Word Problems and Applications

Sales Tax, Total Cost, and Change

Understanding how to calculate total cost, including sales tax, and determining change is a fundamental application of arithmetic and algebra in real life.

• Step 1: Find the total cost of identical items Multiply the price per item by the number of items. Example: 3 pairs of socks at

• Step 2: Find the subtotal before tax Add the cost of all items purchased. Example:

• Step 3: Calculate the sales tax Convert the percent to a decimal and multiply by the subtotal. Example:

• Step 4: Find the total cost Add the sales tax to the subtotal. Example:

• Step 5: Find the change Subtract the total cost from the amount paid. Example:

Unit Rate and Cost per Ounce

Unit rates allow comparison of prices and quantities in different units.

• Step 1: Convert units if necessary 1 pound = 16 ounces.

• Step 2: Find cost per ounce Divide the price per pound by 16. Example: dollars per ounce.

• Step 3: Convert to cents Multiply by 100. Example: cents per ounce.

Geometry and Measurement

Hypotenuse of a Right Triangle

The Pythagorean Theorem relates the lengths of the sides of a right triangle.

• Step 1: Identify the legs Vertical leg: difference in y-values; Horizontal leg: difference in x-values.

• Step 2: Find the lengths For points (2,4), (2,−1), (0,−1): Vertical leg: Horizontal leg:

• Step 3: Apply the Pythagorean Theorem Substitute:

Area of a Composite Figure

Composite figures are broken into simpler shapes for area calculation.

• Step 1: Decompose the figure Break into rectangles, circles, etc.

• Step 2: Use area formulas Rectangle: Circle: (use )

• Step 3: Substitute dimensions and add areas

Volume of a Cylinder

The volume of a cylinder is found using the formula:

• Formula:

• Step 1: Identify radius and height Radius is half the diameter.

• Step 2: Substitute values For diameter 10, ; height

Ratios, Proportions, and Percents

Solving Ratio Problems

Ratios compare quantities and can be used to divide groups proportionally.

• Step 1: Add ratio parts Residents:Visitors = 2:3; Total parts =

• Step 2: Find value of one part Total people total parts:

• Step 3: Find number in each group Visitors:

Percent Equations

Percent problems can be solved using equations.

• Step 1: Convert percent to decimal 75% = 0.75

• Step 2: Write and solve the equation Divide both sides by 0.75:

Roots and Exponents

Square Roots and Cube Roots

Roots are the inverse operation of exponents.

• Square Root: A number that, when squared, gives the original number. Example: and (since and )

• Cube Root: A number that, when cubed, gives the original number. Example: (since )

Exponents

Exponents represent repeated multiplication.

• Zero exponent: Any nonzero number to the zero power is 1. Example:

• Negative exponent: Example:

• Evaluate:

Scientific Notation

Scientific notation expresses very large or small numbers as a product of a coefficient and a power of 10.

• Step 1: Multiply coefficients Multiply:

• Step 2: Add exponents

• Step 3: Write in proper form

Equations and Inequalities

Literal Equations

Literal equations involve solving for a variable in terms of others.

• Example (a): Subtract from both sides:

• Example (b): Divide both sides by :

Solving Linear Inequalities

Solving inequalities is similar to solving equations, but the solution is a range of values.

• Example:

• Subtract 8:

• Divide by 6:

• Graph: Open circle at 1, shade to the left (values less than 1)

Order of Operations

Evaluating Expressions

Order of operations ensures consistent results in calculations.

• Step 1: Parentheses

• Step 2: Exponents

• Step 3: Multiplication/Division

• Step 4: Addition/Subtraction

• Example: First, Then, Then, Add: Subtract:

Functions and Graphing

Graphing Linear Equations

Linear equations can be represented as straight lines on the coordinate plane.

• Equation:

• Choose x-values and find y-values:

• Plot these points and draw a straight line through them.

Number Systems

Rational Numbers and the Number Line

Rational numbers can be written as fractions or decimals that terminate or repeat.

• Step 1: Convert all numbers to decimals

• Step 2: Order from least to greatest and plot on a number line

• Step 3: Identify rational numbers Numbers that can be written as , where and are integers and .

Summary Table: Key Formulas and Concepts

Additional info: These notes synthesize and expand upon the provided material, ensuring each topic is self-contained and includes definitions, examples, and step-by-step procedures relevant to beginning and intermediate algebra students.