1. Number and
Operations Review
The SAT doesn’t include
•tedious or long computations
•matrix operations

2. Properties of Integers
Integers
• Whole numbers and their negatives
(including zero)
..., -3, -2, -1, 0, 1, 2, 3, 4,...
• Extend infinitely in both negative and
positive directions
• Do not include fractions or decimals

3. Integers
• Negative integers
-4, -3, -2, -1
• Positive integers
1, 2, 3, 4
• The integer zero (0) is neither positive
nor negative.

4. • Odd Numbers
..., -5, -3, -1, 1, 3, 5,...
• Even Numbers
..., -4, -2, 0, 2, 4,...
• The integer zero (0) is an even number.

5. Consecutive Integers
• Integers that follow in sequence, where the difference
between two successive integers is 1, are consecutive
integers.
• Here are three examples of some consecutive integers:
-1, 0, 1, 2, 3
1001, 1002, 1003, 1004
-14, -13, -12, -11
• The following is an expression representing consecutive
integers:
n, n + 1, n + 2, n + 3,..., where n is any integer.

6. Addition of Integers
• even + even = even
• odd + odd = even
• odd + even = odd
• Adding zero (0) to any number doesn’t change
the value:
8 + 0 = 8
-1 + 0 = -1

7. Multiplication of Integers
• even x even = even
• odd x odd = odd
• even x odd = even
• Multiplying any number by one (1) doesn’t
change the value:
3 x 1 = 3
10 x -1 = -10

8. Arithmetic Word
Problems
• The SAT includes questions which test
your ability to correctly apply arithmetic
operations in a problem situation.
• To solve the problem,
1. you need to identify which quantities
are being given
2. what is being asked for
3. which arithmetic operations must be
applied to the given quantities to
get the answer

9. Example 1
Ms. Hall is preparing boxes of books for children in the
hospital. If she puts 3 books in each box, she will make
25 boxes of books with no books left over. If instead she
puts 5 books in each box, how many boxes of books will
she have?
• Multiplying 3 times 25, there are 75 books.
• If she puts 5 boxes in each box, she can make 15
boxes of books. (75 divided by 5)
• The correct answer is 15 boxes of books.

10. Example 2
Jack bought 6 doughnuts at the bakery with a five-dollar
bill and got back three quarters in change. He
realized he had gotten too much change, so he
gave one of the quarters back to the cashier. What
was the price of each doughnut?
• Jack originally paid $5.00 - 0.75 = $4.25.
• When he gave back the quarter, he actually paid
$4.25 + 0.25 = $4.50 for the 6 doughnuts.
• Each doughnut cost $4.50/6 = $ . 75.

11. Number Lines
Number line is used to graphically represent the
relationships between numbers like integers,
fractions, or decimals.
• Numbers increase as you move to the right and
tick marks are always equally spaced.
• Number lines are drawn to scale. You will be
expected to make reasonable approximations of
positions between labeled points on the line.

12. Number line questions may ask:
• Where a number should be placed in
relation to other numbers;
• The difference or product of two numbers;
• The lengths and the ratios of the lengths
of line segments represented on the
number line.

13. Example 1
• The ratio of AC to AG is equal to the ratio of CD
to which of the following?
(A)AD (B)BD (C)CG (D)DF (E)EG
• From the number line, AC = 2, AG = 6, CD = 1
• AC is to AG as CD is to what?
AC CD
•
AG x
2 = =
1 2 x =6 x
=3
6
x
• Of the answer choices, only AD = 3 so the
answer is A.

14. Example 2
Find the coordinate of P.
• Subtract the points we know and divide by the
number of intervals to find the distance of each
interval.
•
- =
.430 .425 .001
5
• Add .001 x 3 (number of intervals from lowest
point to point P)
• .425 + .003 = .428 so Point P is at 0.428.

