Example #1 (easy)
If the remainder is 7 when positive integer n is divided by 18, what is the
remainder when n is divided by 6?


Example #2 (easy)
If n is a prime number greater than 3, what is the remainder when n^2 is
divided by 12 ?


Example #3 (easy)
What is the tens digit of positive integer x ?
(1) x divided by 100 has a remainder of 30.
(2) x divided by 110 has a remainder of 30.
(1) x divided by 100 has a remainder of 30. We have that : 30, 130, 230,
... as you can see every such number has 3 as the tens digit. Sufficient.
(2) x divided by 110 has a remainder of 30. We have that : 30, 140, 250,
360, ... so, there are more than 1 value of the tens digit possible. Not sufficient.


Example #5 (medium)
If s and t are positive integers such that s/t = 64.12, which of the following
could be the remainder when s is divided by t ?


Example #6 (medium)
Positive integer n leaves a remainder of 4 after division by 6 and a remainder of
3 after division by 5. If n is greater than 30, what is the remainder that n leaves
after division by 30?


Example #8 (hard)
When 51^25 is divided by 13, the remainder obtained is:


Example #9 (hard)
When positive integer x is divided by 5, the remainder is 3; and when x is
divided by 7, the remainder is 4. When positive integer y is divided by 5, the
remainder is 3; and when y is divided by 7, the remainder is 4. If x > y, which of
the following must be a factor of x - y?


Example #10 (hard)
If p, x, and y are positive integers, y is odd, and p = x^2 + y^2, is x divisible by
4?
(1) When p is divided by 8, the remainder is 5
(2) x – y = 3
(1) When p is divided by 8, the remainder is 5. This implies that .
Since given that , then -->
.
So, . Now, if then
 and if then
, so in any case
 --> --> in order to be multiple of 4 must be
multiple of 16 but as we see it's not, so is not multiple of 4. Sufficient.
(2) x – y = 3 --> --> but not sufficient to say whether it's
multiple of 4.


GMAT Club Math Book
Algebra
https://gmatclub.com/forum/algebra-101576.html
 Algebra 101
Scope
Manipulation of various algebraic expressions
Equations in 1 & more variables
Dealing with non-linear equations
Algebraic identities
Notation & Assumptions
In this document, lower case roman alphabets will be used to denote variables such as a,b,c,x,y,z,w
In general it is assumed that the GMAT will only deal with real numbers ( ) or subsets of such as Integers ( ),
rational numbers ( ) etc
Concept of variables
A variable is a place holder, which can be used in mathematical expressions. They are most often used for two
purposes :
(a) In Algebraic Equations : To represent unknown quantities in known relationships. For eg : "Mary's age is 10
more than twice that of Jim's", we can represent the unknown "Mary's age" by x and "Jim's age" by y and then
the known relationship is
(b) In Algebraic Identities : These are generalized relationships such as , which says for any
number, if you square it and take the root, you get the absolute value back. So the variable acts like a true
placeholder, which may be replaced by any number.
Basic rules of manipulation

When switching terms from one side to the other in an algebraic expression + becomes - and vice versa.

When switching terms from one side to the other in an algebraic expression * becomes / and vice versa.

you can add/subtract/multiply/divide both sides by the same amount. Eg.

you can take to the exponent or bring from the exponent as long as the base is the same.


b) --> . --> . x e { , } and both solutions satisfy the
condition.
(Optional) The following illustration may help you understand how to open modulus at different conditions.


Example 4
For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by
(-1)^(k+1) *(1/2^k). If T is the sum of the first 10 terms in the sequence then T is
a)greater than 2
b)between 1 and 2
c)between 1/2 and 1
d)between 1/4 and 1/2
e)less than 1/4.
Solution
The sequence given has first term 1/2 and each subsequent term can be obtained by
multiplying with -1/2. So it is a GP. We can use the GP summation formula
1023/1024 is very close to 1, so this sum is very close to 1/3
Answer is d
Example 5
The sum of the fourth and twelfth term of an arithmetic progression is 20. What is the
sum of the first 15 terms of the arithmetic progression?


The ‘how much is left’ construction
Sometimes, the question will give you a total amount that is made up of a number of
smaller amounts of unspecified sizes. In this case, just assign a variable to the unknown
amounts and the remaining amount will be what is left after deducting this named amount
from the total.
Consider the following:
A hundred-pound order of animal feed was filled by mixing products from Bins A, B and C,
and that twice as much was added from Bin C as from Bin

Then the amount from Bin C was "2a", and the


calculate the rate at which B works: Rate at which B works = .
‘B prints 5 pages a minute more than printer A’ This means that the difference between
the amount of work B and A complete in one minute corresponds to 5 pages. So, let us
calculate that difference. It will be
‘How many pages does the task contain?’ If of the job consists of 5 pages, then the 1
job will consist of pages.
Example 4.
Machine A and Machine B are used to manufacture 660 sprockets. It takes machine A ten
hours longer to produce 660 sprockets than machine

How many sprockets per hour does machine A


Question:: how many students play exactly two of these sports?
Apply first formula:
{Total}={Hockey}+{Cricket}+{Football}-{HC+CH+HF}+{All three}+{Neither}
50=20+15+11-(7+4+5)+{All three}+18 --> {All three}=2;
Those who play ONLY Hockey and Cricket are 7-2=5;
Those who play ONLY Cricket and Football are 4-2=2;
Those who play ONLY Hockey and Football are 5-2=3;
Hence, 5+2+3=10 students play exactly two of these sports.


