Carolyn Makes And Sells Necklaces. The Material To Make Each Necklace Costs $ 3.20 \$3.20 $3.20 . She Has 12 Necklaces To Sell And Wants To Make A Total Profit Of $ 54 \$54 $54 . The Following Equation Describes The Situation: $12(x - 3.20) =

Introduction

Carolyn is a skilled artisan who makes and sells beautiful necklaces. With a passion for creating unique pieces, she has a steady demand for her products. However, like any business, Carolyn needs to ensure that she makes a profit from her sales. In this article, we will explore the mathematical concept behind Carolyn's necklace business and how she can achieve her desired profit.

The Problem

Carolyn has 12 necklaces to sell, and each necklace costs $\$3.20$ to make. She wants to make a total profit of $\$54$. To understand how she can achieve this, we need to set up an equation that represents the situation.

The Equation

Let $x$ be the selling price of each necklace. The profit made from selling one necklace is the selling price minus the cost price, which is $x - 3.20$. Since Carolyn has 12 necklaces to sell, the total profit is given by the equation:

$$12(x - 3.20) = 54$$

Solving the Equation

To solve for $x$, we need to isolate the variable. We can start by distributing the 12 to the terms inside the parentheses:

$$12x - 38.4 = 54$$

Next, we can add 38.4 to both sides of the equation to get:

$$12x = 54 + 38.4$$

$$12x = 92.4$$

Now, we can divide both sides of the equation by 12 to solve for $x$:

$$x = \frac{92.4}{12}$$

$$x = 7.7$$

Interpretation

The solution to the equation, $x = 7.7$, represents the selling price of each necklace that Carolyn needs to achieve her desired profit of $\$54$. This means that Carolyn should sell each necklace for at least $\$7.70$ to make a total profit of $\$54$.

Conclusion

In this article, we have explored the mathematical concept behind Carolyn's necklace business. By setting up an equation that represents the situation, we were able to solve for the selling price of each necklace that Carolyn needs to achieve her desired profit. This example demonstrates the importance of mathematical problem-solving in real-world business scenarios.

Real-World Applications

The concept of setting up and solving equations can be applied to various real-world scenarios, such as:

• Business: Understanding the cost and revenue of products or services to determine the optimal pricing strategy.
• Finance: Calculating interest rates, investment returns, and loan repayments.
• Science: Modeling population growth, chemical reactions, and physical systems.

By applying mathematical concepts to real-world problems, we can gain a deeper understanding of the underlying principles and make informed decisions.

Additional Resources

For those interested in learning more about mathematical problem-solving, here are some additional resources:

• Online Courses: Websites like Coursera, edX, and Khan Academy offer a wide range of courses on mathematics and problem-solving.
• Books: Textbooks and books on mathematics, such as "A First Course in Linear Algebra" by Robert A. Beezer, can provide a comprehensive understanding of mathematical concepts.
• Practice Problems: Websites like Brilliant and Mathway offer practice problems and exercises to help improve mathematical problem-solving skills.

Introduction

In our previous article, we explored the mathematical concept behind Carolyn's necklace business. We set up an equation to represent the situation and solved for the selling price of each necklace that Carolyn needs to achieve her desired profit. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the cost price of each necklace?

A: The cost price of each necklace is $\$3.20$.

Q: What is the desired profit of Carolyn?

A: Carolyn wants to make a total profit of $\$54$.

Q: What is the equation that represents the situation?

A: The equation that represents the situation is:

$$12(x - 3.20) = 54$$

Q: How do we solve for x in the equation?

A: To solve for x, we need to isolate the variable. We can start by distributing the 12 to the terms inside the parentheses:

$$12x - 38.4 = 54$$

Next, we can add 38.4 to both sides of the equation to get:

$$12x = 54 + 38.4$$

$$12x = 92.4$$

Now, we can divide both sides of the equation by 12 to solve for x:

$$x = \frac{92.4}{12}$$

$$x = 7.7$$

Q: What does the solution to the equation represent?

A: The solution to the equation, $x = 7.7$, represents the selling price of each necklace that Carolyn needs to achieve her desired profit of $\$54$.

Q: Can Carolyn sell each necklace for less than $\$7.70$ and still make a profit?

A: No, Carolyn cannot sell each necklace for less than $\$7.70$ and still make a profit of $\$54$. If she sells each necklace for less than $\$7.70$, her total profit will be less than $\$54$.

Q: Can Carolyn sell each necklace for more than $\$7.70$ and still make a profit?

A: Yes, Carolyn can sell each necklace for more than $\$7.70$ and still make a profit. If she sells each necklace for more than $\$7.70$, her total profit will be more than $\$54$.

Q: What is the minimum number of necklaces Carolyn needs to sell to make a profit?

A: Carolyn needs to sell at least 12 necklaces to make a profit of $\$54$. If she sells fewer than 12 necklaces, her total profit will be less than $\$54$.

Q: Can Carolyn make a profit if she sells each necklace for $\$7.70$ and only sells 10 necklaces?

A: No, Carolyn cannot make a profit if she sells each necklace for $\$7.70$ and only sells 10 necklaces. Her total revenue will be $10 \times 7.70 = 77$, and her total cost will be $10 \times 3.20 = 32$. Her total profit will be $77 - 32 = 45$, which is less than her desired profit of $\$54$.

Conclusion

In this article, we have answered some frequently asked questions related to Carolyn's necklace business. We have explored the mathematical concept behind the problem and provided solutions to the equation. By understanding the underlying principles, we can make informed decisions and achieve our desired outcomes.

Real-World Applications

The concept of setting up and solving equations can be applied to various real-world scenarios, such as:

• Business: Understanding the cost and revenue of products or services to determine the optimal pricing strategy.
• Finance: Calculating interest rates, investment returns, and loan repayments.
• Science: Modeling population growth, chemical reactions, and physical systems.

By applying mathematical concepts to real-world problems, we can gain a deeper understanding of the underlying principles and make informed decisions.

Additional Resources

For those interested in learning more about mathematical problem-solving, here are some additional resources:

• Online Courses: Websites like Coursera, edX, and Khan Academy offer a wide range of courses on mathematics and problem-solving.
• Books: Textbooks and books on mathematics, such as "A First Course in Linear Algebra" by Robert A. Beezer, can provide a comprehensive understanding of mathematical concepts.
• Practice Problems: Websites like Brilliant and Mathway offer practice problems and exercises to help improve mathematical problem-solving skills.

By exploring these resources, you can develop your mathematical problem-solving skills and apply them to real-world scenarios.