# SAT Math: Numbers and Operations

Numbers and operations is an area of SAT Math (and PSAT Math) used in most of the word problems and problems involving percentages, averages, and sequences.

### Numbers and Integers on the SAT

• #### Number Classifications

In the SAT Math section, you will only deal in whole numbers and fractions. While all fractions can be represented in decimal form, it is generally advisable to keep them as fractions. All multiple choice selections will be presented in either fraction or whole number form.

• #### Integers

Integers are whole numbers. All integers greater than zero are known as positive integers, and all integers less than zero are known as negative integers. Zero is neither positive nor negative.

• #### Consecutive Integers

Consecutive Integers are integers in sequence, without skipping any integers. An example of this is the series {9,10,11,12,13,14}. A consecutive integer series may also include zero, for example the series {-2,-1,0,1,2,3).

• #### Odd Integers

Odd Integers are all of the integers in the set of n=2k+1, where k is any integer. In simpler terms, it is every other number starting with 1. So, {1,3,5,7,9,11,13,…}. The negatives of any of these numbers is also odd, so another example of a set of odd numbers is {…,-3,-1,1,3,5,…}.

• #### Even Integers

Even Integers are all integers in the set of n=2k, where k is any integer. This is also all numbers divisible by 2 with no remainder. Zero is considered an even number, despite the fact that it is neither positive nor negative and not divisible by 2.

### Some Rules Regarding even and odd numbers:

When adding, subtracting, or multiplying even and odd numbers:
even ± even = even
even ± odd = odd
odd ± odd = even
even × even = even
even × odd = even
odd × odd = odd

• #### Prime Numbers

Prime Numbers are numbers that are divisible with no remainder by only 2 and itself. By definition 1 is not considered a prime number, the smallest prime number is 2. The set of prime numbers includes {2,3,5,7,11,13,17,…}. All integers in the positive real domain can be broken down into factors of prime numbers.

• #### Rational Numbers

Rational Numbers are numbers that can be expressed by a fraction of integers. For example 0.25 is rational because it can be expressed as 1/4. In decimal form, rational numbers will either terminate after a finite amount of digits or begin to repeat the same sequence of digits. All numbers which do not fit these criteria are called irrational numbers. An example of an irrational number is π.

• #### Average

The average of a set of numbers is defined as the sum of that set of numbers divided by the number of elements in the set.

Example:  if you have the set of numbers {3,7,4,9,2,3}, there are 6 numbers in the set and the average is

average=(3+8+5+9+2+3)/6

=30/6

=5

The average does not necessarily have to be part of the set of numbers, although in our example it was.

• #### Average Speed

The average speed of a time period is defined as the total distance traveled divided by the time it takes to travel that distance.

average speed=(distance traveled)/time.

If you are dealing with a situation where you travel at two different speeds for different amounts of time, you can calculate the total distance traveled by multiplying the time of each leg with its respective speed. From there, you can find the average speed of the entire trip by dividing the total distance traveled by the amount of time it took to travel that entire distance.

• #### Percent Increase/Decrease

Percent is hundredths, which means out of a hundred.

Example: 60%= 60/100= 3/5 = 0.6

All percentages can be represented as a number with a % sign after it, as a fraction, or as a decimal. Percent increase means the percent of the original that a value increases. Percent decrease means the percent of the original that a value decreases. The definition is:

% increase=  increase/original
% decrease= decrease/original

Example: A dress on sale is marked down from \$20 to \$14. What percent discount (decrease) has been applied to the price of the dress?

The difference (decrease) is equal to \$6. Dividing that by the original price (\$20) gives you the equation (\$6/\$20) which can be simplified to (3/10) or (30/100). Stated as a percentage, the discount (decrease) is equal to 30%.

• #### Sequences

Sequences are sets of numbers where each number is found by performing arithmetic operations involving the term or terms preceding it. Two common sequences are arithmetic and geometric sequences.

• #### Arithmetic Sequences

Arithmetic Sequences are sequences in which each term differs from the term before it by a constant amount, that can be positive or negative. The sequence {3,9,15,21,27,…} is an arithmetic sequence because each term is 6 greater than the term before it.

• #### Geometric Sequences

Geometric Sequences are sequences where each term is a constant ratio of the term before it. What that means is that each term is equal to the term before it multiplied by a constant. The sequence {4,12,36,108,…} is a geometric sequence because each term is equal to the term before it multiplied by 3.

Sequences are not limited to arithmetic and geometric. Each term can have any relation to the term before it, be it addition, subtraction, multiplication or division. A term may also be defined by multiple terms preceding it.

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