Problem 61 - College Algebra with Corequisite Support

Question

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. At the same market, the three most popular fruits make up \(37 \%\) of the total fruit sold. Strawberries sell twice as much as oranges, and kiwis sell one more percentage point than oranges. For each fruit, find the percentage of total fruit sold.

Step-by-Step Solution

Answer

Strawberries: 18%, Oranges: 9%, Kiwis: 10%.

Step 1: Define Variables and Equations

Let's define the variables based on the problem statement. Let \( s \) be the percentage of strawberries sold, \( o \) the percentage of oranges, and \( k \) the percentage of kiwis.From the problem:1. The sum of the percentages of strawberries, oranges, and kiwis is 37%. So, we have: \[ s + o + k = 37 \]2. Strawberries sell twice as much as oranges, which gives us: \[ s = 2o \]3. Kiwis sell one percentage point more than oranges: \[ k = o + 1 \]

Step 2: Substitute and Simplify Equations

Substitute the expressions for \( s \) and \( k \) from the given relationships into the first equation.Starting with \( s = 2o \) and \( k = o + 1 \), substitute into the equation:\[ 2o + o + (o + 1) = 37 \]Simplifying gives:\[ 4o + 1 = 37 \]

Step 3: Solve for Oranges

Solve the equation \( 4o + 1 = 37 \) for \( o \).Subtract 1 from both sides:\[ 4o = 36 \]Divide both sides by 4:\[ o = 9 \]

Step 4: Find Strawberries and Kiwis

Using the value of \( o = 9 \):- For strawberries, \( s = 2o = 2 \times 9 = 18 \).- For kiwis, \( k = o + 1 = 9 + 1 = 10 \).

Step 5: Verify the Solution

Verify that the percentages add up to 37. Check that:- \( s + o + k = 18 + 9 + 10 = 37 \)This confirms our solution is consistent with the information given.

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. In solving real-world problems, these equations help establish relationships between different quantities. For instance, in the exercise, we use three variables to represent strawberries (\( s \)), oranges (\( o \)), and kiwis (\( k \)) as percentages.

To capture the problem's conditions:

• We know these fruits together constitute 37% of total sales, resulting in the equation: \( s + o + k = 37 \).
• Strawberries are twice the sales of oranges, giving us this equation: \( s = 2o \).
• Kiwis sell one percentage point more than oranges: \( k = o + 1 \).

These equations form the system we need to solve. As linear equations, they can be graphically interpreted as straight lines, and solving them finds where the lines intersect, indicating a common solution for all equations.

Solving Linear Systems

The step to solve a system of linear equations is to find the values of the variables that satisfy all the equations simultaneously. In many cases, algebraic methods like substitution or elimination are used. In this exercise, we utilize Cramer's Rule, a handy tool when the number of equations equals the number of variables.

Here's a quick recap on the substitution method used here:

• Substitute expressions derived from certain equations into others. For instance, by substituting \( s = 2o \) and \( k = o + 1 \) into the equation \( s + o + k = 37 \), it becomes \( 4o + 1 = 37 \).
• Resolve the simplified equation to find the variable value. For oranges, \( o \) simplifies to \( o = 9 \).
• Use this value to find other variables: strawberries \( s = 18 \) and kiwis \( k = 10 \).

Cramer's Rule, specifically, utilizes determinants for systems where the variables correspond with a square matrix, providing an elegant solution form.

Mathematical Modeling

Mathematical modeling refers to representing real-world problems through mathematical formulations. It allows comprehending and solving complex scenarios by establishing variables and equations. In the case of our exercise, the real-world scenario revolves around fruit sales.

We model the percentages of fruit sales using equations that express certain relationships:

• The proportion of strawberries depends on oranges: \( s = 2o \).
• The proportion of kiwis depend on oranges as well: \( k = o + 1 \).

By redefining this real-world scenario into a mathematical language, it becomes manageable to solve using mathematical tools like Cramer’s Rule. This transformation simplifies complex decisions by allowing predictions and understanding relationships between various factors.

Percentages Problem

Dealing with percentages often simplifies complex numerical relationships as it offers a comparative view. In the context of the exercise, different fruits' sales are represented using percentages of the total sales, making it easier to understand the impact of each type of fruit.

Here's the breakdown:

• We directly solve how strawberries, oranges, and kiwis contribute to the total fruit sales.
• Expressing sales in terms of percentages allows an easy reference to the portion each contributes to the total.
• The equations incorporate typical percentage-based scenarios we might encounter, and solving them gives insight into each fruit's share of the market at 18%, 9%, and 10% respectively.

Using percentages aids in comprehending and communicating proportional relationships clearly, making these problems more intuitive.