There are 6 equally possible outcomes when we roll a die. The probability of getting any
number out of 1-6 is .
All possibilities: {1,2,3,4,5,6}
Marbles, Balls, Cards...
Let's assume we have a jar with 10 green and 90 white marbles. If we randomly choose a
marble, what is the probability of getting a green marble?
The number of all marbles: N = 10 + 90 =100
The number of green marbles: n = 10
Probability of getting a green marble:
There is one important concept in problems with marbles/cards/balls. When the first
marble is removed from a jar and not replaced, the probability for the second marble
differs ( vs. ). Whereas in case of a coin or dice the probabilities are always the
same ( and ). Usually, a problem explicitly states: it is a problem
with replacement or
without replacement.
Independent events
Two events are independent if occurrence of one event does not influence occurrence of
other events. For n independent events the probability is the product of all probabilities of
independent events:
p = p1 * p2 * ... * pn-1 * pn
or
P(A and B) = P


Example #3
Q:There are two sets of integers: {1,3,6,7,8} and {3,5,2}. If Robert chooses randomly
one integer from the first set and one integer from the second set, what is the probability
of getting two odd integers?


Example #4
Q: There is a set A of 19 integers with mean 4 and standard deviation of 3. Now we form
a new set B by adding 2 more elements to the set


Try-It #0-1
A jar is filled with red, white, and blue tokens that are equivalent
except for their color. The chance of randomly selecting a red
token, replacing it, then randomly selecting a white token is the
same as the chance of randomly selecting a blue token. If the
number of tokens of every color is a multiple of 3, what is the
smallest possible total number of tokens in the jar?
 9


Try-It #0-3
In the diagram to the right, the value
of x is closest to which of the
following?
21


In the very first equation above, each fraction on the
le is less than 1, so their product is even smaller.
The denominators of the three fractions are all the
same.
So the numerator of the product

are out now. The smallest possible total is


A jar is filled with red, white, and blue
tokens …
BU is alert—what about 0?
TD: “What about 0? Hmm…the wording at the
beginning assumes that there actually are tokens
of each color. So there can’t be 0 tokens of any
kind.”


Select a white:
  , which is not “select a
blue”


Try-It #0-2
Arrow
 , which is a line segment exactly 5 units long with an
arrowhead at A, is constructed in the xy-plane. The x- and y-
coordinates of A and B are integers that satisfy the inequalities 0 ≤
x ≤ 9 and 0 ≤ y ≤ 9. How many different arrows with these
properties can be constructed?


There are 10 identical columns: x = 0
through x = 9. 5 × 10 = 50 possible positions
for the arrow pointing straight up.
50 × 2 = 100 possible positions for the
arrow if it points straight up or to the right.
TD: “Great. I’ve Solved one part. Other
possibilities?”
BU notices the square is the same vertically as
horizontally. Go right.
TD: “I get the same result for arrows pointing
right. 50 more positions. Is that it? Am I done?”
50 × 4 = 200 possible positions
BU wonders about “down” and “le .”
TD: “These arrows can point straight down or
straight le , too. Those would have the same
result. So there are 50 positions in each of the
four directions. Calculate at this point and
evaluate answers. Eliminate


Try-It #0-3
In the diagram to the right, the value of x is closest to which of the
following?


BU has no idea how to take the square root of
this.
TD: “Neither do I. Let’s try estimating. If x2 is
about 3.5, then the square root must be a bit
less than 2 (since the square root of 4 is 2). 182 is
324 and 192 is 361, so the answer is around 1.8
or 1.9.”
TD: “Answer


Alternatively, the question stem asks for an approximate answer, so you
can also try estimating from the start. Draw the triangle carefully and start
with the same perpendicular line as before. This line is a little shorter than
the side of length
 (which is about 1.4). Call the two shorter legs 1.2
and calculate the hypotenuse. It equals 1.2 multiplied by 1.4, or
approximately 1.7. (Bonus question: How can you estimate that math
quickly? Answer below.)
Now, examine the answer choices using 1.4 for
 and 1.7 for
 :
They’re all close, but you can pretty confidently eliminate answers


Try-It #1-1
x = 910 − 317 and
 is an integer. If n is a positive integer that has
exactly two factors, how many different values for n are possible?
Glance. This is a PS problem. The answers are numbers but in written form;
this format is reserved for problems that ask for the number of numbers or
number of possibilities for something. The numbers are small.
Read. Dive into the text. Here are some possible answers to the Pólya
questions:
What
exactly is
the
problem
asking
for?
The number of possible values for n.
This means that n might have multiple possible values. In fact, it probably can
take on more than one value.
I may not need these actual values. I just need to count them.
What are
the
quantities
I care
about?
I’m given x and n as variables. These are the quantities I care about.
One