15. Squares and Square
Roots
• Squares of
Integers

16. Your knowledge of common squares and square
roots may speed up your solution to some math
problems. The most common types of problems
for which this knowledge will help you will be those
involving:
• Factoring and/or simplifying expressions
• Problems involving the Pythagorean theorem
(a2 + b2 = c2 )
• Areas of circles or squares

17. Squares of Fractions
• Remember that if a positive fraction with a value less than 1 is
squared, the result is always smaller than the original fraction:
If 0 < n < 1
Then n 2 < n
• What are the values of the following fractions?
1
64
4
9
• The answers are and respectively. Each of these is less
than the original fraction.
• For example, .
4 <
2
9 3

18. Fractions and Rational
Numbers
Basic operations with fractions:
• Adding, subtracting, multiplying, and dividing fractions
• Reducing to lowest terms
• Finding the least common denominator
• Expressing a value as a mixed number and as an
improper fraction
2 1
3
æ ö
çè ÷ø 7
3
æ ö
çè ÷ø
• Working with complex fractions—ones that have
fractions in their numerators or denominators

19. Rational Number
• A number that can be represented by a
fraction whose numerator and denominator are
both integers (and the denominator must be
nonzero).
• The following are rational numbers:
⎛ ⎞
⎜⎝ ⎟⎠
3 7 or 3 1 − 5 3
(or 3)
4 2 2
7 1
• Every integer is a rational number.

20. Decimal Fraction Equivalents
• Recognize common fractions as decimals and
vice versa.
• To change any fraction to a decimal, divide the
numerator by the denominator.

21. You can figure out the decimal equivalent of any
fraction with a calculator, however you’ll save
time if you know the following:

22. Reciprocals
• The reciprocal of a number is 1 divided by that
number. For example, the reciprocal of 5
1
is 5
. Note that 5 ´ 1 =
.
1
5
• The product of a number and its reciprocal is
always 1.
• You can find the reciprocal of any nonzero
fraction by switching its numerator and
denominator.

23. Examples of reciprocals:
3 11
1
2
4 4
4
1
-2® - 1
= ® 2
The reciprocal of 1 is 1.
The number zero has no reciprocal.

24. Scientific Notation
2,300,000,000 = 2.3´109
.0000000000000076 = 7.6´10-15

25. Elementary Number
Theory
Factors
The factors of a number are positive integers
that can evenly be divided into the number
—that is, there is no remainder.
• Example: 24 - The numbers 24, 12, 8, 6, 4,
3, 2, and 1 are all factors of the number 24.

26. Divisible By
Divisible without any remainder or with a
remainder of zero.
• 15 is divisible by 5 because 15 divided by 5
is 3 with a remainder of 0.
• 15 is not divisible by 7, because 15 divided
by 7 is 2 with a remainder of 1.

27. Common factors
Factors that two (or more) numbers have
in common.
• 3 is a common factor of 12 and 18.
• The largest common factor of two (or
more numbers) is called their greatest
common factor (GCF).
• 6 is the GCF of 12 and 18.

28. Multiples
Those numbers that can be divided by a
given number without a remainder.
• You can find the multiples of a number by
multiplying it by 1, 2, 3, 4, and so on.
• 8, 16, 24, 32, 40, and 48 are some of the
multiples of 8.

29. • The multiples of any number will always be
multiples of all the factors of that number.
• The numbers 30, 45, 60, and 75 are all
multiples of the number 15.
• Two factors of 15 are the numbers 3 and 5.
• That means that 30, 45, 60, and 75 are all
multiples of 3 and 5.

30. Common multiples
Any number that is a multiple of all the given
numbers.
• 48 and 96 are both common multiples of 8 and
12.
• The smallest multiple of two (or more) numbers
is called their least common multiple (LCM).
• 24 is the LCM of 8 and 12.

31. Example 1
What is the least positive integer divisible by the
numbers 2, 3, 4, and 5?
• One possible answer is to multiply those numbers
together. You could multiply and the result
would be divisible by all those factors.
2g3g4g5
• However, to find the least positive number divisible by all
four (in other words, the LCM of the four numbers), you
have to eliminate any extra factors.

