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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2y
y y22 + 1
Factor and reduce:
Factor the numerator of 1 . We get
 
2
( 5)( 2)
5
2
2 2
2
y
y y
y= 2
Substitute 8.000001 in 2 . We have
 8.000001 2 5 5
 3.000001 \u2248 3 (Answer)
GRE2015_P04.indd 122 4/18/14 11:29 AM
STRATEGY SECTION \u2022 123
Introduction
In the quantitative comparison question, you are presented with quantities under two columns. 
You have to determine whether the Quantity A is less than, greater than, or equal to the 
Quantity B. There is also a possibility that a definite comparison cannot be made. In very, very 
few cases will you ever have to calculate quantities. Since you just have to compare the relative 
values, you can usually manipulate the columns to get a simple comparison, as you will see.
6 Quantitative Comparison 
Strategies
Things You Must Know First
 1. Memorize directions for quantitative comparison 
questions. The following reword ing of the standard 
directions given on the GRE tells you precisely how 
to choose an answer:
	 \u2022	 If	Quantity	A	is	always greater than Quantity B (or 
if Quantity B is always less than Quantity A), select 
Choice A.
	 \u2022	 If	 Quantity	 A	 is	 always less than Quantity B (or 
if Quantity B is always greater than Quantity A), 
select Choice B.
	 \u2022	 If	Quantity	A	is	always equal to Quantity B, select 
Choice C.
	 \u2022	 If	there	is	no	way	to	make	any	of	the	above	definite	
comparisons, select Choice D. (In other words, 
if a definite comparison or relationship cannot be 
made, select Choice D.)
Note: If there is a quantity centered between the columns, 
you may use that information in the columns. A symbol 
such as x, a, etc., means the same thing in both columns. 
All variables like x, n, a, represent real numbers. Make 
sure that you memorize these direc tions.
example
 Quantity A Quantity B
 1 2
1
In this example you would select Choice A, since 
Quantity A . Quantity B.
 2. When there\u2019s a quantity between the columns, use 
the middle quantity in both columns.
example
 Quantity A Quantity B
x 5 2
 x x 1 1
 In this example, 
 substitute 2 for x:
 2 (2 1 1)
 Quantity A , Quantity B, so Choice B is correct.
 3. You can add or subtract the same quantity to 
(or from) both columns and still get the same 
comparison.
example
 Quantity A Quantity B
 8 2 3 7 (Quantity A , 
 Quantity B)
 Add 3
 to both
 columns: 8 7 1 3 (Quantity A ,
 Quantity B)
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124 \u2022 GRUBER\u2019S COMPLETE GRE GUIDE 2015
 4. You can multiply or divide both col umns by the same 
positive number and still get the same comparison.
example
 Quantity A Quantity B
 
