Mathematics Review for Exam Preparation

Real Number Set

Includes all numbers used in everyday life and is comprised of rational and irrational numbers.

Rational Numbers

Numbers that can be expressed as ratios (i.e., fractions) where the denominator is not zero.

Examples of Rational Numbers

2/3, 4/1, or even 1/½.

Irrational Numbers

Numbers like pi or the square root of 2 that have non-repeating, non-terminating decimal expressions.

Place Value

Refers to the value that each digit in a number has, based on its position.

Units of Ten

It takes ten ones to make one ten, ten tens to make one hundred, and so on.

Example of Place Value

In the number 1,234,567, the rightmost digit represents ones, the digit to the immediate left represents tens, and so on.

Decimal Representation

Decimals represent part of a whole, the entire whole, or more than the whole.

Tenths Place

The first digit after the decimal point.

Hundredths Place

The next digit after the decimal point.

Fraction Definition

A fraction is another way to write a division problem, consisting of a numerator on top and a denominator on the bottom.

Numerator

Shows how many equal parts of the whole or collection are taken.

Denominator

Shows the total number of equal parts the whole is divided into or the total number of the same objects in a collection.

Percent Definition

A percent or percentage is a fraction with a denominator equal to 100.

Percentage Symbol

The symbol (%) is another way of writing a denominator of 100.

Example of Percentage

25% is equal to 25/100, which reduces to 1/4.

Example of Percentage Greater than 1

150% = 150/100, which is greater than 1 whole.

Example of Decimal

In 8.123456, 8 is the ones place, 1 is the tenths place, 2 is the hundredths place, and so on.

Thousandths Place

The third digit after the decimal point.

Ten-Thousandths Place

The fourth digit after the decimal point.

Hundred-Thousandths Place

The fifth digit after the decimal point.

Millionths Place

The sixth digit after the decimal point.

Proper fraction

The numerator is less than or equal to the denominator.

Improper fraction

The numerator is greater than or equal to the denominator; when the numerator equals the denominator, the fraction is equal to 1 whole.

Mixed number

The sum of a whole number and a fraction, or just another way of writing an improper fraction.

Complex fraction

The numerator and/or the denominator are fractions.

Equal fractions

Fractions that represent the same value.

Equivalent fractions

Fractions that have different numerators and denominators but represent the same value.

Least common multiple (LCM)

The smallest number that two or more numbers divide into evenly.

Greatest common factor (GCF)

The largest number that divides evenly into two or more numbers.

Reciprocal

The resulting fraction when you switch the numerator and denominator; when you multiply a number and its reciprocal, the result is 1.

Converting Improper Fractions to Mixed Numbers

Divide the denominator into the numerator; the result is the whole number part, and the remainder becomes the numerator part.

Converting a Mixed Number into an Improper Fraction

Multiply the denominator by the whole number and add the result to the numerator; the answer becomes the numerator of the improper fraction.

Reducing Fractions

Dividing the numerator and denominator by the largest number that divides evenly into both.

Adding and Subtracting Fractions

All fractions must have a common denominator; add or subtract the numerators and do not add or subtract the denominators.

Lowest common denominator (LCD)

The smallest number that divides evenly into all denominators in the problem.

LCD

The smallest number that two or more denominators go into.

Equivalent fractions

Fractions that represent the same value, achieved by multiplying the numerator and denominator by the same number.

Improper fraction

A fraction where the numerator is greater than or equal to the denominator.

Mixed number

A whole number combined with a proper fraction.

Multiplying fractions

Multiply the numerators and multiply the denominators (straight across).

Reciprocal

The inverse of a fraction, obtained by flipping the numerator and denominator.

Decimal to percentage conversion

To convert a decimal to a percentage, multiply by 100.

Percentage to decimal conversion

To convert a percentage to a decimal, divide by 100.

Fraction to percentage conversion

Change the fraction to a decimal by dividing the numerator by the denominator, then multiply by 100.

Common equivalencies

Connections between fractions, decimals, and percents, such as 1/4 = 0.25 = 25%.

Basic percentage problems

Problems that involve a percent, a whole, and a part.

Percent

A ratio expressed as a fraction of 100.

Whole

The total amount or quantity in a percentage problem.

Part

The portion of the whole in a percentage problem.

Percentage equation

An equation formed by converting percentage problems into numerical form.

What is 10% of 50?

Translate to x = 0.1 * 50 = 5.

5 is 10% of what number?

