Problem 32 - Precalculus

Question

At The Crispy Critter's Head Shop and Patchouli Emporium along with their dried up weeds, sunflower seeds and astrologieal posteards they sell an herbal tea blend. By weight, Type I herbal tea is \(30 \%\) peppermint, \(40 \%\) rose hips and \(30 \%\) chamomile, Type II has percents \(40 \%\), \(20 \%\) and \(40 \%\), respectively, and Type III has percents \(35 \%, 30 \%\) and \(35 \%\), respectively. How much of each Type of tea is needed to make 2 pounds of a new blend of tea that is equal parts peppermint, rose hips and chamomile?

Step-by-Step Solution

Answer

Use 0.6667 pounds each of Type I, Type II, and Type III tea.

Step 1: Define Variables

Let \( x \), \( y \), and \( z \) represent the amounts (in pounds) of Type I, Type II, and Type III tea needed, respectively.

Step 2: Set Up the System of Equations

We know the blend will be 2 pounds total, so \( x + y + z = 2 \). We need equations for the percentages: \(0.3x + 0.4y + 0.35z = \frac{1}{3} \cdot 2 \) for peppermint, \(0.4x + 0.2y + 0.3z = \frac{1}{3} \cdot 2\) for rose hips, \(0.3x + 0.4y + 0.35z = \frac{1}{3} \cdot 2\) for chamomile.

Step 3: Simplify the Equations

The equations are: \(0.3x + 0.4y + 0.35z = 0.6667\), \(0.4x + 0.2y + 0.3z = 0.6667\), and \( x + y + z = 2 \). Notice the equations for peppermint and chamomile are the same, reducing the effective number of equations.

Step 4: Solve the System of Equations

Solve \( x + y + z = 2 \), \( 0.4x + 0.2y + 0.3z = 0.6667 \) and \(0.3x + 0.4y + 0.35z = 0.6667\). Using substitution or elimination methods, after simplification, one solution is \( x = 0.6667, y = 0.6667, z = 0.6667 \).

Step 5: Verify the Solution

Substitute \( x = 0.6667, y = 0.6667, z = 0.6667 \) back into the original percentage equations to check all spice proportions equal \(\frac{2}{3}\) pounds and total \(2\) pounds.

Key Concepts

Linear Equations

In algebra, linear equations are equations of the first degree, meaning their highest exponent is one. They are powerful tools to solve problems involving unknowns. Linear equations can represent lines in a coordinate plane. For the tea blend problem, we use linear equations to represent the total weight and percentages of each ingredient.
We formulate equations based on given conditions. For instance, if we need a mixture weighing 2 pounds, one equation is formed as: \(x + y + z = 2\). Each variable represents the weight of a different tea type in the mixture.
Linear equations can often be solved using substitution or elimination methods to find the values of unknown variables. Understanding how to set up these equations correctly is key to solving various algebraic problems.

Mixture Problems

Mixture problems are a common application of linear equations, often requiring you to combine different substances to meet certain criteria. In the herbal tea exercise, we aim to find the right amounts of three types of tea to achieve a specific blend.
This requires setting up equations that account for different components and their ratios in the mixture.

• By using weights or percentages, you translate the problem's requirements into mathematical equations.
• You consider the contribution of each component to the total mixture to ensure the resulting blend meets the specified conditions.

Properly constructing and solving these equations allows you to determine the exact ratio of each component needed.

Percentages

Understanding the concept of percentages is crucial in solving the herbal tea mixture problem. Percentages represent parts per hundred and are essential for comparing different quantities. For this problem, each tea type consists of different percentage compositions of peppermint, rose hips, and chamomile.
In order to achieve an equal parts blend, the percentage values are crucial. The conditions specify that \(30\%\), \(40\%\), and \(35\%\) of each type contribute to the mixture, depending on what component you look at.
It is important to use percentages to form the basis for equations that govern the proportions of each type of tea in the final blend, ensuring each ingredient contributes exactly as needed to reach equal parts.

Problem-Solving with Algebra

Algebra allows us to solve complex problems systematically. In the case of creating an herb tea blend, algebraic techniques are employed to handle the intricate requirements of weight and composition. This process involves several steps:

• Define variables to represent unknown quantities.
• Set up equations that represent the problem conditions.
• Simplify the equations, if possible, to reduce complexity.
• Use algebraic methods like substitution or elimination to find solutions to the equations.

Solving the system of equations provides insights into how much of each tea type is needed. By cross-verifying our solution with the given percentage conditions and total weight, algebra ensures an accurate and logical resolution to the problem.