2.4 Solve Mixture and Uniform Motion Applications

Mixture and uniform motion problems are all about combining things or tracking movement. These applications use simple equations to solve real-world scenarios involving coins, tickets, solutions, and travel.

You'll learn to set up equations using given info, like coin values or speeds. Then, you'll solve for unknowns using algebra. This helps you tackle practical math problems you might encounter in daily life or future careers.

Mixture and Uniform Motion Applications

Combinations of coins and currency

• Identify the types of coins or bills involved in the problem
  • Common U.S. coins include pennies (1¢), nickels (5¢), dimes (10¢), and quarters (25¢)
  • Common U.S. bills include $1,$5, $10,$20, $50, and$100 denominations
• Determine the quantity of each type of coin or bill given in the problem
• Set up an equation based on the given information
  • Let variables represent the unknown quantities such as letting $x$ be the number of quarters
  • Use the total value or total number of coins/bills to create an equation (e.g., if the total value is $5.75 and there are 3 types of coins, set up an equation like$0.01x + 0.05y + 0.25z = 5.75$)
• Solve the equation to find the unknown quantities using algebra techniques (isolating variables, substitution)
• Verify the solution by substituting the values back into the original problem to ensure it satisfies all conditions

Scenarios with tickets and stamps

• Identify the types of tickets or stamps involved in the problem (e.g., adult tickets, child tickets, first-class stamps, second-class stamps)
• Determine the price or value of each type of ticket or stamp provided in the problem
• Set up an equation based on the given information
  • Let variables represent the unknown quantities such as letting $x$ be the number of adult tickets
  • Use the total value or total number of tickets/stamps to create an equation (e.g., if the total cost is $100 and there are 2 types of tickets, set up an equation like$10x + 5y = 100$)
• Solve the equation to find the unknown quantities using algebra techniques (isolating variables, substitution)
• Check the solution by substituting the values back into the original problem to verify it meets all requirements

Concentrations in mixture problems

• Identify the substances being mixed and their initial concentrations or quantities (e.g., salt solutions, alcohol solutions, alloys)
• Determine the final concentration or quantity of the mixture desired or given in the problem
• Set up an equation using the mixture formula: $C_1V_1 + C_2V_2 = C_fV_f$
  • $C_1$ and $C_2$ are the initial concentrations of the substances (e.g., 20% salt solution, 5% alcohol solution)
  • $V_1$ and $V_2$ are the initial volumes or quantities of the substances (e.g., 3 liters of 20% salt solution, 2 liters of 5% alcohol solution)
  • $C_f$ is the final concentration of the mixture (e.g., desired final concentration of 15% salt solution)
  • $V_f$ is the final volume or quantity of the mixture (e.g., total volume of 5 liters)
• Solve the equation for the unknown variable using algebra techniques (isolating variables, substitution)
• Verify the solution by substituting the values back into the mixture formula to ensure it balances
• Use proportions to express concentrations as ratios of volume or mass

Linear equations for uniform motion

• Identify the given information: distance traveled, rate (speed), and/or time elapsed
• Use the uniform motion formula: $D = RT$
  • $D$ is the distance traveled (e.g., 240 miles)
  • $R$ is the rate or speed (e.g., 60 miles per hour)
  • $T$ is the time (e.g., 4 hours)
• If needed, convert units to ensure consistency (e.g., convert minutes to hours, feet to miles)
• Set up an equation using the uniform motion formula based on the given information (e.g., if distance and rate are known, set up $D = 60T$ to solve for time)
• Solve the equation for the unknown variable (distance, rate, or time) using algebra techniques (isolating variables)
• Check the solution by substituting the values back into the uniform motion formula to verify it holds true
• For problems involving multiple moving objects:
  1. Determine the relative positions and directions of the objects (e.g., two trains starting 300 miles apart and traveling towards each other)
  2. Set up equations for each object using the uniform motion formula (e.g., $D_1 = 60T$ and $D_2 = 80T$)
  3. Use additional information, such as total distance or meeting times, to create a system of equations (e.g., $D_1 + D_2 = 300$ and $T_1 = T_2$)
  4. Solve the system of equations to find the unknown variables using algebra techniques (substitution, elimination)

Algebraic Modeling and Problem Solving

• Identify key information in word problems related to mixtures and uniform motion
• Translate problem statements into mathematical expressions and equations
• Use variables to represent unknown quantities in the problem
• Apply appropriate formulas and equations based on the problem context
• Utilize conversion factors when necessary to ensure consistent units
• Solve the resulting equations or systems of equations using algebraic techniques
• Interpret the solution in the context of the original problem and verify its reasonableness