Translate to (0.1) * (x) = 5, then x = 50.

What percentage of 50 is 5?

Translate to x = (5 / 50) * 100.

Decimal point movement

To convert a decimal to a percentage, move the decimal point two places to the right.

Fraction representation

A fraction can be represented as a decimal and a percentage.

Multiplying a fraction by a whole number

Convert the whole number to a fraction with a denominator of 1.

Reducing fractions

Simplifying a fraction to its lowest terms.

Adding fractions

Convert to equivalent fractions with a common denominator before adding.

Subtracting fractions

The same procedure as adding fractions applies.

x%

Represents a percentage where 50 is the whole and 5 is the part.

Percentage Calculation Formula

x% * 50 = 5.

Decimal Conversion

To convert a percentage to a decimal, divide by 100.

Discount Calculation Steps

1. Change the percentage to a decimal. 2. Multiply by the original cost. 3. Add or subtract accordingly.

Example of Discount Calculation

A $250 stereo discounted by 18% results in a sale price of $205.

Sales Tax Calculation Steps

1. Convert the sales tax percentage to a decimal. 2. Multiply by the original cost. 3. Add the tax to the original cost.

Example of Sales Tax Calculation

A $230 camera with an 8.5% sales tax costs $249.55.

Percentage Increase Calculation Steps

1. Write the amount of increase as the numerator. 2. Write the original amount as the denominator. 3. Change the fraction to a percentage.

Example of Percentage Increase Calculation

A height increase from 60 to 66 inches is a 10% increase.

Percentage Decrease Calculation Steps

1. Find the amount of decrease. 2. Write the original amount as the denominator. 3. Change the fraction to a percentage.

Example of Percentage Decrease Calculation

A bank account decrease from $150.00 to $120.00 is a 20% decrease.

Rounding Definition

Rounding means replacing a number with an approximate value that has a shorter or simpler representation.

Rounding Steps

1. Underline the place value to which you are rounding. 2. Examine the number to the right. 3. Round up if 5 or greater, leave unchanged if less than 5. 4. Change all numbers to the right to zero.

Example of Rounding to Nearest Hundred

8,248 rounded to the nearest hundred is 8,200.

Example of Rounding to Nearest One

84.71 rounded to the nearest one is 85.

Example of Rounding to Nearest Tenths

458.296 rounded to the nearest tenths is 458.3.

Variable Definition

A variable is a symbol that takes the place of a number in an algebraic expression or equation.

Multiplication Notation

Multiplication can be denoted using (), *, · , X, or by placing variables next to each other.

Algebraic Expression Definition

An algebraic expression is a collection of numbers and variables connected by signs and symbols.

Example of Multiplication Notation

3z X 4, 3z * 4, (3z)4 are all equivalent.

Algebraic Expression Example

An example of an algebraic expression is 3x + 5y - 2.

Term

A term is an individual piece of an expression. It can be a number, a variable, or a product of numbers and variables.

Coefficient

A coefficient is the numerical factor of a term.

Like terms

Like terms exist when two or more terms have the same variables raised to the same power (exponent).

Unlike terms

Unlike terms are terms that do not have the same variables raised to the same power.

Equation

An equation is a mathematical sentence formed by joining two expressions with an equal sign stating that the two expressions are equal to each other.

Inequality

An inequality is a mathematical sentence stating that one expression is greater than ( > ) or less than ( < ) the other.

Exponents

Exponents are a simple way of notating that a number or variable (the base) is multiplied by itself a specific number of times.

Multiplying Exponents

To multiply similar bases, simply add the exponents.

Dividing Exponents

To divide similar bases, subtract the exponents.

Raising a Power to a Power

When raising a power to another power, multiply the exponents.

Negative Exponents

When the exponent is negative, move its base to the denominator and make the exponent positive.

Zero Exponent

Any base (except zero) raised to the zero power is equal to 1.

Exponential Roots

Exponential roots are the exponent's base.

Square Root

The square root of r2 (often written as √r2) is r.

Cube Root

The cube root of x3 (often written as √x3) is x.

Distributive Property

The distributive property is a multiplication property used in algebra to create and eliminate groups of terms.

Evaluating Expressions

To evaluate means to find the value of something by substituting a number for a variable into the expression.

Order of Operations

The specific order in which mathematical operations are performed when simplifying expressions.

Grouping Symbols

Start from the innermost set of grouping symbols (parenthesis, brackets, or braces) and perform the operations within.