A School Spent $\$150$ On Advertising For A Breakfast Fundraiser. Each Plate Of Food Was Sold For $\$8.00$ But Cost The School An Unknown Amount To Prepare. After All Expenses Were Paid, The School Raised

Feb 26, 2025 by ADMIN 211 views

A School's Breakfast Fundraiser: A Mathematical Analysis of Profit and Loss

Introduction

In this article, we will delve into the world of mathematics and explore the concept of profit and loss in the context of a school's breakfast fundraiser. The scenario presented involves a school spending $150 on advertising for a breakfast fundraiser, with each plate of food sold for $8.00, but at an unknown cost to prepare. Our goal is to determine the amount of profit or loss incurred by the school after all expenses were paid.

The Problem

Let's break down the problem step by step:

The school spent $150 on advertising.
Each plate of food was sold for $8.00.
The cost to prepare each plate of food is unknown.
The school raised a certain amount of money from selling the plates of food.
After all expenses were paid, the school incurred a profit or loss.

Variables and Assumptions

To solve this problem, we need to introduce some variables and make a few assumptions:

Let x be the number of plates of food sold.
Let y be the cost to prepare each plate of food.
The total revenue from selling the plates of food is 8x (since each plate was sold for $8.00).
The total cost of preparing the plates of food is xy (since each plate costs y dollars to prepare).
The total cost of advertising is $150.
The total cost of the fundraiser is the sum of the cost of preparing the plates of food and the cost of advertising, which is xy + 150.

Setting Up the Equation

Now that we have introduced the variables and assumptions, we can set up an equation to represent the situation:

8x - (xy + 150) = Profit or Loss

Simplifying the equation, we get:

8x - xy - 150 = Profit or Loss

Solving for x and y

To solve for x and y, we need to make some additional assumptions. Let's assume that the school incurred a profit, which means that the revenue from selling the plates of food was greater than the total cost of the fundraiser. Mathematically, this can be represented as:

8x > xy + 150

Simplifying the inequality, we get:

8x > xy + 150

Subtracting xy from both sides, we get:

8x - xy > 150

Factoring out x, we get:

x(8 - y) > 150

Now, we need to find the values of x and y that satisfy this inequality. Since we don't know the value of y, we can't find a specific value for x. However, we can express x in terms of y:

x > 150 / (8 - y)

Conclusion

In conclusion, the school's breakfast fundraiser resulted in a profit or loss, depending on the number of plates of food sold and the cost to prepare each plate. The amount of profit or loss can be calculated using the equation:

Profit or Loss = 8x - (xy + 150)

Simplifying the equation, we get:

Profit or Loss = 8x - xy - 150

To determine the values of x and y that satisfy this equation, we need to make some additional assumptions. Let's assume that the school incurred a profit, which means that the revenue from selling the plates of food was greater than the total cost of the fundraiser.

Example

Let's consider an example to illustrate the concept. Suppose the school sold 20 plates of food, and the cost to prepare each plate was $4.00. In this case, the total revenue from selling the plates of food is 8(20) = $160.00. The total cost of preparing the plates of food is 4(20) = $80.00. The total cost of advertising is $150. The total cost of the fundraiser is $80.00 + $150 = $230.00. The profit or loss incurred by the school is:

Profit or Loss = 8(20) - (4(20) + 150) = $160.00 - $230.00 = -$70.00

In this example, the school incurred a loss of $70.00.

Real-World Applications

The concept of profit and loss is a fundamental aspect of business and economics. In the real world, businesses and organizations use mathematical models to determine their profit and loss, and make informed decisions about their operations. The school's breakfast fundraiser is a simple example of how mathematical concepts can be applied to real-world problems.

Future Research Directions

There are several areas of future research that can be explored in the context of the school's breakfast fundraiser:

Determining the optimal number of plates of food to sell in order to maximize profit.
Analyzing the impact of different advertising strategies on the school's profit and loss.
Investigating the relationship between the cost to prepare each plate of food and the school's profit and loss.

