2.2 Use a Problem Solving Strategy

Problem-solving in algebra is all about breaking down complex issues into manageable steps. By understanding the given info, assigning variables, and translating problems into equations, you can tackle even the toughest questions. This systematic approach helps you solve and interpret solutions effectively.

Percentages and financial applications are crucial real-world uses of algebra. From calculating discounts and markups to computing simple interest, these skills are essential for personal finance. Understanding how to work with percentages and apply them to various scenarios will serve you well in everyday life.

Problem Solving Strategy

Systematic problem-solving approach

Top images from around the web for Systematic problem-solving approach

• 

  Introduction to Problem Solving Skills | CCMIT View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Solving Linear Equations: Part I View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Solving Linear Equations: Part II View original

  Is this image relevant?

• 

  Introduction to Problem Solving Skills | CCMIT View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Solving Linear Equations: Part I View original

  Is this image relevant?

1 of 3

Top images from around the web for Systematic problem-solving approach

• 

  Introduction to Problem Solving Skills | CCMIT View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Solving Linear Equations: Part I View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Solving Linear Equations: Part II View original

  Is this image relevant?

• 

  Introduction to Problem Solving Skills | CCMIT View original

  Is this image relevant?

• 

  OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Solving Linear Equations: Part I View original

  Is this image relevant?

1 of 3

• Understand the given information
  • Identify known quantities (prices, ages, distances)
  • Determine the unknown quantity to solve for (total cost, future age, arrival time)
• Assign variables to unknown quantities
  • Choose meaningful variable names (x for the unknown value, t for time)
  • Define variables clearly to avoid confusion
• Translate the problem into mathematical equations
  • Express relationships between known and unknown quantities using mathematical operations (addition, subtraction, multiplication, division)
  • Set up equations based on the problem statement (if John is 5 years older than Amy, then John's age = Amy's age + 5)
• Solve the equations using appropriate algebraic techniques
  • Isolate the variable of interest by performing inverse operations
  • Apply inverse operations to solve for the unknown quantity (addition and subtraction, multiplication and division)
• Interpret the solution in the context of the original problem
  • Check if the solution makes sense (a negative age or distance is not realistic)
  • Provide a clear answer to the question asked in the appropriate units (years, dollars, miles)
• Use estimation to verify the reasonableness of the solution

Algebraic techniques for word problems

• Identify the type of word problem
  • Recognize patterns and common problem types (age problems, distance-rate-time problems, mixture problems)
• Extract relevant information from the problem statement
  • Identify given numbers and their relationships (John's age is 3 times Amy's age)
  • Determine the quantity to be found (the sum of their ages in 5 years)
• Represent the problem using algebraic expressions or equations
  • Assign variables to unknown quantities (let x represent Amy's current age)
  • Express relationships between quantities using mathematical symbols (John's age = 3x)
• Apply appropriate algebraic techniques to solve the equations
  • Simplify expressions by combining like terms
  • Isolate the variable of interest by performing inverse operations
  • Use inverse operations to solve for the unknown quantity (addition and subtraction, multiplication and division)
• Check the solution and interpret it in the context of the problem
  • Verify that the solution satisfies the given conditions (substitute the value back into the original equations)
  • Provide a clear and concise answer to the question asked in the appropriate units (Amy is currently 10 years old)
• Use visualization techniques (e.g., diagrams or graphs) to better understand complex problems

Enhanced problem-solving strategies

• Apply logical reasoning to break down complex problems into smaller, manageable steps
• Use a step-by-step approach to systematically work through each part of the problem
• Employ critical thinking skills to analyze the problem from different angles and consider alternative solutions

Percentages and Financial Applications

Percentages in real-world scenarios

• Understand the concept of percentages
  • Recognize that a percentage represents a fraction out of 100 (25% is equivalent to 25/100 or 0.25)
  • Convert between percentages, decimals, and fractions (75% = 0.75 = 3/4)
• Calculate discounts
  • Determine the discount amount using the formula: $\text{Discount} = \text{Original Price} \times \text{Discount Percentage}$ (a $50 item with a 20$10)
  • Find the discounted price by subtracting the discount from the original price (the discounted price is $40)
• Compute markups
  • Calculate the markup amount using the formula: $\text{Markup} = \text{Cost} \times \text{Markup Percentage}$ (a product with a cost of $20 and a markup of 50$10)
  • Determine the selling price by adding the markup to the cost (the selling price is $30)
• Apply tax calculations
  • Compute the tax amount using the formula: $\text{Tax} = \text{Pre-tax Amount} \times \text{Tax Rate}$ (a $100 item with an 8$8)
  • Find the total cost by adding the tax to the pre-tax amount (the total cost is $108)

Simple interest calculations

• Understand the components of simple interest
  • Principal ($P$): The initial amount borrowed or invested ($1000 loan,$500 investment)
  • Interest Rate ($r$): The percentage of the principal charged as interest, usually expressed as an annual rate (5% per year, 3.5% annually)
  • Time ($t$): The duration of the loan or investment, typically measured in years (2 years, 18 months)
• Use the simple interest formula: $I = P \times r \times t$
  • $I$ represents the interest earned or paid
  • Multiply the principal by the interest rate and time to calculate the interest ($1000 \times 0.05 \times 2 =$100)
• Solve for the total amount ($A$) using the formula: $A = P + I$
  • Add the principal and interest to find the total amount after the specified time ($1000 +$100 = $1100)
• Apply the simple interest formula to various financial scenarios
  • Calculating interest earned on investments (a $2000 investment at 4$240 in interest)
  • Determining interest paid on loans (a $5000 loan at 6$600 in interest payments)