Problem Solving in College Algebra

Overview of the Problem-Solving Process

Solving applied problems in algebra involves translating real-world situations into mathematical language, formulating equations, and interpreting solutions. The process is systematic and requires careful analysis at each stage.

• Real Problem: The initial scenario or question from a real-life context.

• Verbal Description: Restating the problem in clear, everyday language.

• Language of Mathematics: Expressing the problem using variables, equations, and mathematical terms.

• Mathematical Problem: The formal algebraic representation to be solved.

• Solution: The answer, checked for consistency with the original context.

Steps for Solving Applied Problems

Follow these structured steps to approach and solve applied algebraic problems efficiently:

1. Read the problem carefully to understand what is being asked.

2. Assign a variable to represent the unknown quantity.

3. Translate all known facts into mathematical expressions or equations.

4. Solve the equation for the variable and answer the question.

5. Check the solution in the context of the problem.

Interest Problems

Simple Interest Formula

Interest problems involve calculating the amount earned or paid on a principal over time at a given rate. The simple interest formula is:

Where:

• I = Interest earned or paid

• P = Principal (initial amount)

• r = Annual interest rate (as a decimal)

• t = Time (in years)

Example: If you borrow $1000 for 6 months at 6% per annum, the interest is calculated as:

The total amount owed after 6 months is $1000 + $30 = $1030.

Mixture Problems

Mixture problems involve combining two or more substances with different characteristics to achieve a mixture with a desired property.

Example: Mixing two types of coffee to achieve a blend at a specific price.

The equation for the mixture is:

Solving for gives the required pounds of each type of coffee.

Uniform Motion Problems

Uniform Motion Formula

Uniform motion problems involve objects moving at constant speeds. The fundamental formula is:

Where:

• d = distance traveled

• r = rate (speed)

• t = time

Example: Tanya runs at 8 mi/hr, and you follow at 40 mi/hr after 2 hours. To find when you catch up:

Set distances equal to solve for :

Motion with Currents

When moving in a medium with a current (like a river), adjust the rate by adding or subtracting the current's speed.

The total time for the round trip is 6 hours:

Constant Rate Job Applications

Work Problems

Work problems involve finding how long it takes for people or machines working together to complete a task. The rate is typically given as "part of the job done per unit time."

The equation for their combined work is:

Solving for gives the time to complete the job together. In this example, hours, or 2 hours and 24 minutes.

Summary Table: Key Formulas