Download Basic Mathematics Review

Basic Math Review
You have access to a calculator now, so you don’t have to keep reviewing your
multiplication table (but you can if you want to).
And I’m not going to spend time showing you how to add fractions by hand, do long
division by hand, or estimate square roots.
Your “State of Mind” when doing Math
I’m just telling you something you already know, but it’s always helpful to be reminded:
“You can choose your thoughts!”
Here is the one we will choose as we embark upon a math problem:
“Gee! This is going to be fun!”
Why does doing this help you so much? Choosing a good thought prevents you from
choosing a bad one (e.g. “I’m terrible at math”, “I just know I will mess this up”, “I’m too
cool for school”).
What is your “State of Mind” when you make a mathematical mistake?
Let’s always choose “Ohhh, so THAT’S how it’s done! Okay! This is great, I just learned
something! I will remember that the next time it comes up!”
Your goal is to practice each day.
Don’t be concerned if you can’t do everything perfectly the first time you try it.
You learn from your mistakes. So making those mistakes is a good thing.
This is why you start preparing early for the GED exam, so you have LOTS of time to
learn from your mistakes. By the time the exam rolls around, you’re not making mistakes
anymore (or only a few).
Page 1 of 7 Flash Cards
In my early years as an instructor, I began to notice what my best students often did while
they were waiting for class to begin: They were going through a stack of 3x5 Flash Cards
they had created. They were “thickening the neural pathways in their brain”, so when they
needed that information, it came to them quickly and easily.
As you go through the “Cracking the GRE” (or something similar to it), and you come
across a “good thing to remember”, make a Flash Card for it. I will be giving you
suggestions on how you might word your Flash Card as we cover these “good ideas”.
======================
Let’s practice applying some “good ideas” on some Math Problems.
But before we begin, what is the thought that I should choose, so as to put myself into the
right frame of mind?
“Gee! This is going to be FUN!”
=============================
Exponents
Using exponents is just repeated multiplication:
3 x 3 x 3 x 3 = 34 = 81
(they’re all the same)
When working with Exponents, always remember this:
“When in doubt, expand it out.”
32 x 34 = 32+4 = 36
Experiencing any “doubt”? Then expand it out:
(3)(3) x (3)(3)(3)(3) = 36
___________________________
34 = 34−2 = 32
32
Experiencing any “doubt”? Then expand it out:
(3) (3) (3) (3) (3)(3) 2
=
=3
1
(3) (3)
____________________________
Page 2 of 7 ( )
3
32 = 32×3 = 36
Experiencing any “doubt”? Then expand it out:
(32) (32) (32)
(3) (3) (3) (3) (3) (3) = 36
____________________________
32 = 32−4 = 3−2
34
What the heck is 3-2?
When in doubt, expand it out:
(3) (3)
= 1 = 12 = 3-2
(3) (3) (3) (3) (3)(3) 3
_____________________________
Try this one:
( )
x5
( )
3
( x)( x3 )
x5
3
5×3
15
x
x
= 1+3 = 4 = x15−4 = x11
3
( x)( x ) x
x
=?
_____________________________________
A number raised to the power of zero will be equal to 1.
Why?
First, do we all agree that the following is equal to 1?:
Alright then:
x3
x3
x3 = x3−3 = x0 So x0 = 1
x3
Exception to this rule: 00 is undefined (i.e. 00 does NOT equal zero, one, or anything!)
Why?
Because we can’t divide by zero, this is a big no-no:
03 = 03−3 = 00
03
Practice:
Which of the following is equal to ( 78 × 79 ) ?
10
A)
B)
C)
D)
E)
727
782
7170
49170
49720
Page 3 of 7 Solution: Choice (C)
If we first multiply inside the parentheses: 78 × 79 = 78+9 = 717
Then raise this to the 10th power: ( 717 ) = 7170
10
==========================
If ( 7
A)
a
) (7 )
b
7c
= d , what is d in terms of a, b, and c?
7
c
ab
B) c − a − b
C) a + b − c
D) c − ab
E)
c
a+b
Solution: Choice (B)
(7a ) (7b ) = 7a+b , and
7c
= 7c −d , so:
d
7
(7 a )(7b ) = 77d
c
7 a+b = 7 c−d
Now, since we have the same base “7” being used, these could only be equal to each other
IF their exponents are equal to each other. So our problem simplifies down to:
a+b= c−d
Now solve this equation for d:
d = c −a −b
==================================
For what value of x is 8
A) 2
B) 3
C) 4
D) 6
E) 8
2x-4
x
= 16 ?
Solution: Choice (D)
Page 4 of 7 Simplify both sides so that they will have the same “base”:
82 x−4 = 16 x
(2 )
3
2 x−4
( )
= 24
x
26 x−12 = 24 x
Now, since we have the same base “2” being used, these could only be equal to each other
IF their exponents are equal to each other. So our problem simplifies down to:
6x − 12 = 4x
2x = 12
x=6
===================================
Prime Numbers: They are divisible only by itself and 1. For example:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
Practice:
Quantity A
Quantity B
The number of
The number of
primes that are
primes that are
divisible by 2
divisible by 3
A) Quantity A is greater
B) Quantity B is greater
C) Quantities A and B are equal
D) It is impossible to determine which quantity is greater
Solution: Choice (C)
The only prime divisible by 2 is “2”, and the only prime divisible by 3 is “3”. Quantity A
and Quantity B are each 1.
integers:
..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...
positive integers: 1, 2, 3, 4, 5, ...
negative integers: ... -5, -4, -3, -2, -1
Every integer has a finite set of factors (or divisors):
Factors of 8: -8, -4, -2, -1, 1, 2, 4, 8
Every integer has an infinite set of multiples:
Multiples of 8: ..., -24, -16, -8, 0, 8, 16, 24, ...
Note: Zero is a multiple of every integer
Zero is an even integer
Page 5 of 7 Practice:
Quantity A
Quantity B
The number of
The number of
even positive
odd positive
factors of 30
factors of 30
A) Quantity A is greater
B) Quantity B is greater
C) Quantities A and B are equal
D) It is impossible to determine which quantity is greater
Solution: Choice (C)
List of factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Four of them are odd and four are even.
Every integer greater than 1 that is not a prime can be written as a product of primes.
Here you can see the prime factorization being done for 108 and 240 using a factor tree:
Practice:
Quantity A
Quantity B
The number of
The number of
different prime
different prime
2
factors of n
factors of n
A) Quantity A is greater
B) Quantity B is greater
C) Quantities A and B are equal
D) It is impossible to determine which quantity is greater
Page 6 of 7 Solution: Choice (C)
2
If you make a factor tree for n , the first branches would be n and n.
Now, when you factor each n, you get exactly the same prime factors.
For example, if n = 20:
GCF (Greatest Common Factor) or GCD (Greatest Common Divisor) – The product of all
the primes in each factorization, using each prime the smallest number of times it appears.
It’s the largest integer that is a factor of each of them.
Example: GCF for the numbers: 16 and 12
Prime factorization for 16: 2 x 2 x 2 x 2
Prime factorization for 12: 2 x 2 x 3
GCF: 2 x 2 = 4
LCM (Least Common Multiple) – The product of all the primes that appear in any of the
factorizations using each prime the largest number of times it appears. It’s the smallest
positive integer that is a multiple of each of them.
Example: LCM for the numbers: 16 and 12
Prime factorization for 16: 2 x 2 x 2 x 2
Prime factorization for 12: 2 x 2 x 3
LCM: 2 x 2 x 2 x 2 x 3 = 48
48 is the smallest multiple for both 16 and 12.
Page 7 of 7