Mixture and Investment Word Problems

Word problems involving mixtures and investments are common applications of algebraic equations. These problems require translating real-world scenarios into mathematical models, usually involving systems of linear equations.

Investment Problems

Investment problems typically involve dividing a sum of money into parts, each invested at different interest rates, to achieve a desired total income.

• Key Terms:

  • Principal: The original amount of money invested.

  • Interest Rate: The percentage at which the money grows annually.

  • Annual Income: The total interest earned in one year.

• General Approach:

  1. Let variables represent the unknown amounts invested at each rate (e.g., let be the amount invested at one rate, at another).

  2. Write an equation for the total amount invested:

  3. Write an equation for the total annual income:

  4. Solve the system of equations for and .

• Example:

  An investor has $2900 to invest, part at 8% and the rest at 5%. How much should be invested at each rate to obtain an annual income of $204?

  • Let = amount invested at 8%, = amount invested at 5%.

  • Equation 1:

  • Equation 2:

  • Solution:

    1. From Equation 1:

    2. Substitute into Equation 2:

    3. Solve for :

    Answer: Invest $1966.67 at 8% and $933.33 at 5%.

Mixture Problems

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

• Key Terms:

  • Concentration: The percentage of a substance in a solution.

  • Volume: The amount of solution, usually in cubic centimeters (cc) or liters.

• General Approach:

  1. Let variables represent the unknown volumes of each solution (e.g., cc of one solution, cc of another).

  2. Write an equation for the total volume:

  3. Write an equation for the total amount of substance:

  4. Solve the system of equations for and .

• Example:

  A nurse has two solutions: one is 12% medication, the other is 8%. How many cc of each should be mixed to obtain 20 cc of a 9% solution?

  • Let = cc of 12% solution, = cc of 8% solution.

  • Equation 1:

  • Equation 2:

  • Solution:

    1. From Equation 1:

    2. Substitute into Equation 2:

    Answer: Mix 5 cc of 12% solution with 15 cc of 8% solution.

Solving Systems of Linear Equations

Both mixture and investment problems are solved using systems of linear equations. The most common methods are substitution and elimination.

• Substitution Method:

  1. Solve one equation for one variable.

  2. Substitute this expression into the other equation.

  3. Solve for the remaining variable.

• Elimination Method:

  1. Multiply one or both equations to align coefficients.

  2. Add or subtract equations to eliminate one variable.

  3. Solve for the remaining variable.

• Example:

  Given the system:

  Using substitution, as shown above, leads to , .

Summary Table: Steps for Solving Mixture and Investment Problems

Additional info: The original file contained handwritten and fragmented notes with partial equations and numbers. The above study notes reconstruct the intended problems and provide full academic context for clarity and completeness.