Practice Problems
Ready for some on your own? Good—have a crack at the five questions in
this set, and then thoroughly review the explanations to see how you did.
Directions: Each of the following
questions consists of two quantities, one in Column A and one in
Column B. There may be additional information, centered above
the two columns, that concerns one or both of the quantities. A
symbol that appears in both columns represents the same thing in
Column A as it does in Column B.
You are to compare the quantity in Column A with the
quantity in Column B and decide whether:
(A) The quantity in Column A is greater.
(B) The quantity in Column B is greater.
(C) The two quantities are equal.
(D) The relationship cannot be determined from the
information given.
In a question, there may be additional information,
centered above the two columns, that concerns one or both of the
quantities to be compared. A symbol that appears in both columns
represents the same thing in Column A as it does in Column B.


1. 
p <
q < r <
s < 0 

Column A

Column B


1


 (A) 
The quantity in Column A is greater. 
 (B) 
The quantity in Column B is greater. 
 (C) 
The two quantities are equal. 
 (D) 
The relationship cannot be determined from the
information given. 


2. 
Let a* = aa and 

Column A

Column B

(((3*)#)#)#

27


 (A) 
The quantity in Column A is greater. 
 (B) 
The quantity in Column B is greater. 
 (C) 
The two quantities are equal. 
 (D) 
The relationship cannot be determined from the
information given. 




Column A

Column B

The number of prime
numbers between 1 and 7,
inclusive

The number of prime
numbers between 12 and 29,
exclusive

 (A) 
The quantity in Column A is greater. 
 (B) 
The quantity in Column B is greater. 
 (C) 
The two quantities are equal. 
 (D) 
The relationship cannot be determined from the
information given. 




Column A

Column B

500^{100} –
500^{99}

499 ×
500^{99}

 (A) 
The quantity in Column A is greater. 
 (B) 
The quantity in Column B is greater. 
 (C) 
The two quantities are equal. 
 (D) 
The relationship cannot be determined from the
information given. 


5. 
On tests graded out of 100,
Nancy received an 85 on each of her first three tests and an
average of 80 on her last two tests. Alfonso received a 90
on each of his first three tests and a 100 on each of his
last two tests. 

Column A

Column B

The standard deviation
of Nancy’s five test scores

The standard deviation
of Alfonso’s five test scores


 (A) 
The quantity in Column A is greater. 
 (B) 
The quantity in Column B is greater. 
 (C) 
The two quantities are equal. 
 (D) 
The relationship cannot be determined from the
information given. 

