BackCollege Algebra Study Notes: Mixture and Investment Word Problems
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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:
Let variables represent the unknown amounts invested at each rate (e.g., let be the amount invested at one rate, at another).
Write an equation for the total amount invested:
Write an equation for the total annual income:
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:
From Equation 1:
Substitute into Equation 2:
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:
Let variables represent the unknown volumes of each solution (e.g., cc of one solution, cc of another).
Write an equation for the total volume:
Write an equation for the total amount of substance:
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:
From Equation 1:
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:
Solve one equation for one variable.
Substitute this expression into the other equation.
Solve for the remaining variable.
Elimination Method:
Multiply one or both equations to align coefficients.
Add or subtract equations to eliminate one variable.
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
Step | Description |
|---|---|
1 | Define variables for unknown quantities. |
2 | Write an equation for the total amount (investment or volume). |
3 | Write an equation for the total value (income or substance). |
4 | Solve the system of equations using substitution or elimination. |
5 | Interpret the solution in the context of the problem. |
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.