32. Example 1 (continued)
What is the least positive integer divisible by the numbers
2, 3, 4, and 5?
• Any number divisible by 4 will also be divisible by 2, so
you can eliminate 2 from your initial multiplication. So if
you multiply , you will get a smaller number and
the number will still be divisible by 2.
3 g 4 g 5
• Answer: is the Least Common Multiple
(LCM).
3 g 4 g 5 = 60

33. Example 2
Which of the following could be the remainders
when four consecutive positive integers are
each divided by 3?
(A) 1, 2, 3, 1
(B) 1, 2, 3, 4
(C) 0, 1, 2, 3
(D) 0, 1, 2, 0
(E) 0, 2, 3, 0

34. • When you divide any positive integer by 3, the
remainder must be less than or equal to 2.
• All the choices except (D) include remainders
greater than 2, so (D) is the only possible
correct choice.
• If the first and fourth of the consecutive integers
are multiples of 3, the remainders will be 0, 1, 2,
and 0. example - 3,4,5,6

35. Example 3
Does the equation 3x + 6y = 47 have a solution in which x
and y are both positive integers?
• Notice that 3x + 6y = 3(x + 2y).
• Therefore, the sum 3x + 6y = 3(x + 2y) is a
multiple of 3. But 47 is not a multiple of 3.
• No matter what positive integers you choose for x and
y, the sum will not be 47. Thus, the equation 3x + 6y =
47 does not have a solution in which x and y are both
positive integers.

36. Prime Numbers
A prime number is a positive integer greater
than 1 that has exactly two whole-number
factors—itself and the number 1.
• The number 1 itself is not prime.
• Prime numbers include:
2, 3, 5, 7, 11, 13, 17, 19
• 2 is the only even prime number.

37. Prime factors
Prime factors are the factors of a number
that are prime numbers.
• Prime factors of a number cannot be
further divided into factors.
• The prime factors of 24 are 2 and 3.

38. Ratios, Proportions, and
Percents
Ratio
• A mathematical relationship between two
quantities.
• Ratio is a quotient of those quantities.
• Ratios can be expressed in several ways:
2 2 : 3 2 to
3
3

39. Percent (%)
• A ratio in which the second quantity is 100.
85
100
• 85 percent is equivalent to .

40. Proportion
• An equation in which two ratios are set
equal to each other.

41. Example 1
The weight of the tea in a box of 100 identical tea
bags is 8 ounces. What is the weight, in ounces,
of the tea in 3 tea bags?
• The ratio of the tea in 3 tea bags to the tea in all
of the tea bags is 3 to 100 æ 3
ö
çè ÷ø
.
100
• Let x equal the weight, in ounces, of the tea in 3
tea bags.
• The ratio of the weight of the tea in 3 tea bags to
the total weight of the tea in 100 tea bags is x
ounces to 8 ounces .
æ x ö
çè 8
÷ø

42. x =
3
8 100
100x = 24
x = .24
The answer is .24 ounces in 3 bags of tea.

43. Example 2
The ratio of the length of a rectangular floor to its
width is 3:2. If the length of the floor is 12 meters,
what is the perimeter of the floor, in meters?
• Set 3:2 equal to the ratio of the actual measures of
the sides of the rectangle.
length x x
width x
3 = , 3 = 12 , 3 = 24, = 8 (
width)
2 2

44. • Find the perimeter: 2(length + width) =
2(12 + 8).
• The perimeter of the floor is 40 meters.

