psat mathematics 9-a basic arithmetic concepts. set a collection of “things” that have been...
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PSAT MATHEMATICS
9-A Basic Arithmetic Concepts
Set
A collection of “things” that have been grouped together in some way.
“things” – elements or members
Example
If A is the set of states in the United States, then _____________ is an element of A.
If B is the set of letters in the English alphabet, then _____ is a member of B.
If C is the set of even integers, then ___ is in C.
Union
The union of two sets, A and B, is the set consisting of all the elements that are in A or B or both.
A U B
Intersection
The intersection of two sets, A and B, is the set consisting of only the elements that are in both A and B.
A B U
Example
A = {2, 3, 5, 7}B = {2, 4, 6, 8}
Find the union and intersection of sets A and B.
Solution Set
The set of all numbers that satisfy the equation.
Example
If C={-2, -1, 0, 1, 2} and D is the set that consists of the squares of each of the elements of C, how many numbers are elements of D?
Signed Numbers
Draw a number line from -5 to 5. Mark the following points.
-3 ½ 0.25 π - ¾ |-4|
Example
How many integers are solutions of the inequality
|x|<5?
A. 0 B. 4C. 8 D. 9 E. infinitely many
Example
What is the sum of the product and quotient of 8 and 8?
Example
What is the product of all the integers from -5 to 5, inclusive?
Example
Which of the following are equal to (-1)25?I. (-1)100 II. -(1)100 III. - (-1)100
A. I only B. II only C. III only D. II and III only E. I, II, and III
Integers
Integers Positive integers Negative integers Consecutive integers Consecutive odd integers Consecutive even integers Prime numbers
Example
If 0<x<3 and 2<y<8, what is the largest integer value of x+y?
Example
If the remainder when a is divided by 7 is 2 and the remainder when b is divided by 7 is 4, what is the remainder when ab is divided by 7?
Example
What is the sum of the largest prime factor of 26 and the largest prime factor of 28?
Exponents and Roots
Repeated addition of the same number is indicated by multiplication.
Repeated multiplication of the same number is indicated by an exponent.
Laws of Exponents
54 x 53
56 / 54
(52)3
53 x 63
Example
If 5a x 5b = 550, what is the average (arithmetic mean) of a and b?
Example
What is the value of 16¾?
Perfect Squares
List the squares of 0 to 15.
Example
What is the circumference of a circle whose area is 10π?
A. 5πB. 10πC. π√10D. 2π√10 E. π√20
Order of Operations
Parentheses Exponents Multiplication & Division Addition & Subtraction
Practice
1. 10 + 4 x 2 2. (10 + 4) x 2 3. 16 ÷ 2 x 4 4. 16 ÷ (2 x 4) 5. 10 + 15 ÷ (2 + 3) 6. 100 – 22 (3 + 4 x 5)
Example
Which of the following is equivalent to for all nonzero numbers x?
A. x + 1 B. x5 C. x5 + 1D. x + x4 E. x16
€
x 5 + x 4
x 4
Practice Problems