Transcript SAT Math Review

SAT Math Review
Numbers and Operations
Properties of Integers
 Integers consist of the whole numbers
and their negatives (including 0)
 They extend infinitely (neg. and pos.)
 Do not include fractions or decimals
 Pos.Integers – 5, 6, 10
 Neg. Integers - -5,-6, and -10
 0 is neither neg. or pos.
Properties of Integers
 Odd numbers: -5, -3, -1, 1, 3, 5
 Even numbers: -4, -2, 0, 2, 4
 Consecutive Integers: Integers that follow
in sequence and the difference between
two in a row is 1 (-1, 0, 1, 2)
 n, n +1, n+2, n+3
Properties of Integers
 Addition
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Even + even = even
Odd + odd = even
Odd + even = odd
Adding 0 to any number doesn’t change value
 Multiplication
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Even * even = even
Odd * odd = odd
Odd * even = even
Multiplying any number by 1 doesn’t change the
value
Arithmetic Word Problems
 Testing your ability to correctly apply
arithmetic operations in a problem
situation. You will need to id which
quantities are given, what is being asked,
and which arithmetic operations must be
applied to the given quantities to get the
answer.
Arithmetic Word Problems
 Example 1:
 Ms. Griffen is making bags of Halloween
treats. If she puts 3 treats in each bag, she
will make 30 bags of treats and have no
treats left over. If instead she puts 5 treats in
each bag, how many bags of treats can she
make?
 3 treats in 30 bags means she has 90
treats total. Divide 90 by 5 to see how
many treats she can make with 5 treats
(answer – 18 bags)
Arithmetic Word Problems
 Example 2
 Jorge brought 5 pencils from the store. He
gave the cashier a five dollar bill and got
back 3 quarters in change. Jorge saw that
he had gotten too much change, and he
gave 1 quarter back to the cashier. What
was the price of each pencil?
 5 dollars minus 50 cents = 4.50 (cost for
5 pencils); 4.50/5 = 0.90
Number Lines
 A number line is used to graphically
represent the relationships between
numbers (integers, fractions, or decimals)
 Numbers on a number line always
increase as you move the right and tick
marks are always equally spaced
 Neg. #’s are shown with – sign. But +
sign not on pos. #’s
Number Lines
 Number Line questions usually require
you to figure out the relationships among
#’s placed on the line
 Where a number should be placed in
relation to other #’s
 The difference or product of 2 numbers
 The lengths and ratios of the lengths of line
segments represented on the number line
Number Lines
 Example
•On the number line above, 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
•Answer: AC = 2, AG = 6, CD = 1; The ratio of AC to AG is 2 to 6; 2/6 = 1/x; 6=2x
x= 3; so which line segment equals 3? Answer is A (AD)
Squares and Square Roots
 Squares of Integers
 It’s good to know or at least recognize the
squares of integers between -12 and 12
 Problems
 Factoring or simplifying expressions
 Problems involving the Pythagorean
theorem (a^2 + b^2 = c^2)
 Areas of circles or squares
Squares of Fractions
 If a positive fraction with a value less
than 1 is squared, the result is always
smaller than the original fraction
 (2/3)^2 = 4/9 (which is less than 2/3)
Fractions and Rational
Numbers
 Be able to do basic operations w/
Fractions
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Adding, subtracting, multiplying, and dividing
Reducing to lowest terms
Finding least common denominator
Express as Mixed Number or improper
fraction
 Work with complex fractions (ones with
fractions in the numerator or denominator)
Fractions and Rational
Numbers
 A Rational Number is a number that can be
represented by a faction whose numerator and
denominator are both integers (and the
denominator is nonzero)
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½
15/4 or 3 ¾
-12/13
5/1 or 5
 Every integer is a rational number
Fractions and Rational
Numbers
 Decimal Fraction Equivalents
 Be able to recognize common fractions as decimals
and vice versa
 To change a fraction to a decimal, divide the
numerator by the denominator
 Good ones to have memorized
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¼ = .25
1/3 = .33333
1/5 = .5
2/3 = .66666
¾ = .75
Fractions and Rational
Numbers
 Reciprocals
 The reciprocal of a number is 1 divided by
the number.
