3.3 Solve Mixture Applications

Mixture applications in algebra involve combining different elements to solve real-world problems. These range from calculating coin combinations to determining concentrations in chemical solutions. Understanding these concepts helps you tackle everyday math challenges and prepares you for more advanced problem-solving.

In this section, we'll explore various mixture scenarios, including coin calculations, ticket pricing, and interest calculations. You'll learn how to set up equations, solve for unknowns, and apply algebraic techniques to find solutions. These skills are crucial for many practical situations and future math courses.

Solving Mixture Applications

Coin combination calculations

• Identify the types of coins involved and their respective values
  • Common U.S. coins: penny ($0.01), nickel ($0.05), dime ($0.10), quarter ($0.25)
  • Other coins: half-dollar ($0.50), dollar coin ($1.00)
• Determine the total value of the coin mixture
  • Multiply the number of each coin type by its value and sum the results
  • Example: 5 quarters, 10 dimes, 20 nickels = $5 \times$0.25 + $10 \times$0.10 + $20 \times$0.05 = $3.25
• Set up an equation based on the given information
  • Let variables represent the unknown quantities of each coin type (e.g., let $x$ be the number of quarters, $y$ be the number of dimes)
  • The sum of the coin quantities should equal the total number of coins
  • Example: $x + y = 15$ coins in total
• Solve the equation to find the quantities of each coin type
  • Use algebra techniques such as substitution or elimination to solve for the variables
  • Example: If the total value is $2.50 and there are 15 coins, solve$0.25x + 0.10y = 2.50$and$x + y = 15$

Multi-item quantity and pricing

• Identify the different types of tickets or stamps and their respective prices
  • Example: adult tickets at $12 each, child tickets at$8 each
• Determine the total cost or revenue from the sale of tickets or stamps
  • Multiply the quantity of each item by its price and sum the results
  • Example: 50 adult tickets and 30 child tickets sold = $50 \times$12 + $30 \times$8 = $840
• Set up an equation based on the given information
  • Let variables represent the unknown quantities of each item type (e.g., let $a$ be the number of adult tickets, $c$ be the number of child tickets)
  • The sum of the item quantities should equal the total number of items
  • Example: $a + c = 80$ tickets sold in total
• Solve the equation to find the quantities of each item type
  • Use algebra techniques such as substitution or elimination to solve for the variables
  • Example: If the total revenue is $840 and there are 80 tickets sold, solve$12a + 8c = 840$and$a + c = 80$
• Consider using proportions to solve problems involving ratios of different item types

Mixture problem solutions

• Identify the substances or concentrations being mixed and their respective quantities
  • Example: mixing a 20% alcohol solution with a 50% alcohol solution
• Determine the concentration of the final mixture
  • Use the formula: $\text{Final Concentration} = \frac{\text{Total Amount of Substance}}{\text{Total Volume of Mixture}}$
  • Example: If 2 liters of 20% solution and 3 liters of 50% solution are mixed, the final concentration is $\frac{2 \times 0.2 + 3 \times 0.5}{2 + 3} = 0.38$ or 38%
• Set up an equation based on the given information
  • Let variables represent the unknown quantities or concentrations (e.g., let $x$ be the volume of 20% solution, $y$ be the volume of 50% solution)
  • The sum of the individual substance amounts should equal the total amount in the mixture
  • Example: $0.2x + 0.5y = 0.38(x + y)$
• Solve the equation to find the unknown quantities or concentrations
  • Use algebra techniques such as substitution or elimination to solve for the variables
  • Example: If the final mixture is 10 liters with a 38% concentration, solve $0.2x + 0.5y = 0.38(10)$ and $x + y = 10$
• Consider using the weighted average method for problems involving multiple components with different concentrations

Simple interest mixture models

• Identify the principal (initial investment), interest rate, and time period
  • Example: $1000 invested at 5% annual interest for 3 years
• Calculate the simple interest earned using the formula:
  • $\text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time}$
  • Rate is typically expressed as a decimal (e.g., 5% = 0.05)
  • Time is usually expressed in years
  • Example: $\text{Interest} =$1000 \times 0.05 \times 3 = $150
• Determine the total amount (principal + interest) after the specified time period
  • $\text{Total Amount} = \text{Principal} + \text{Interest}$
  • Example: $\text{Total Amount} =$1000 + $150 =$1150
• Solve for unknown variables in the simple interest formula
  • Rearrange the formula to solve for the principal, rate, or time when given the other values
  • Example: If the interest earned is $150 over 3 years and the rate is 5$150 = P \times 0.05 \times 3$

Additional Concepts in Mixture Applications

• Systems of equations are often used to solve complex mixture problems with multiple unknowns
• Unit conversion may be necessary when dealing with different units of measurement in mixture problems
• Algebra techniques such as combining like terms and distributing are essential for simplifying and solving mixture equations