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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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
explanatory answers for examples 4\u20137
 Quantity A Quantity B
4. (A) 12
7 2 14
1 14
6
 Add 14
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
 Add 3b to both columns:
 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) 
 Add a to both columns; then add b to both 
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+
explanatory answers for examples 8\u201314
 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
explanatory answers for examples 16\u201318
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
explanatory answers for examples 20\u201322
 Quantity A Quantity B
20. (D) c . b . a 
 bc ab
 Don\u2019t cancel b since b may be negative. 
 Suppose b 5 0. Then the columns are