity B.
Quantity A Quantity B
12. (A) 7 3 812 8 3 712
Use logic. It\u2019s obviously too hard to calculate
7 3 812 and 8 3 712. So let\u2019s try to take advantage
of the curious form of the numbers. Divide both
columns by 7 and then divide by 8:
7 7
83 12 7
8 73 12
812 8 3 711
Now divide by 8:
8
812
8
8 73 11
811 711
Quantity A . Quantity B.
Quantity A Quantity B
35 3 65 34 3 66
13. (A) You do not have to multiply 35 3 65 and
34 3 66! Note the relationship of the numbers in the
columns: That is, between 35 and 34 and 65 and
66. Divide 34 into the quantities in both columns
and divide 65 into the quantities in both columns.
We get
Quantity A Quantity B

34 65
35 65
3
3
34 65
34 66
3
3
Simplified, this becomes
Quantity A Quantity B

34
35
65
66
Now
34
35 is 1 34
1 and
65
66 is 1 65
1 . Since 34
1 . 65
1 , the
quantity in Quantity A is greater than the quantity
in Quantity B, and Choice A is correct.
14. (A) Quantity A Quantity B
a . b . 0
a b
ab2
+
a b2
1
Multiply both columns by a 1 b and by 2 to get rid
of fractions:
Quantity A Quantity B
2( )a b a b
ab21
1
3 2( )a b1 a b2
1
\u2193 \u2193
2( )a b a b
ab21
1
3 2( )( )a b a b2
1 1
\u2193 \u2193
4ab (a 1 b)(a 1 b)
\u2193 \u2193
4ab a2 1 2ab 1 b2
\u2193 \u2193
Subtract
4ab: 4ab 2 4ab a2 1 2ab 1 b2 2 4ab
\u2193 \u2193
0 a2 2 2ab 1 b2
\u2193 \u2193
0 (a 2 b)(a 2 b)
Since a . b . 0, (a 2 b)(a 2 b) . 0
So Quantity B > Quantity A.
For Examples 15\u201318, square both columns to get rid of
square roots.
GRE2015_P04.indd 133 4/18/14 11:29 AM
134 \u2022 GRUBER\u2019S COMPLETE GRE GUIDE 2015
In Comparing Square Roots, Instead of Calculating
or Substituting Numbers, You Can Usually Just
Square the Square Roots to Get Rid of Square
Root Signs.
example 15
Quantity A Quantity B
a . 0
a 31 3a1
Choice A is correct.
Square both columns:
Quantity A Quantity B
a 31 2^ h 3a1 2^ h
\u2193 \u2193
3 2a a 31 1 a 1 3
Cancel a 1 3:
3 2a a 31 1 a 1 3
\u2193 \u2193
2 a 3 0
2 a 3 . 0
examples 16\u201318
Quantity A Quantity B
16. 19 6 131
17. x . y . 0
x y1 x y1
18. 17 32 14
16. (B) Quantity A Quantity B
19 6 131
Square both columns:
19 3 19 6 13 6 131 1^ ^h h
19 6 13 2 13 61 1
Cancel 19:
0 , 2 13 6
Quantity A Quantity B
x . y . 0
17. (B) x y1 x y1
Square both columns:
x y x y1 1^ ^h h x y x y1 1^ ^h h
x 1 y 2x y x y1 1
Cancel common x 1 y from both columns:
0 x y2
Quantity A , Quantity B.
Quantity A Quantity B
18. (B) 17 32 14
First, add 3 to both columns to get rid of 32^ h
from Quantity A. (It is usually easier to work
with 1 than with 2.)
17 3 32 1 14 31
Now square both columns:
17 14 31 2^ h
17 14 3 2 14 31 1
17 , 1 27 14 31
For Examples 19\u201322, make sure you don\u2019t divide by 0 or
by a negative number.
When Canceling a Quantity from Both Columns,
Make Sure You Are Not Actually Dividing by 0 or
by a Negative Number.
example 19
Quantity A Quantity B
0 . a
b . 2
b
a ab
Choice A is correct.
Don\u2019t divide by a, because a , 0. But you can
multiply by b:
Quantity A Quantity B
b
a 3 b ab 3 b
Now since a is negative and b . 2, you can see
that a . ab2, and so Quantity A is greater than
Quantity B.
examples 20\u201322
Quantity A Quantity B
20. bc
c . b . a
ab
21. ab
0 . a . b
b
22. a2
a , 1
a
GRE2015_P04.indd 134 4/18/14 11:29 AM
STRATEGY SECTION \u2022 135
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