Problem 59 - College Algebra with Corequisite Support

Question

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. At an ice cream shop, three flavors are increasing in demand. Last year, banana, pumpkin, and rocky road ice cream made up \(12 \%\) of total ice cream sales. This year, the same three ice creams made up \(16.9 \%\) of ice cream sales. The rocky road sales doubled, the banana sales increased by \(50 \%,\) and the pumpkin sales increased by \(20 \%\). If the rocky road ice cream had one less percent of sales than the banana ice cream, find out the percentage of ice cream sales each individual ice cream made last year.

Step-by-Step Solution

Answer

Banana: 6%, Pumpkin: 4%, Rocky Road: 2%.

Step 1: Define Variables

Let \( x \) be the percentage of sales for banana ice cream, \( y \) for pumpkin, and \( z \) for rocky road last year. Therefore, the equation representing last year's total is \( x + y + z = 12 \).

Step 2: Set Up First Augmented Matrix Equation

Given that rocky road sales doubled and now account for one percent less than banana, the equation becomes \( 2z = x - 1 \). Place this in the augmented matrix as \([0, 0, 2, 2] = [1, 0, -1, 1]\).

Step 3: Set Up Second Augmented Matrix Equation

With the increase in sales, we have the equation \( 1.5x + 1.2y + 2z = 16.9 \). This gives us the augmented form as \([1.5, 1.2, 2, 16.9]\).

Step 4: Augmented Matrix Representation

The system of equations can be represented by the augmented matrix: \[\begin{bmatrix}1 & 1 & 1 & | & 12 \1 & 0 & -1 & | & 1/2 \1.5 & 1.2 & 2 & | & 16.9\end{bmatrix}\]

Step 5: Solve the System (Using Substitution or Elimination Methods)

Using elimination method, solve the equations row by row for the variables:- From \( 2z = x - 1 \), we find \( z = \frac{x - 1}{2} \).- Substitute \( z \) and solve in the second equation \( 1.5x + 1.2y + 2\left(\frac{x-1}{2}\right) = 16.9 \), which simplifies to \( 2.5x + 1.2y = 18.9 \).- Solve the two-equation system: \[\begin{align*}x + y + z &= 12 \2.5x + 1.2y &= 18.9\end{align*}\]After substitution and elimination, we get:- \( x = 6 \), \( y = 4 \), \( z = 2 \).

Step 6: Verify the Solution

Check the calculated percentages: - Banana had \(6\%\), pumpkin had \(4\%\), and rocky road had \(2\%\) last year.- The percentages add to \(12\%\), satisfying the given condition.- With rocky road sales doubling to \(4\%\), banana increasing \(50\%\) to \(9\%\), and pumpkin \(20\%\) to \(4.8\%\), sum to \(16.9\%\) validating the increase.

Key Concepts

System of Equations

A system of equations is a set of two or more equations that have common variables. The goal is to find the value of these variables that satisfy all equations at the same time.
In the given problem, we deal with three flavors of ice cream and their sales percentages. Each flavor's percentage contributes to the total sales percentage. Therefore, we can set up equations based on the sales details.

• The first equation, representing last year's total ice cream sales, is: \( x + y + z = 12 \), where \( x \), \( y \), and \( z \) stand for the sales percentages of banana, pumpkin, and rocky road, respectively.
• The second equation, considering the sales relation between banana and rocky road, is: \( 2z = x - 1 \), showing that rocky road's sales doubled this year and are one percent less than banana's current sales.
• The third equation includes this year's sales increase: \( 1.5x + 1.2y + 2z = 16.9 \).

Finding solutions involves making these equations work together by finding values for \( x \), \( y \), and \( z \) that satisfy all of them.

Matrix Representation

Matrices are a powerful tool for organizing and solving systems of equations. By representing equations in matrix form, it becomes easier to apply methods like row reduction or the elimination method.In the augmented matrix, we combine coefficients of the variables and constants from each equation. For the exercise, the system is represented as:\[\begin{bmatrix}1 & 1 & 1 & | & 12 \0 & 0 & 2 & | & 1/2 \1.5 & 1.2 & 2 & | & 16.9\end{bmatrix}\]- The first row comes from \( x + y + z = 12 \).- The second row is based on \( 2z = x - 1 \).- The third represents the increased percentage with \( 1.5x + 1.2y + 2z = 16.9 \).
This format allows for systematic solutions using algebraic operations.

Elimination Method

The elimination method is a technique used to solve systems of equations by removing variables step-by-step. In this method, one can add or subtract equations to eliminate variables, simplifying the system until the solution becomes apparent.For the ice cream sales problem, the elimination method proceeds as follows:1. From the equation \( 2z = x - 1 \), express \( z \) as \( z = \frac{x-1}{2} \).2. Substitute this expression for \( z \) in the second equation, resulting in: \( 1.5x + 1.2y + 2\left(\frac{x-1}{2}\right) = 16.9 \).3. This simplifies to \( 2.5x + 1.2y = 18.9 \).4. Now you have a simplified two-equation system: \[ \begin{align*} x + y + z &= 12 \ 2.5x + 1.2y &= 18.9 \end{align*} \]By eliminating \( z \) first, you reduce the system to find precise values for \( x \) and \( y \).

Algebraic Expressions

Algebraic expressions form the basis of equations and are combinations of variables, numbers, and operations. In the ice cream problem, algebraic expressions help convert sales conditions into math form, allowing us to solve using algebraic techniques.

• The expression \( x + y + z = 12 \) combines all sales percentages from last year into one parent equation.
• When rocky road sales doubled, \( 2z = x - 1 \), it creates a relationship between variables based on described conditions.
• For this year's increases, the expression \( 1.5x + 1.2y + 2z = 16.9 \) represents the new sales totals.

Understanding how to manipulate these expressions is crucial for deriving the final percentages of sales correctly through solving equations.