 5 = 1/5
 2/3 = 1/(2/3) = 3/2 (you can switch the numerator
and denominator for any nonzero fraction)
 2 1/3 = 7/3 (write as improper fraction first) = 3/7
is reciprocal
 The reciprocal of a neg number is neg
 0 has no reciprocal
Fraction and Rational
Numbers
 Place Value
 The number 123 can be written as 100+20+3 or as
(1*10^2)+(2*10^1)+(3*1)
 The digit 1 stands for 1 times 100 (hundreds place), 2 is 2
times 10 (tens place), 3 is 3 times 1(ones place)
 0.56 = 0.5+0.06 = (5*10^-1)+(6*10^-2)
 5 is in the tenths place and 6 is in the hundredths place
 Scientific Notation
 Write a big number in shorter form
 2,300,000,000,000 = 2.3 * 10^12
 .0000000007 = 7*10^-10
Elementary Number Theory:
Factors, Multiples, and
Remainders
 Factors
 Positive integers that can be evenly divided
into the number (with no remainder)
 Factors of 24 = 24, 12, 8, 6, 4, 3, 2, 1
 Common Factors are factors that 2 numbers
have in common
 3 is a common factor of 12 and 18
 The Greatest Common Factor is the largest
common factor of two or more numbers
 6 is the GCF of 12 and 18
Elementary Number Theory:
Factors, Multiples, and
Remainders
 Multiples
 Numbers that can be divided by that the given number without a
remainder
 You can find the multiples of any number by multiplying it by 1, 2, 3, 4,
and so on
 Multiples of 8 = 8, 16, 24, 32, etc.
 Multiples of any number will always be multiples of all the factors of
that number
 30, 45, 60, & 75 are all multiples of 15
 3 & 5 are factors of 15
 30, 45, 60, & 75 are multiples of 3 & 5 too
 Common Multiples
 Any number that is a multiple of all the given numbers (48 & 96 are both
common multiples of 8 & 12)
 Least Common Multiple is smallest multiple of 2 numbers (24 is LCM of 8
& 12)
Elementary Number Theory:
Factors, Multiples, and
Remainders
 Examples
 What is the least positive integer divisible by the
numbers 2, 3, 4, and 5?
 It’s looking for the LCM (least common multiple) of all 4
numbers
 To get any common multiple you could multiply them all:
2x3x4x5 (but we can’t do it on this one – lcm)
 Any number divisible by 4 is also divisible by 2, so 2
doesn’t need to be in equation
 Out of 3, 4, & 5, there is no common factor, so you can
multiply the 3 together to get the answer: 3x4x5 = 60 (that’s
the answer)
Elementary Number Theory:
Factors, Multiples, and
Remainders
 Examples
 Which of the following could be the remainders when 4
consecutive positive integers are each divided by 3?
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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
 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 only correct choice
 If the 1st and 4th of the consec. integers are multiples of 3, the
remainders will be 0, 1, 2, and 0
Elementary Number Theory:
Factors, Multiples, and
Remainders
 Example
 Does the equation 3x+6y=47 have a solution
in which x and y are both positive integers?
 3x+6y=3(x+2y) so for any positive integers x
and y, the sum would be a multiple of 3, but 47 is
not a multiple of 3 so there NO solution in which
they could both be positive integers.
Elementary Number Theory:
Factors, Multiples, and
Remainders
 Prime Numbers
 Is a positive integer greater than 1 that has
exactly 2 whole number factors (itself and
the number 1). The number 1 is not prime
though.
 2,3,5,7,11,13,17,19
 2 is the only even prime number
 Prime Factors
 The factors of a number that are prime numbers
 2 and 3 are the prime factors of 24
Ratios, Proportions, and
Percents
 Ratio
 Expresses a mathematical relationship
between 2 quantities. Specifically, a ratio is a
quotient of those quantities.