Try-It #1-2
 for all integer values of n greater than 1. If S1 = 1,
what is the sum of the first 61 terms in the sequence?
Nobody is going to write out all 61 terms and then add them up in two minutes.
There must be a pattern. The recursive definition of Sn doesn’t yield any secrets
upon first glance. So write out the early cases in the sequence, starting at n = 1
and looking for a pattern:
−48


Try-It #1-3
Truck A is on a straight highway heading due south at the same time
Truck B is on a different straight highway heading due east. At 1:00
p.m., Truck A is exactly 14 miles north of Truck B. If both trucks are
traveling at a constant speed of 30 miles per hour, at which of the
following times will they be exactly 10 miles apart?
1:10 p.m.


Try-It #1-4
If x and y are positive integers and
 is the square of an odd
integer, what is the smallest possible value of xy ?
As you read, jot down the given information.
Note that you might not immediately write down the square of an odd integer
info if you still have to puzzle out what it means:
What does the square of an odd integer look like? List out a few examples, on
paper or in your head:
 1


4. A rectangular solid is changed such that the width and length are
each increased by 1 inch and the height is decreased by 9 inches.
Despite these changes, the new rectangular solid has the same
volume as the original rectangular solid. If the width and length of
the original rectangular solid are equal and the height of the new
rectangular solid is 4 times the width of the original rectangular
solid, what is the volume of the rectangular solid?
  18


Andra, Elif, and Grady each invested in a certain stock. Andra
invested q dollars, which was 40% more than Elif invested. If Elif
invested 25% less than Grady invested, what was the total amount
invested by all three, in terms of q ?
This problem can be solved algebraically: write a couple of equations and
solve for all three variables, then add them up. A glance at the answers,
though, indicates that the algebra is likely to get messy. Instead, choose a real
number and solve the problem arithmetically.
If you’ve made it to the GMAT Advanced Quant book, then you have likely
used this strategy before (or at least learned about it). At times, you may have
been frustrated because this technique didn’t actually seem easier than doing
the math algebraically. If so, here’s the missing piece: you need to learn how
to choose smart numbers in the best possible way.
Most of the time, you’re going to choose for the variable given (in this case, q).
In some cases, though, starting with the given variable doesn’t make your life
any easier. The problem above actually has three unknowns: one for Andra,
one for Elif, and one for Grady. Take a look at the relationship between those
unknowns before you decide which one is the best starting point.
2q


Andra invests 40% more than Elif. For just these two, it would be easier to pick
a number for Elif and then calculate Andra’s amount.
Elif invests 25% less than Grady. For these two, it is easier to start with Grady
and then calculate Elif. As a result, start with Grady, then find Elif, then find
Andra.
If Grady invests $100, then Elif invests 25% less, or $75. Andra invests 40%
more than Elif, or $75 + $30 = $105. Make sure to note on your scrap paper
that q = 105.
Collectively, the three invest $100 + $75 + $105 = $280. Find the answer choice
that matches $280:
The correct answer is


Try-It #2-2
If
 , which of these integers could be the value of z ?
Look at all of those fractions! Solving for z algebraically in this problem would
not be easy. Instead, notice two important clues: the problem asks for the
value of a single variable and the answer choices oﬀer nice-and-easy integers.
Work backwards! People o en start at the beginning, with choice


Try-It #2-3
A certain college party is attended by both male and female students.
The ratio of male-to-female students is 3 to 5. If 5 of the male
students were to leave the party, the ratio would change to 1 to 2.
How many total students are at the party?
Of course, you could set up equations for the unknowns in the problem and
solve them algebraically. However, the numbers in the answers are pretty
straightforward integers. Try Working Backwards.
Again, begin with answer


3. TEST CASES
In the original ratio, you can use an unknown multiplier to represent
the total number of students: 3x + 5x = 8x. If there are 30 students
total, then the unknown multiplier is
 = 3.75, leaving you with a
non-integer number of students. This is impossible, so this answer
must be incorrect.

If there are 24 total students, then the unknown multiplier is 3: there

If there are 80 students total, then the unknown multiplier is 10, so


Some problems allow you to choose real numbers to solve, but you can’t
choose just one set of numbers as you do when you choose smart numbers.
Rather, you have to test multiple scenarios to get yourself to the one correct


n n3 − n
Case 1 1 0
Try your next case, ignoring answers

Yes, 0 is a multiple of 6. (0 is a multiple of any number.)

Yes, 0 does not have more than two prime factors.