4
7 
4
9 (Quantity A , 
 Quantity B)
 Multiply
 by 4: 7 9 (Quantity A ,
 Quantity B)
 5. REMEMBER THE FOLLOWING:
	 \u2022	 Any	variable	like	x, a, etc., can be negative, positive, 
0, or a fraction unless otherwise specified.
	 \u2022	 If	 you	 get	 different	 comparisons	 when	 you	 try	
different num bers (for example, if you were to find 
that in one case Quantity A . Quantity B and in 
another Quantity A , Quantity B), then Choice D 
is correct.
	 \u2022	 Never	 divide	 or	 multiply	 columns	 by	 a	 negative	
num ber or by 0.
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
Cancel Numbers or Expressions Common to Both Columns 
by addition or Subtraction
If the same expression or number appears in both columns, we can then subtract it from 
both columns.
Examples 1, 4, and 5 can also be solved with the aid of a calculator. However, to illustrate 
the effectiveness of Math Strategy A, we did not use the calculator method of solution in 
these examples.
Math 
StratEg
y a
example 1
 Quantity A Quantity B
 2
1 1 6
1 1 17
1 17
1 1 2
1 1 7
1
Choice A is correct.
Don\u2019t add fractions in columns! Cancel common 
quantities:
 Quantity A Quantity B
 2
1 1 6
1 1 17
1 17
1 1 2
1 1 7
1
 6
1 7
1
Quantity A . Quantity B.
example 2
 Quantity A Quantity B
y . 0
 w 1 x w 1 x 1 y
Choice B is correct. Since w 1 x appears in both columns, 
we can subtract w 1 x from both columns to get
 Quantity A Quantity B
 0 y
and from the given information we know that y . 0.
example 3
 Quantity A Quantity B
y is an integer 
y , 0
 21 y
1 2 1
Choice A is correct.
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STRATEGY SECTION \u2022 125
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
 21 y
1 2 1
Cancel 21: 21 y
1 2 1
 \u2193 \u2193
 0 y
1
Since y , 0, y
1 , 0, and Quantity A . Quantity B.
example 4
 Quantity A Quantity B
 5 2 0.005 5 2 0.0055
Choice A is correct.
 Quantity A Quantity B
 5 2 0.005 5 2 0.0055
 Cancel 5:
 5 2 0.005 5 2 0.0055
 2 0.0050 2 0.0055
 2 0.0050 . 2 0.0055
example 5
 Quantity A Quantity B
 a # 15 
 7 1 3 1 15 7 1 3 1 a
Choice D is correct.
 Quantity A Quantity B
 a # 15 
 7 1 3 1 15 7 1 3 1 a
 Cancel 7 1 3:
 7 1 31 15 7 1 3 1 a
 15 a
Since 15 $ a (given), you can\u2019t make a definite 
comparison.
example 6
 Quantity A Quantity B
 
l1
l3
a°
b°l2
l1 \ue069 l2; l3 intersects l1 and l2
 a 1 a 1 b a 1 2b
Choice C is correct.
 Quantity A Quantity B
 a 1 a 1 b a 1 2b
 Subtract a, b from
 both columns:
 a 1 a 1 b a 1 2b
 2 a 2 b 2 a 2 b
 \u2193 \u2193
 a b
Now a 5 b because if l1 \ue069 l2, alternate interior angles a 
and b are equal.
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126 \u2022 GRUBER\u2019S COMPLETE GRE GUIDE 2015
Cancel Numbers or Expressions (Positive Quantities Only!!) 
Common to Both Columns by Multiplication or Division
If the same expression or number (positive quantities only which may be multiplied by other 
expressions) appears in both columns, we can then divide it from both columns. NEVER divide 
both columns by zero or a negative number.
Examples 1 and 3 can also be solved with the aid of a calculator. However, to illustrate 
the effectiveness of Math Strategy B, we did not use the calculator method of solution in 
these examples.
Math 
StratEg
y B
example 1
 Quantity A Quantity B
 24 3 46 3 35 46 3 24 3 36
Choice B is correct.
Don\u2019t multiply out! Cancel 24 3 46 from both columns 
(by dividing both columns by 24 3 46).
 Quantity A Quantity B
 24 3 46 3 35 46 3 24 3 36
Quantity A , Quantity B.
example 2
 Quantity A Quantity B
m . 1
n . 0
 mn n
Choice A is correct.
Since n . 0 and n appears in both columns, we can 
divide it from both columns to get
 Quantity A Quantity B
 m 1
and we are given that m . 1.
example 3
 Quantity A Quantity B
 14
3 3 7
5 3 3
2 7
5 3 14
3 3 4
3
Choice B is correct.
 Quantity A Quantity B
 14
3 3 7
5 3 3
2 7
5 3 14
3 3 4
3
 Cancel common fractions:
 14
3 3 7
5 3 3
2 7
5 3 14
3 3 4
3
 3
2 4
3
 3
2 , 4
3
example 4
 Quantity A Quantity B
 
A B
D C
h
60°
 ABCD is a parallelogram
 h 3 AB AD 3 CD
Choice B is correct.
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STRATEGY SECTION \u2022 127
When a Comparison of the two Columns Is Difficult, Use 
Numbers in Place of Variables
Sometimes by using numbers in place of variables, you can show that different comparisons 
exist, making choice D correct.
Math 
StratEg
y C
example 1
 Quantity A Quantity B
 b . 1 
 a . 1 
 a fi b 
 
b
a a