Conclusion

In conclusion, the school's breakfast fundraiser is a simple example of how mathematical concepts can be applied to real-world problems. By using mathematical models, we can determine the profit or loss incurred by the school, and make informed decisions about their operations. The concept of profit and loss is a fundamental aspect of business and economics, and has numerous real-world applications.

References

[1] "Mathematics for Business and Economics" by John N. Franklin
[2] "Business Mathematics" by James R. Thompson
[3] "Economics for Business" by David B. Spiegelman

Glossary

Profit: The amount of money earned by a business or organization after deducting all expenses.
Loss: The amount of money lost by a business or organization due to expenses exceeding revenue.
Revenue: The amount of money earned by a business or organization from selling goods or services.
Expenses: The amount of money spent by a business or organization on goods, services, and other costs.
Advertising: The promotion of a product, service, or idea through various media channels.
Fundraiser: An event or activity organized to raise money for a cause or organization.

Introduction

In our previous article, we explored the concept of profit and loss in the context of a school's breakfast fundraiser. We introduced variables and assumptions, set up an equation, and solved for x and y. In this article, we will answer some frequently asked questions (FAQs) related to the school's breakfast fundraiser.

Q&A

Q1: What is the main goal of the school's breakfast fundraiser?

A1: The main goal of the school's breakfast fundraiser is to raise money for the school while also promoting a healthy breakfast option for students.

Q2: How much money did the school spend on advertising for the breakfast fundraiser?

A2: The school spent $150 on advertising for the breakfast fundraiser.

Q3: What is the cost to prepare each plate of food?

A3: The cost to prepare each plate of food is unknown.

Q4: How much money did the school raise from selling the plates of food?

A4: The amount of money raised from selling the plates of food depends on the number of plates sold and the cost to prepare each plate.

Q5: What is the equation to represent the situation?

A5: The equation to represent the situation is:

8x - (xy + 150) = Profit or Loss

Q6: How can we determine the values of x and y that satisfy the equation?

A6: We can determine the values of x and y that satisfy the equation by making some additional assumptions. Let's assume that the school incurred a profit, which means that the revenue from selling the plates of food was greater than the total cost of the fundraiser.

Q7: What is the relationship between the cost to prepare each plate of food and the school's profit and loss?

A7: The cost to prepare each plate of food has a direct impact on the school's profit and loss. If the cost to prepare each plate is high, the school's profit will be lower.

Q8: How can we maximize the school's profit?

A8: To maximize the school's profit, we need to determine the optimal number of plates of food to sell and the cost to prepare each plate.

Q9: What are some real-world applications of the concept of profit and loss?

A9: The concept of profit and loss has numerous real-world applications in business and economics. It is used to determine the financial performance of a company, make informed decisions about investments, and evaluate the effectiveness of marketing strategies.

Q10: What are some future research directions in the context of the school's breakfast fundraiser?

A10: Some future research directions in the context of the school's breakfast fundraiser include:

Determining the optimal number of plates of food to sell in order to maximize profit.
Analyzing the impact of different advertising strategies on the school's profit and loss.
Investigating the relationship between the cost to prepare each plate of food and the school's profit and loss.

Conclusion

References

[1] "Mathematics for Business and Economics" by John N. Franklin
[2] "Business Mathematics" by James R. Thompson
[3] "Economics for Business" by David B. Spiegelman

Glossary

Profit: The amount of money earned by a business or organization after deducting all expenses.
Loss: The amount of money lost by a business or organization due to expenses exceeding revenue.
Revenue: The amount of money earned by a business or organization from selling goods or services.
Expenses: The amount of money spent by a business or organization on goods, services, and other costs.
Advertising: The promotion of a product, service, or idea through various media channels.
Fundraiser: An event or activity organized to raise money for a cause or organization.