Yes, 6 does not have more than two distinct prime factors. (The prime


n  n3 − n
Case 3 5 120
The correct answer is

No, 120 does have more than two distinct prime factors. (The prime


Try-It #2-5
The percent change from 29 to 43 is approximately what percent of
the percent change from 43 to 57 ?
In this case, the question stem straight up tells you that you can estimate. Any
time you see the word approximately (or a synonym), definitely do not try to
solve for the exact


Try-It #2-6
 is equivalent to which of the following?
At first glance, it appears that precise long division is necessary. The answer
choices are very close together, making estimation diﬀicult. However, with
some manipulation and factoring, the solution is much more straightforward.
The key to factoring this fraction is to move the decimals of both the
numerator and denominator three places to the right so that you’re dealing
with integers. Then you might notice that 3,507 is divisible by 7 (3,500 and 7
are both divisible by 7, providing a clue that you may be able to factor out 7).
Moreover, 10,020 is divisible by 10 and by 2 (10,020 ends in a 0, and 1,002 is
even):
The correct answer is


Alternatively, you might observe that 10.02 is very slightly larger than 10.
Therefore, the correct answer will be slightly smaller than
 Guess between choices


Try-It #2-7
What is the value of
Since 102 is extremely small compared to 105, and the choices are somewhat
spread out, estimate:
Notice that by ignoring the 102 term, you made the denominator slightly
larger than it originally was. Therefore, 0.81918 is slightly smaller than the
correct answer, 0.82. The correct answer is


Try-It #2-8
A shoe cobbler charges n dollars to repair a single pair of loafers.
Tomorrow, he intends to earn $240 repairing loafers. If he were to
reduce his fee per pair by $20, he would have to repair an additional
pair of the loafers to earn the same amount of revenue. How many
pairs of loafers does he intend to repair tomorrow?
The problem may not seem too bad . . . until you try to set it up algebraically.
Assign x to represent the number of pairs of loafers the cobbler intends to
repair tomorrow. Using the equation for revenue gives you the following:
Furthermore, reducing his fee by $20 would result in the need to repair an
additional pair of shoes for the same amount of revenue gives you the
following:
1


Try-It #2-9
At a certain high school, the junior class is twice the size of the
senior class. If
 of the seniors and
 of the juniors study
Japanese, what fraction of the students in both classes do not study
Japanese?
Note that two pairs of answers each add up to 1:
 , and
. Answer


This strategy can sometimes be used in combination with an answer pairs
strategy, as pairs of answers are o en composed of a high and a low value.
In the previous problem, for example,
 of the seniors and
 of the
juniors study Japanese. Therefore, somewhere between
   and
 of the
students overall, or less than half, must study Japanese. This implies that
the fraction of students who do not study Japanese must be more than
half. You could eliminate answer choices


Try-It #2-10
The eSoroban device is available in two colors, orange and green.
In 2013, 60% of the eSoroban devices sold were purchased by
women,
 of whom purchased the orange device. If an equal
number of orange and green eSoroban devices were sold in 2013,
what fraction of men who purchased an eSoroban in
2013 purchased the green device?


Try-It #2-11
If x □ y is defined to equal
 for all x and y, then (−1 □ 2) □ 3 is
equivalent to which of the following?


Try-It #2-12
In the diagram to the right, equilateral
triangle ADE is drawn inside square
BCDE. A circle is then inscribed inside
triangle ADE. What is the ratio of the area
of the circle to the area of the square?
If a Problem Solving question does not say that the diagram is not drawn to
scale, then the diagram is drawn to scale. The circle is about
 the height
of the square and about
 the width. Therefore, the area of the circle
should be approximately
 of the area of the square. Eliminate any answer
choices that are far away from that estimate:
Ok
On the high side
Too high


2. If x < 10, y < 8, and y < x, what must be true?
xy < 80I.
II.
x 2 + y 2 > 1III.
None


7. In a certain clothing store, the most expensive pair of socks sells for
$1 less than twice the price of the cheapest pair of socks. A
customer notices that for exactly $18, she can buy three fewer pairs
of the most expensive socks than the cheapest socks. What could
be the number of pairs of the cheapest socks she could have
purchased?
  3


Without solving problems 13–16, which answers could you confidently
eliminate and why?
None


12. The integer k is positive but less than 400. If 21k is a multiple of 180,
how many unique prime factors does k have?
One


15. In the 7-inch square to the right, another
square is inscribed. What fraction of the
larger square is shaded?


does not have to be true. Eliminate answers


# Cheap Pairs
(c)
$c Pairs
 # e Pairs (c −
3)
$e Pairs
 Match?
$e = 2($c) − 1?

is incorrect. Note that you don’t actually have to figure out

are also probably not correct answers. Try answer

makes the final equation work, so it is the correct


, or
 . Therefore, x = −1.
Testing choices is fast, too. In this case,


does not have to be true. Eliminate answer


This tactic may seem identical to the previous one, but it is not. Notice in
this example that statement (1) does not determine whether y is positive or
negative, and statement (2) does not even include z. Therefore, neither
statement is a subset of the other.
That said, the fact that y is negative does not change anything in statement
(1), because regardless of the value of y, z will remain the same if you swap
y and −y. If y = 4, then you’d get 4x + (−4)x = z. If y = −4, then you’d get (−4)x +
4x = z. Those equations are the same! The sign of y doesn’t matter because
y and −y are symmetric. So knowing the sign of y adds no information to
statement (1).
Note that in this example,


