Subtraction
The definition of subtraction is: A - B = A + (-B)
A minus B is the same as A plus (the opposite of B)
X > 0, means that X is a positive number
X < 0, means that X is a negative number
-(A - B) = -A + B = B - A
(-X) 2 = X2
If X - 0, X 2 > 0
If, on the number line, one number occurs to the left of another
number, the number on the left is the smallest number.
Therefore, when studying the line above, you will know that X < Y and
Y < Z.
For example:
Use the number line to make conclusions with regards to whether each
number is positive or negative.
In this situation, you will have an easier time if you implement specific
numbers to fit the problem. For example, let X = -7, Y = -2, and Z =
3. Be certain to utilize some negative numbers while substituting.
The following is an example of a subtraction question:
Y - X
Solution: Positive Y is greater than X.
-2 - (-7) = -2 + 7 = 5
Evens and Odds
An even number is any word that is divisible by 2: numbers that are
within the set {¼-6, -4, -2, 0, 2, 4, 6,¼}. Remember, though, that an
even number is divisible by 2 and not have any remainder. Keep in
mind also that 0 is an even number. Consecutive even numbers are all
located 2 units apart. For example, if x is an even number, then the
next consecutive even number would be represented as X + 2.
Odd numbers, on the other hand, are numbers within the set {¼-5, -3,
-1, 1, 3, 5,¼}.
The following charts demonstrate the properties of odd and even
numbers. To check the property of a number, you can simply
substitute the appropriate numbers.
Properties of odd and even numbers with Addition
Property
Even + Even = Even
Odd + Odd = Even
Odd + Even = Odd
Example
2 + 8 = 10
3 + 9 = 12
3 + 8 = 11
Properties of odd and even numbers with Addition
Property
Even x Even = Even
Even x Odd = Even
Odd x Odd = Odd
Example
4 x 6 = 24
4 x 5 = 20
3 x 9 = 27
Consider the following example:
If R is an odd integer, what are the next two consecutive odd integers?
A) T and V
B) R and R+1
C) R+1 and R+2
D) R+2 and R+4
E) R+1 and R+3
Note: the correct answer is (D)
Here's another example:
If x is an odd integer and y is an even integer, tell whether each
expression is odd or even.
A. x2
B. xy
C. y2
D. x + y
E. 2x + y
Note (A) is odd. (B) is even. (C) is even. (D) is odd, and (E) is even.
Prime Numbers
A prime number is defined as an integer that is greater than 1, and
has only two positive factors, 1 and itself.
For example, 7 is a prime number, as its only factors are 1 and 7.
However, 6 is not a prime number, because its factors are 1, 2, 3, 6
The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Note, though that 1 is not a prime number, and both the smallest and
the only even prime number is 2.
Prime factorization is the process by which you express a number as a
result of only prime numbers.
For example:
To create the prime factorization of 24, you'd represent it as:
2 x 2 x 2 x 3 or 2 3 × 3
To create the prime factorization of 15, you'd represent it as:
5 x 3
An example of a factor question is:
If xy = 13 and both x and y are positive integers, then what is the sum
of x + y?
A. 13
B. 14
C. 16
D. 20
E. 26
Note: the answer is B
Here is another example:
What is the sum of the first 5 prime numbers?
A. 18
B. 28
C. 30
D. 34
E. 38
Note: The first five prime numbers are 2, 3, 5, 7, 11 and their sum is
28. The answer is B.
Percents
The word percent means "hundredths" or a number which is divided by
100. Converting a number into a percentage involves multiplying the
number by 100.
A percent can be determined by performing the division of the part by
the total and multiplying it by 100:
Percent = Part x 100
Total
For example, if Wendy missed 12 out of 80 examination questions,
what is the percent of questions she missed?
Percent = missed questions x 100 = 12/80 x 100 = 0.15 x 100 =
15%
Total
The phrase "X is N percent of Y" can also be written mathematically as
X = N
x Y
100
The word "is" means equal (=), while the word "of" means "multiply"
However, before multiplying, you must change a percent into a
decimal or fractional format.
For example:
5 is 20% of 25, means 5 = 0.20 x 25
To change the fraction into the percent, you must first change the
fraction into a decimal, and then multiply by 100 (or move the decimal
point by 2 places to the right)
For example:
Change the fraction 1/5 into a percent.
First, change the fraction 1/5 into the decimal 0.2, and multiply by 100
(move the decimal 2 places to the right). Therefore:
1/5 x 100 = 20%
The following table provides the common percentages that you will use
on a regular basis, and may wish to memorize.
Fraction
- 1/100 = 0.01
- 1/10 = 0.1
- 1/7 = 0.1428571
Percent
