Problem 54 - Intermediate Algebra

Question

Room Freshener. A florist sells mixtures of dricd, fragrant plant material that provides a gentle natural scent for houses. She wants to mix lilac (that sells for \(\$ 18.25\) a pound) with lavender (that sells for \(\$ 12.25\) a pound) to create 30 pounds of a blend that sells for \(\$ 15\) a pound. How many pounds of each should the florist use?

Step-by-Step Solution

Answer

Use 13.75 pounds of lilac and 16.25 pounds of lavender.

Step 1: Define Variables

Let \( x \) represent the number of pounds of lilac, and \( y \) represent the number of pounds of lavender. We know the total weight is 30 pounds, so we have: \( x + y = 30 \).

Step 2: Set Up Cost Equation

We want to create a mixture that costs \\(15 per pound from lilac costing \\)18.25 per pound and lavender costing \$12.25 per pound. The equation for the total cost is: \( 18.25x + 12.25y = 15 \times 30 \). Simplifying, we find: \( 18.25x + 12.25y = 450 \).

Step 3: Solve the System of Equations

We have a system of equations now:1. \( x + y = 30 \)2. \( 18.25x + 12.25y = 450 \)Substitute \( y = 30 - x \) into the second equation:\[ 18.25x + 12.25(30 - x) = 450 \]Simplify and solve for \( x \).

Step 4: Simplify and Solve for x

Substituting \( y = 30 - x \) into our cost equation, we have:\[ 18.25x + 12.25(30 - x) = 450 \]Distribute the \(12.25\):\[ 18.25x + 12.25 \times 30 - 12.25x = 450 \]\[ 18.25x - 12.25x + 367.5 = 450 \]Combine like terms:\[ 6x = 82.5 \]Thus, \( x = \frac{82.5}{6} = 13.75 \) pounds of lilac.

Step 5: Calculate y

Using \( y = 30 - x \), substitute the value for \( x \):\[ y = 30 - 13.75 = 16.25 \]Therefore, \( y = 16.25 \) pounds of lavender.

Key Concepts

Linear Equations

Linear equations are fundamental tools in algebra that allow us to express relationships using variables and constants. In this exercise, the florist's problem of mixing lilac and lavender can be modeled using linear equations. Linear equations present themselves in the form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \), \( y \) are the variables.

The problem provides a perfect example with two linear equations:

• The first equation \( x + y = 30 \) relates to the total weight of the mixture.
• The second equation \( 18.25x + 12.25y = 450 \) represents the total cost of the blend, derived from each component's price per pound and the price for the final mixture.

Solving systems like these helps one determine the quantities of each item to achieve a desired output, whether it's a specific weight or cost.

Understanding how linear equations work and setting up equations to solve real-world problems make them indispensable in cost analysis and various fields of study.

Cost Analysis

Cost analysis involves evaluating the total cost associated with a particular business decision, such as producing a blend of products. In this scenario, the florist must determine how much of each type of plant to use to create a mixture that sells for a target price per pound. The detailed cost equation laid out in the problem is a critical step in this analysis.

Understanding cost analysis through the cost equation \( 18.25x + 12.25y = 450 \) focuses on:

• Cost per unit: Each pound of lilac costs \( \\( 18.25 \), and each pound of lavender costs \( \\) 12.25 \).
• Total desired cost: The florist wants the 30 pounds of the mixture to cost \( \$ 450 \).

By setting an equation like this, businesses can determine how to mix various components of differing costs to achieve a specific benchmark price for their products. This process exemplifies the importance of mathematics in making informed financial decisions.

Problem Solving Steps

Problem-solving is an essential skill, especially when dealing with real-world applications like this florist's dilemma. Addressing such problems requires a structured approach:

1. **Define Variables**: Identify the unknowns, in this case, the pounds of lilac \( x \) and lavender \( y \). Effectively defining variables aids in setting up realistic and logical equations.
2. **Set Up Equations**: Use the given relationships and constraints to write equations. We established two equations related to weight and cost in this scenario.
3. **Substitute**: Simplify the equations by substituting expressions, such as replacing \( y \) with \( 30 - x \). This helps in reducing the number of variables to solve for.
4. **Solve**: Perform the algebraic mission of solving the equation for one variable. This often involves combining like terms, performing subtraction/addition, or multiplication/division to isolate the desired variable.
5. **Recalculate**: Once one variable is found, use it to determine the other. This checks the consistency and accuracy of your defined equations.

Following these systematic steps not only provides the solution but reinforces logical thinking and analytical problem-solving skills.