Try-It #4-13
If
 and ab ≠ 0, what is the value of K ?
You can prove that the two statements combined are suﬀicient without
actually doing any math. If you wanted to, you could plug the value of a
from statement (1) into statement (2) to solve for the value of b. Then, you
could plug those values into the question to solve for K.
Immediately be skeptical. Answer

. In some problems, the individual statements may


Try-It #4-15
Is xy < 1 ?
Statement (1) might imply that
 , but only if x is positive. If it is
negative, you would need to flip the sign:
 . You need to know the
sign of x—information that is provided in statement (2). The correct answer
is


2.
Spot One Statement Inside the Other (High): If a narrow statement is
completely contained within a broader statement, then eliminate


9.
For x = 7, 8, or 9, set A contains two integers that are within the range 1
to 100: x and x2. In all three cases, x3 and x4 are outside of the range.
Also, set B contains four integers that are within the range, but one of
them, x itself, is also contained in set A. So in every case in which x is
between 5 and 10, there are exactly five distinct integers in the range
that are contained in one or both sets.
The correct answer is


6. In sequence A, A1 = 1, A2 = 100, and the value of An is strictly
between the values of An – 1 and An – 2 for all n ≥ 3. Which of the
following must be true?
A100 < A200 < A300 < A400


The sum of the first 10 terms of S is
Between −1 and


In contrast, when you subtract these two templates, the end terms cancel,
leaving the middle term:
  Factored
 Distributed
Square of a Sum (a + b)2 = a2 + 2ab + b2
+ Square of a Diﬀerence (a − b)2 = a2 − 2ab + b2

Subtraction: (a + b)2 − (a − b)2 = 0 + 4ab + 0
This is handy for simplification. Also, whenever you see the sum of two
squares (a2 + b2), which is not itself a Quadratic Template, remember that it
can be derived from this sum of two templates.
Try It #6-5
What is the sum of 9,9992 and 10,0012?
99,980,001


Try-It #7-1
A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch.
If the dowel is painted on all surfaces and then cut into
-inch
cubes, what fraction of the resulting cube faces are painted?
If you draw a picture, this problem becomes a matter of counting:


The monthly sales (in thousands of $) at a certain restaurant for the past two
years are given in the chart above. If the standard deviation of the monthly
sales is greater this year than last year, which of the following are possible
values for x and y ?
Except for x and y, the two lists of monthly sales numbers are identical, so focus
exclusively on those terms that changed: 9.5 and 12.5 from last year were replaced by x
and y this year. If this year’s standard deviation is greater, then this year’s numbers
must be more spread out from the mean than last year’s. The numbers are close
enough together to indicate that the average should be somewhere around 11.
Visually, here are the interesting terms from last year:
This problem does not require actual computation of the standard deviation using the
new x and y values. The math would be too complex to complete in two minutes.
Instead, determine visually which x and y values increase the standard deviation: the
pair of x and y values that are farther from the mean than are 9.5 and 12.5 will increase
the standard deviation.
9 and 12.5


Try-It #7-11
A list of 12 test scores has an average (arithmetic mean) of 500 and a standard
deviation of 50. Which of the following lists of additional test scores, when
combined with the original list of 12 test scores, must result in a combined list
with a standard deviation less than 50 ?
6 test scores with average of 450 and standard deviation of 50

6 test scores with average of 500 and standard deviation of 25

6 test scores with average of 550 and standard deviation of 25

12 test scores with average of 450 and standard deviation of 25

2 test scores with average of 550 and standard deviation of 50


It is not generally true that all of the terms in a list are within 1 standard deviation of the
mean. However, standard deviation is a measure of the spread of the terms of a list, so
you could represent the original list of scores this way:
The oval spans ±1 standard deviation from the mean, where many of the scores will
likely be. This simplification is acceptable as long as you represent all of the other lists
the same way so that you can compare the relative eﬀects of the new test scores
systematically.
For each of the answer choices, overlay the representative ovals for the new data on top
of the oval for the original data:


In a certain lottery drawing, five balls are selected from a tumbler in which
each ball is printed with a different two-digit positive integer. If the average
(arithmetic mean) of the five numbers drawn is 56 and the median is 60, what
is the greatest value that the lowest number selected could be?
The goal is to maximize the value of the lowest-numbered ball. All balls contain a two-
digit positive integer, and none of the balls have the same number. The problem
provides enough information to calculate the sum and to lay out a visual listing of the
numbers:
In order to maximize the value of the first (lowest) number in the set, what do you need
to do to the other numbers?
You’d want to minimize them. Select the smallest numbers that you can for the
remaining slots:
In some problems, you might actually make diﬀerent slots equal to each other (e.g., the
three largest numbers could be 60, 60, and 60). This problem, though, specifies that the
43


Try-It #7-14
The average (arithmetic mean) of six numbers is 18 and the median of the six
numbers is 16. What is the minimum possible value for the greatest number in
the list?
19