Guided Explanations
1. A
Step 1: Get the Specs. We’ve got variables in Column A,
which means choice D remains on the table. It’s also worth
noticing the restrictions on the variables set out in the additional
information: All four variables must be negative and decrease in size from
s to p.
Step 2: Plan the Attack. FONZing seems to be a promising
route, since making up numbers for the variables might help us to compare
the expression in Column A to the number 1 in Column B. Just make sure to
obey the restrictions in the additional information.
Step 3: Mine the Math. The rules of negative
numbers come into play here: A negative × a negative = a
positive, and a negative ÷ a negative equals a positive
too. Those are really the only math concepts we’ll need, other than
some simple arithmetic, to test out some numbers.
Step 4: Make the Comparison. Ready . . . set . . . FONZ!
Since the variables must be negative, 0 and 1 are out, so let’s make
s our simplest negative number, –1, and decrease from
there:
r = –2,
q = –3, and
p = –4. This gives
. This is greater than 1 in Column B, so the answer could be
A. If we pick other negative integers, such as –2, –3, –4, and
–5, the result is the same: a positive number greater than 1. Just to be
sure, though, we’d better try some fractions, because they sometimes cause
funny things to happen. Let’s go with these:
. This maintains the relationship in
the additional information since
is closer to 0 than
, and hence
bigger, and so on down the line. Filling in these for the variables yields
the numerator
, and the denominator
. The entire
fraction is then
. Again, we get a
positive number greater than 1 and can conclude that
A is
indeed correct.
2. B
Step 1: Get the Specs. Even though this may look
intimidating, it’s nothing more than a madeup symbol problem. The
expression in Column A and the whole number in Column B suggest that we can
treat this as a disguised Problem Solving question testing whether we
understand how madeup symbol problems work.
Step 2: Plan the Attack. There are no real shortcuts to
madeupsymbol problems; we simply need to follow the rule to calculate the
quantity in Column A and then compare what we get to the number in Column B.
Step 3: Mine the Math.
Exponents and
fractions seem to be the relevant
math concepts, and
is a special
kind of fraction, called a
reciprocal. We form the reciprocal of a
number by flipping its numerator and denominator. If you’re up on
these arithmetic concepts, you should have little problem following the
rules to arrive at an answer.
Step 4: Make the Comparison. Let’s crunch through Column
A and then compare what we get to 27. The first rule, a* = aa, means that
whenever we see a number followed by an asterisk, we need to take that
number to its own power. So 3* means we need to calculate 33 = 27. Anyone
who stopped there would have been tempted to choose
C, but of
course we need to continue through the rest of the expression containing the
# signs. The second rule,
, means
that whenever we see a number followed by a #, we need to take the
reciprocal of that number. The expression 27# therefore means
, which takes care of the first #.
Performing the same for the second # brings us back to plain old 27, and
doing it for the third and final time yields
yet again as our final answer. That’s
a lot of back and forth, but if you kept things straight, and took the
reciprocal the right number of times (three), you’d find that 27 in Column B
is bigger than
in Column A. That
means the second oval (what we call choice
B) is the oval to
click.
3. C
Step 1: Get the Specs. All of the information is found in
the columns themselves, and the problem deals with prime numbers. There’s
also this “inclusive/ exclusive” business, but that’s about all we’re up
against. That means we’re basically looking at another Problem Solving
question in disguise.
Step 2: Plan the Attack. They’re not asking us to come up
with dozens of primes, so we may as well just do the work and count ‘em up.
Step 3: Mine the Math. Of course, to count them, we have
to know what a prime number is, something we covered in the Math 101
chapter. A prime number is a number that has exactly two positive
factors, 1 and itself. We also need to know that
inclusive means we are to include the outer
values of the set (if they apply), while exclusive means we
must exclude the outer values of the set (even if they apply).
Step 4: Make the Comparison. Let’s get counting,
beginning with Column A. As we just saw, the word inclusive
means we have to include the 1 and the 7, if they apply. Seven is prime, so
it counts, but 1 isn’t a prime number because it
doesn’t have two distinct factors. So the numbers in the set
defined in Column A are 2, 3, 5, and 7—four numbers total. The set defined
in Column B contains these numbers: 13, 17, 19, and 23. The word
exclusive means we have to exclude prime number 29 from
our list, so there are four numbers in this set as well. Four primes in A,
four primes in B, so the columns are equal, choice C.
4. C
Step 1: Get the Specs. No computer—let alone person—in
its right mind would be happy crunching these numbers, which means there
must be a logical way into it. There’s no additional information, and the
columns contain pure numbers, no variables, so we can chop D
from the getgo. But we’d better find a shortcut, since there’s no way we
can possibly calculate these expressions. The shortcut to employ is
suggested by the look of the quantities in the two columns. They look kind
of similar, don’t they, what with all those 500s and a couple of 99th powers
floating around? That can only mean one thing . . .
Step 2: Plan the Attack. Mirroring! These expressions
look so similar that there simply has to be a way to make them resemble one
another even further, making a comparison possible. But that’s gonna require
some math chops, which is another way to say it’s time for Step 3:
Step 3: Mine the Math. Clearly exponents is the name of
the game, and here’s the main thing you need to know about them to get over
the hump: x^{
y
} =
(x)(x^{y
– 1}). For example, since
x^{10} is the same as x
multiplied by itself 10 times, it’s also the same as
(x) × (x multiplied by itself 9 times).
Either way, you get x times itself 10 times. Why does this
help us, you ask? Here’s why:
Step 4: Make the Comparison.
500^{100} = (500)(500^{99}).
Substituting that for 500100 in Column A gives us:
(500)(500^{99}) –
500^{99}
Factoring out the common 500^{99} term results
in:
(500^{99})(500 – 1)
which equals:
(500^{99})(499)
That’s the same as Column B, even though the terms are reversed.
(The order of terms doesn’t matter for multiplication: A × B = B ×
A.) The quantities are equal, so C is correct.
5. D
Step 1: Get the Specs. Next up is a standard deviation
question. As we discussed in our Math 101 chapter, you’ll never be asked to
actually calculate standard deviation since the formula is too complex to be
tested on the GRE. We have to assume therefore that the question is merely
testing our understanding of the mechanics—that is, the
principle behind it. We’ll need to mine our reservoir of math concepts for
that principle, but first let’s get our plan of attack in place.
Step 2: Plan the Attack. We’ll use the numbers given but
not expect to calculate the actual standard deviations for the two columns.
Instead, we’ll rely on math logic to determine the relationship between
them. First we have to know what standard deviation is, and that’s the
purpose of Step 3.
Step 3: Mine the Math.
Standard deviation is a measure of the spread of a group of
numbers. In other words, a group whose numbers are all close to the
same value has a smaller standard deviation than a group whose numbers
differ widely from one another.
Step 4: Make the Comparison. Based on the definition in
Step 3, our task is to figure out which set of numbers is more spread out.
Alfonso’s performance is better defined: We know that his test scores are
90, 90, 90, 100, and 100. That gives us a good sense of the spread of his
scores, but we’ll need to compare that to the spread of Nancy’s scores to
get our answer. Her scores are less defined. While we know she definitely
received 85, 85, and 85 on her first three tests, we don’t know exactly what
she got on the last two; all we know is that they average out to 80.
Now, anyone who assumed that means her last two scores were 80 and 80
would be likely to choose B as the answer, since the spread of
85, 85, 85, 80, and 80 is less than the spread of 90, 90, 90, 100, and 100.
That’s because 80 is closer to 85 than 100 is to 90. The problem is, we
don’t know that Nancy’s final two scores are 80 and 80. They could be, in
which case her standard deviation would be smaller than Alfonso’s, and
B would be correct. But we have to use some cleverness and
ask ourselves how else her final two scores could possibly average out to
80. The other extreme, given that the tests are out of 100, is that she got
100 on one test and 60 on the other. Those numbers also average out to 80,
keeping with the requirement in the additional information, but change her
test score distribution to 85, 85, 85, 100, and 60. Now Nancy’s last two
scores differ from her previous scores by 15 and 25 points, respectively.
This is greater than the 10point spread in Alfonso’s scores, so in this
scenario Nancy has the greater standard deviation. The answer must be
D, since we get two different results from two different
but equally plausible scenarios.
So there you have it: the wonderful world of Quantitative Comparisons,
in a bit more than a nutshell. Sure, we covered a lot of ground in this
chapter, but the fundamentals and step method will sink in as you get more
practice with this question type. A little extra work on QCs is warranted
since QCs make up the majority of GRE math questions: approximately 14
compared to 10 Problem Solving and only 4 Data Interpretation questions. It
is to this final math question type, Data Interpretation, that we turn our
attention now.