Solving Mixture and Distance-Rate-Time Problems Using Two Variables 3.3

Solving Mixture Problems Using Two Variables

Introduction

Mixture problems in algebra involve combining two or more solutions with different concentrations to achieve a desired concentration. These problems are commonly solved using systems of equations with two variables.

Key Concepts

• Assign Variables: Let x and y represent the unknown quantities (e.g., ounces of each solution).

• Create a Table: Organize the information about each solution, including volume, concentration, and amount of pure substance.

• Set Up Equations: Use the total volume and total amount of pure substance to write two equations.

• Solve the System: Use substitution or elimination to solve for the variables.

Example: Mixing Hydrochloric Acid Solutions

Problem: How many ounces of 5% hydrochloric acid and 20% hydrochloric acid must be combined to obtain 10 oz of solution that is 12.5% hydrochloric acid?

• Let x: ounces of 5% solution

• Let y: ounces of 20% solution

Solution	Ounces	Concentration	Amount of Pure Acid
5% Solution	x	0.05	0.05x
20% Solution	y	0.20	0.20y
Mixture	10	0.125	0.125(10) = 1.25

• Equation 1 (Total Volume): $x + y = 10$

• Equation 2 (Total Acid): $0.05x + 0.20y = 1.25$

Solving the System

• Multiply Equation 2 by 100 to clear decimals: $5x + 20y = 125$

• Use elimination:

  • Multiply Equation 1 by 5: $5x + 5y = 50$

  • Subtract: $(5x + 20y) - (5x + 5y) = 125 - 50$

  • $15y = 75$

  • $y = 5$

  • Substitute into Equation 1: $x + 5 = 10 \Rightarrow x = 5$

Solution

• 5 oz of 5% solution

• 5 oz of 20% solution

Verification

• Total acid: $0.05(5) + 0.20(5) = 0.25 + 1.00 = 1.25$

• Concentration: $\frac{1.25}{10} = 0.125 = 12.5\%$

Solving Distance-Rate-Time Problems Using Two Variables

Introduction

Distance-rate-time problems involve finding unknown rates or times when two objects travel different distances at different speeds but share a common time or other relationship. These problems are solved using systems of equations.

Key Concepts

• Distance Formula: $d = r \times t$

• Assign Variables: Let x and y represent unknown rates or times.

• Set Up Equations: Use the relationships given in the problem to write two equations.

• Solve the System: Use substitution or elimination to find the unknowns.

Example: Car and Truck Travel Problem

Problem: A car travels 250 km in the same time that a truck travels 225 km. If the rate of the car is 8 km/hr faster than the rate of the truck, find both rates.

• Let x: rate of car (km/hr)

• Let y: rate of truck (km/hr)

Vehicle	Distance (km)	Rate (km/hr)	Time (hr)
Car	250	x	$\frac{250}{x}$
Truck	225	y	$\frac{225}{y}$

• Relationship: $x = y + 8$

• Equal time: $\frac{250}{x} = \frac{225}{y}$

Solving the System

• Substitute $x = y + 8$ into the time equation:

  • $\frac{250}{y + 8} = \frac{225}{y}$

  • Cross-multiply: $250y = 225(y + 8)$

  • $250y = 225y + 1800$

  • $25y = 1800$

  • $y = 72$

  • $x = y + 8 = 80$

Solution

• Rate of car: 80 km/hr

• Rate of truck: 72 km/hr

Verification

• Time for car: $\frac{250}{80} = 3.125$ hr

• Time for truck: $\frac{225}{72} \approx 3.125$ hr

Both times are equal, confirming the solution.