Try-It #7-15
Jake 51
Keri 63
Luke 15
Mia 38
Nora 22
The table above shows the number of points held by five players of a certain
game. If an integer number of Keri’s points were taken from her and given to
Luke, and the median score of the five players increased, how many points
were transferred from Keri to Luke?
The key to this problem is that by taking enough points from Keri and giving them to
Luke, the median of the list can change.
23


Anton and Lena start at the same point on a circular track,
measuring 10 meters in diameter, and begin walking
counterclockwise at the same time, with Anton walking more
quickly than Lena. When Anton has traveled exactly halfway around
the track, they both stop walking. They then observe that the
distance between them along a straight line measures exactly
 meters, as shown above. What fraction of the track has Lena
covered?


10.
A square is drawn in the coordinate plane with its vertices at the
points (–2, –2), (–2, 6), (6, 6), and (6, –2), and a non-vertical line is
drawn that passes through the point (0, 4). The portion of the
coordinate plane that lies within the square, but above the line, is
then shaded as shown above. If A is the area of the shaded region in
square units, which of the following specifies all the possible values
of A ?
8 ≤ A ≤ 16


There is a repeating cycle of four numbers. Starting at 5 on the number line,
the ant moves le 1, then right 6, then le 2, then right 7, which takes it to 15
on the number line. That is, the ant walks a total of 1 + 6 + 2 + 7 = 16 units to
move from 5 to 15. Then it walks another 16 units to move from 15 to 25, from
25 to 35, and from 35 to 45. In total, the ant walks 4(16) = 64 units to move
from 5 to 45.
To get from 45 to 50, the ant first walks one unit le to 44, then six units right
to 50. In total, the ant walks 64 + 1 + 6 = 71 units to move from 5 to 50.
Finally, add the five units it walks from the beginning: 71 + 5 = 76.
It is also possible to estimate the


Where to Start
STARTING AT THE BEGINNING
As you contemplate the logical order of steps, you might feel less confident
with the second stage than with the first. If so, go ahead and start at the
beginning. Just articulate very clearly, “What intermediate result will I get
once I’m finished with the first part of the problem?”
Begin by defining set

Set A consists of four consecutive integers, such as


Set A Scenarios Set B Range
1, 2,
3, 4
All values increased by 10% 1.1, 2.2, 3.3,
4.4
4.4—1.1 =
3.3
1, 2,
3, 4
Smallest value increased by 10%, other values
decreased by 10%
1.1, 1.8, 2.7,
3.6
3.6—1.1 =
2.7
Since diﬀerent ranges are possible, the statement is insuﬀicient. Eliminate
answers


Now, put the two statements together. One option is to try a few diﬀerent
cases, making sure to increase the smallest integer in set A and decrease
the greatest integer in set

One of the two cases used for statement (1)


5. If x and y are positive integers greater than 1 such that x − y and
are both even integers, which of the following numbers must be
non-prime integers?
xI.
x + yII.
III.
I only


2. A batch of widgets costs p + 15 dollars for a company to produce
and each batch sells for p(9 − p) dollars. For which of the following
values of p, does the company make a profit?
3


6. Half an hour a er Car A started traveling from Newtown to
Oldtown, a distance of 62 miles, Car B started traveling along the
same road from Oldtown to Newtown. The cars met each other on
the road 15 minutes a er Car B started its trip. If Car A traveled at a
constant rate that was 8 miles per hour greater than Car B’s
constant rate, how many miles had Car B driven when they met?
14


9. If y is 20% less than 90% of x and z is 25% more than 130% of y,
then z is what percent of x ?
  72%


integers in set A is equal to the sum of the integers in set B, what is
the least number that could be a member of set A ?
69


Profit will be zero if p = 5 or p = 3, which eliminates answers


Try Working Backwards from the answers, starting with


8.
 Since there are variables in the
answer choices, choose a smart number to solve. If k = 2, then
,
which is less than 1. Now, test the answer choices and try to match the
target; stop if you can tell that an answer won’t equal the target.


Try the answer choices; since the problem asks for the least value, start
with answer

{69, 72, 75} sums to 3(72) = not a multiple of 5.


21. Let f(x) equal the sum of all of the integers from 1 to x, inclusive,
where x ≥ 1. If
, which of
the following is true of g ?
0 ≤ g < 5


25. At a certain store, all notebooks have the same price and all pencils
have the same price. The price of four notebooks and three pencils
is more than $12 and less than $13. The price of two notebooks and
five pencils is more than $8 and less than $9. If a notebook costs x
more than a pencil, which of the following could be the value of x ?
$0.60


26. A certain sequence is defined by the following rule: Sn = k(Sn − 1),
where k is a constant. If S1 = 2 and S13 = 72 , what is the value of S7  ?
  6


28. In a certain sequence, each term a er the first is twice the previous
term. If the first term of the sequence is 3, what is the sum of the
14th, 15th, and 16th terms in the sequence?
3(216)


The correct answer will yield a value of z = 30 mph when x = 40 and y =
40 are plugged in.

is a fraction that is significantly less

is not an integer, so it cannot equal 30. Eliminate.