45. Sequences
A sequence is an ordered list of numbers.
• Infinite sequence goes on forever and
indicated by …
2,5,8,11,14…
• Finite sequence terminates.
3,6,9,…21,24

46. Arithmetic Sequences
• Sequences for which there is a constant
difference (d) between consecutive terms.
• 4, 7, 10, 13,... is an arithmetic sequence in which
the first term is 4 and the constant difference (d)
is 3.
• Notice that 7-4=3 10-7=3 13-10=3 so d =3.
• To find the nth term use:
1 1 ( 1) a + n - d where a = first term

47. Geometic Sequences
Sequences for which there is a constant ratio (r)
between consecutive terms.
• 4, 12, 36, 108,... is a geometric sequence in
which the first term is 4 and the constant ratio (r)
is 3.
• Notice that 12 = 3 36 = 3 108
=
3
4 12
36
so r = 3.
• To find the nth term use:
a grn- 1
where a = first term
1 1

48. • The mathematics section does not ask you to figure out
the rule for determining the numbers in a sequence
without giving the information in some way.
• For instance, in the preceding example, if you were given
that the nth term of the sequence was 4g3n-,
1
you may be asked to determine that each term after the
first was 3 times the term before it.
• Or if you were given that the first term in the sequence
was 4 and that each term after the first was 3 times the
term before it, you might be asked to find that the nth
term of the sequence was 4 g 3 n - 1 .

49. Number sequence questions might ask you
for the following:
• The sum of certain terms in a sequence
• The average of certain terms in a
sequence
• The value of a specific term in a sequence

50. Sets (Union, Intersection, Elements)
Set
• A collection of things, and the things are called
elements or members of the set.
Union
• Union of two sets is the set consisting of the
elements that are in either set or both sets.
Intersection
• Intersection of two sets is the set consisting of
the common elements.

51. Given: set A = {1, 3, 7, 10, 12} and
set B = {8, 10, 12, 15}
• the union of sets A and B would be
{1, 3, 7, 8, 10, 12, 15}
• the intersection of sets A and B would be
{10, 12}

52. Counting Problems
Counting problems involve figuring out how many
ways you can select or arrange members of
groups, such as letters of the alphabet, numbers, or
menu selections.
Fundamental counting principle
• The principle by which you figure out how many
possibilities there are for selecting members of
different groups.
• If one event can happen in n ways, and a second,
independent event can happen in m ways, the total
ways in which the two events can happen is n times
m.

53. Example
On a restaurant menu, there are three appetizers
and four main courses, five beverages and three
desserts. How many different dinners can be
ordered if each dinner consists of one appetizer,
one main course, one beverage and one
dessert?
3g4g5g3 = 180
• There are 180 possible
dinners.

54. Permutations
• A way to arrange things in which order is
important.
Example
A security system uses a four-digit password, but
no number can be used more than once. How
many possible passwords are there?
10 9 8 7 5,040 or
use your calculator and enter P
10 4
g g g =

55. Combinations
• A way to arrange things in which order is
not important.

56. Example
There are 12 students in a statistics class. Three
students will be selected to attend a conference on
statistical research. How many different groups of three
can be selected from the class?
• Order does not matter
• or
= = g g
g
12 11 10 permutation of 12 things 3 at a time 220
3 2 permutation of 3 things 3 at a time
12C3
• Use the calculator = 220 different groupings

57. Logical Reasoning
Logical thinking questions
• You have to figure out how to draw
conclusions from a set of facts.

58. Example
• In the figure above, circular region A represents the set
of all numbers of the form 2m, circular region B
represents the set of all numbers of the form n 2, and
circular region C represents the set of all numbers of the
form 10k , where m, n, and k are positive integers.
• Which of the following numbers belongs in the set
represented by the shaded region?
• (A)2 (B)4 (C)10 (D)25 (E)100

59. • The shaded region is part of all of the circles.
Therefore, any numbers in the shaded region
have to obey the rules for all the circles.
• Circle A represents even numbers (2m).
• Circle B represents perfect squares (n 2).
• Circle C represents numbers with a whole-number
power of 10 (10k) like 10, 100, 1000,
10000.
• The only choice that represents all the circles is
100. Thus the answer is E (100).