 Examples:
 The ratio of my serving of pizza to the whole pie
is 1 to 4. (1/4 or 1:4 or 1 to 4)
 The ratio of chocolate to vanilla cookies is 2 to 1.
(2/1 or 2:1 or 2 to 1)
Ratios, Proportions, and
Percents
 Percent
 Is a ratio in which the second quantity is
100.
 I got a 75 on my test or 75% of the questions
right (ratio is 75 to 100)
 Proportion
 Is an equation in which 2 ratios are set equal
to each other. You may be asked to answer
questions that require you to set up a
proportion and solve it.
Ratios, Proportions, and
Percents
 Example
 The weight of the tea in a box of 100
identical tea bags is 8 oz. What is the
weight, in oz, of the tea in 3 tea bags?
 Use a proportion – set up 2 ratios.
 Ratio of tea in 3 bags to tea in all bags is 3:100
 x is weight in oz of tea in 3 bags; Ratio of weight of 3
bags to weight of 100 bags is x:8
 Set ratios equal and solve for x
 x/8 = 3/100 (100x=24; x=24/100 or .24)
Ratios, Proportions,
Percents
 Example
 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 ratio of floor to width equal to ratio of actual
measures: 3/2 = 12/x
 3x=24, x=8 (the width)
 Now find the perimeter: 2(length + width) =
2(12+8); perim = 40 meters
Sequences
 A sequence is an ordered list of numbers.
Some follow specific patterns (like adding
the same number to the one before to get
the next).
 Number sequence questions might ask:
 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
Sets (Union, Intersection,
Elements)
 Set
 Is a collection of things, and the things are
called Elements or Members of the set.
 Questions might ask about the union of 2
sets (the set consisting of the elements that
are in either set or both sets) or the
intersection of 2 sets (the set consisting of
the common elements).
 Set A = [2,4,6,8,10] and Set B = [8,10,12,14]
 Union of A,B = [2,4,6,8,10,12,14]
 Intersection of A,B = [8,10]
Counting Problems
 Counting Problems involve figuring out how
many ways you can select or arrange
members of groups (letters, alphabet,
numbers, or menu selections)
 Fundamental Counting Principle
 How to 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 2 events can happen is M times N.
Counting Problems
 Example
 On a restaurant menu, there are three
appetizers and four main courses. How
many different dinner can be ordered if each
dinner consists of one appetizer and one
main course?
 3 choices app and 4 choices course: 3 x 4 = 12
Counting Problems
 Permutations
 If you select member after member from the same
group, the number of possible choices will decrease
by 1 for each choice.
 Example:
 A security system uses a 4 letter password, but no letter
can be used more than once. How many possible
passwords are there?
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For first number, there are 26 possibilities (letters of abc)
2nd number decreases by 1, so 25
3rd number decreases by 1, so 24
4th number decreases by 1, so 23
 26x25x24x23 = 358800
Counting Problems
 Combinations
 If the order in which the members are chosen
makes no difference
 Example
 There are 12 students in the school theater class. Two
students will be responsible for finding the props needed
for the skit the class is performing. How many different
pairs of students can be chosen to find the props?
 1st student can be any of the 12, 2nd can be any of 11 left
 So 12 x 11, but then multiply answer by 2 (because there are
2 ways that the student can be chosen – as either the first or
the second student)
Logical Reasoning
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Figure out how to draw conclusions from a set of facts
B
A
C
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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 10^k, where m, n, and k are positive
integers. Which of the following numbers belongs in the set represented by the shaded
region? (where all 3 circles intersect)
 A: 2 B: 4 C: 10 D: 25 E: 100
 Understand Logic: Question is asking about shaded region, shaded region is part of
all circles, so any numbers in it have to obey the rules for all the circles
 Rule for A: The numbers must be of the form 2m, which means that they must
all be even numbers
 Rule for B: The numbers must be of the form n^2, so it must be a perfect
square
 Rule for C: Numbers must also be of the form 10^k, which means they have to
some whole number power of 10 (10, 100, 1000, 10000, etc)
 Look at the Answer choices – 100 is the only one that obeys all the rules (so E)