32. If Sn = 5n + 94 and K = (S80 + S82 + S84) − (S81 + S83 + S85), what is the
value of K ?
−282


34. If y ≠ x, then
 is the equivalent of
which of the following?
(x − 1)2y


39. In a certain sequence, the term Sn is given by the formula Sn = (n +
1)! for all integers n ≥ 1. Which of the following is equivalent to the
diﬀerence between S100 and S99 ?
101!


Alternatively, you could choose smart numbers. If x = 2 and y = 3, then.
Plug the selected values into the choices. The choice that equals 7 is the
correct


Alternatively, choose smart numbers. For example, if x = 2 and y = 3, then
the final answer x + y = 5.
For values to plug into the choices, first compute a and b.
Next, test each answer choice; the one that equals 5 is the correct


52. Three of the four vertices of a rectangle in the xy-coordinate plane
are (–5, 1), (–4, 4), and (8, 0). What is the fourth vertex?
(–4.5, 2.5)


element in the set. If the median of the set equals the mean, what is
the greatest possible value in the set?
85


60. Four diﬀerent children have jelly beans: Aaron has 5, Bianca has 7,
Callie has 8, and Dante has 11. How many jelly beans must Dante
give to Aaron to ensure that each child has within 1 jelly bean of all
the other children?
2


Similarly, point B can be anywhere on the line from point A to point C.
Maximize E by minimizing B, so make B = A.
Therefore, E = 5 + 3A = 5 + 3(29) = 5 + 87 = 92.
The correct answer is


62. A group of friends charters a boat for $540 and each person
contributes equally to the cost. They determine that if they can get
three more of their friends to join them, every person in the group
will pay $9 less. If they find three more friends to join them, what is
the total number of people renting the boat?
  6


64. The sum of the interior angle measures for any n-sided polygon
equals 180(n − 2). If Polygon A has interior angle measures that
correspond to a set of consecutive integers, and if the median angle
measure for Polygon A is 140°, what is the least angle measure in
the polygon?
130°


66. If a and b are odd integers, a ∆ b represents the product of all odd
integers between a and b,inclusive. If y is the least prime factor of (3
∆ 47) + 2, which of the following must be true?
y > 50


67. Set S is the set of all prime integers between 0 and 20. If three
numbers are chosen randomly from set S, and no number is chosen
more than once, what is the probability that the sum of all three
numbers is odd?


71.
Energy usage (units) 11 10 8 7
Number of days   4   5 n 3
The table above shows daily energy usage for an oﬀice building and
the number of days that amount of energy was used. If the average
(arithmetic mean) daily energy usage was greater than the median
daily energy usage, what is the least possible value for n ?
2


75. If (s × 10q) − (t × 10r) = 10r, where q, r, s, and t are positive integers
and q > r, then what is the units digit of t ?
0


78. A decimal is called a “shrinking number” if its value is between 0
and 1 and each digit to the right of the decimal is not less than the
digit to its immediate right. For instance, 0.86553221 is a shrinking
number. If x is a shrinking number, which of the following must be
true?
 is a shrinking number.I.
 is a shrinking number.II.
 is a shrinking number. III.
I only


80. An (x, y) coordinate pair is to be chosen at random from the xy-
plane. What is the probability that y ≥ |x| ?


20 houses painted at a rate of 4 houses/week = 5 weeks
60 houses painted at a rate of 5 houses/week = 12 weeks
Total time for 80 houses = 5 + 12 = 17 weeks
The correct answer is


number is relatively easy to do math with, start by using x = 0.1 to test
cases.
I. As a fraction,
 , so
. However, this is not a
shrinking number, since 9 is greater than 0. So statement I is not
necessarily true. Eliminate choices


81.
As shown in the figure, square ABCD is inscribed in a circle with
circumference
 . What is the area of the shaded region in the
diagram above?
2x


If B started out with 80 sheets, then a 20-sheet increase would still be
less than 30% of the starting number, and again B would end up with
more sheets than

Algebraically, use a for the original number of sheets in Printer A and use

A er the transfer, A has a – 0.2a = 0.8a sheets. B has b + 0.2a sheets.


93. A cylinder of height h is
 full of water. When all of the water is
poured into an empty cylinder whose radius is 25% larger than that
of the original cylinder, the new cylinder is
 full. The height of the
new cylinder is what percent of h ?
  25%


In valid codes, the second digit in the code is exactly twice the first
digit. How many valid codes are there?
  42


As an alternative to testing all five choices, you could use a hybrid
approach to determine the formula using variables.
The number of possible groups was
 , but remember that this
formula took Sally into account.
The 3 came from the diﬀerence between these numbers: 7 − 4 = (x − 1) − (y
− 1) = (x − y).


104.
In the figure above, the trapezoid ABCD is inscribed in a circle.
Parallel sides AB and CD are 7 inches apart and 6 and 8 inches long,
respectively. What is the radius of the circle in inches?
4


107. For how many values of x from 1 to 300, inclusive, is the sum of the
integers from 1 to x, inclusive, divisible by 3 ?
  30


110. If p = (22)(3x) and r = (22)(3y), where x and y are prime numbers and
x ≠ y, which of the following represents the least common multiple
of p and r ?
12xy


Since (x + y)(x − y) = 7, (x − y) = 1.
Solve for x and y.
The radius of the circle is 5, because r2 = 32 + 42 = 25.
The correct answer is

: First, translate the question stem: Is πr2 ≥ r ? Simplify.


111.
Which of the following equations represents a line parallel to line ℓ in the
figure above?
2y − 3x = 0


117. Three boys are ages 4, 6, and 7, respectively. Three girls are ages 5, 8, and
9, respectively. If two of the boys and two of the girls are randomly
selected and the sum of the selected children’s ages is z, what is the
diﬀerence between the probability that z is even and the probability that
z is odd?


119. Positive integers a and b are less than or equal to 9. If a and b are
assembled into the six-digit number ababab, which of the following must
be a factor of ababab ?
3


3. On the test, stop here and select answer

4: An integer is divisible by 4 if its last two digits represent a two-digit

5: An integer is divisible by 5 if the last digit is 0 or 5. It is uncertain


121. If x and y are positive integers such that x2 – y2 = 48, how many
diﬀerent values of y are possible?
Two


122. If the number 200! is written in the form p × 10q, where p and q are
integers, what is the maximum possible value of q ?
40


127.
Which of the following equations represents a line perpendicular to
line k in the figure above?
3y + 2x = −12


129. K-numbers are positive integers with only 2’s as their digits. For
example, 2, 22, and 222 are K-numbers. The K-weight of a number n
is the minimum number of K-numbers that must be added together
to equal n.For example, the K-weight of 50 is 5, because 50 = 22 + 22
+ 2 + 2 + 2. What is the K-weight of 600 ?
10


Thus, the slope of a line perpendicular to k is −2.
Use the desired slope to create a slope-intercept equation and look for a match among
the answers.
Only answer

oﬀers a match for the le side of the equation.


Plug x = 3 into the answers, and find the one that equals
 .
The correct answer is


131. The sequence A is defined as follows: A1 = 1, and An = An − 1 + (−1)n +
1(n2) for all integer values n > 1. What is the value of A15 − A13 ?
  14


134. A chain is comprised of 10 identical links, each of which
independently has a 1% chance of breaking under a certain load. If
the failure of any individual link means the failure of the entire
chain, what is the probability that the chain will fail under the load?
(0.01)10


138. If f(x) = (x + 6)2 and g(x) = 9x, which of the following specifies all the
possible values of x for which f(g(x)) < g(f(x)) ?
–2 < x < 2


139.
In the figure above, ABCD is a rectangle, and each of AP and CQ is
perpendicular to BD. If DP = PQ = QB = 1, what is the length of AB  ?


140.
In the rectangular coordinate system above, point A is not shown. If
the area of triangle OAB is at least 16, which of the following could
NOT be the coordinates of point A  ?
(1, 8)


Finally, use the Pythagorean theorem to determine the length of AB.
Alternatively, since the diagram is drawn to scale, you can try to estimate. Side
AB appears to be longer than BP, but shorter than BD. Its length should be
between 2 and 3. Estimate the values of the answer choices.
Only answers

(1, 8): First, identify any answer choices that will be easy to eliminate before


141. If
 , where a and b are integers and a
does not equal 1 or −2, which of the following could be the value of
b  ?
1I.
2II.
3III.
I only


144. Each digit 1 through 5 is used exactly once to create a five-digit
integer. If the 3 and the 4 cannot be adjacent digits in the integer,
how many five-digit integers are possible?
48


146. A soccer competition consisted of two rounds. In each round, the 20
teams competing were divided randomly into 10 pairs, and each
pair played a single game. None of the games resulted in a tie, and
of the teams that lost in the first round, 7 also lost in the second
round. How many teams won both of their games?
3


rate of y milligrams per hour, how many hours will it take for the
concentration of salt to reach (x + 10)%, in terms of x and y ?


160. An unpainted wall with a total area of 100 square feet needed to be
painted with three coats of paint. Nelson began the job by painting
an area of 60 square feet. Then, Jamaica painted a total area of 70
square feet. Finally, Bryan painted a total area of 80 square feet.
A er Bryan finished painting, an area of q square feet of the wall
was covered in three coats of paint. If the entire surface area of the
wall had received at least one coat of paint, which of the following
specifies all the possible values of q  ?
0 ≤ q ≤ 50


Thus, if set A contains the integers from 1 to 5, set B consists of 1, 2, 3, 4, 5, 6, 7, 8,
and 9. The question asks for the greatest integer in set

(1) SUFFICIENT: The range of a set is the diﬀerence between the greatest and the

Those numbers are x + 3 and x + 4, and their sum is 2x + 7.