Equation For Total Price Of Books And Cost Of 13 Books

Jul 12, 2025 by Admin 55 views

Understanding the Equation for Total Price of Books and Calculating the Cost of 13 Books

In this article, we will delve into the mathematical equation that represents the total price (p) paid when purchasing a certain number of books (b), each priced at $3.99. We will explore how to derive this equation and then use it to calculate the cost of buying 13 books. This problem is a practical application of basic algebra, particularly the concept of direct variation, which is fundamental in various real-world scenarios. Understanding the relationship between the number of items purchased and the total cost is crucial not only in mathematics but also in everyday financial transactions. We will also analyze why certain equations might be incorrect and how to identify the correct relationship between the variables. This step-by-step approach will help clarify the underlying mathematical principles and enhance your problem-solving skills.

Deriving the Equation for Total Cost

To begin, let's break down the problem. We know that each book costs $3.99, and we want to find the total price (p) for any number of books (b). The total price is directly related to the number of books purchased; as the number of books increases, the total price also increases proportionally. This relationship can be expressed using a simple linear equation. The key here is to identify the correct operation that links the number of books and the price per book to the total price. The total price is the product of the price per book and the number of books. Therefore, we need to multiply the price of one book ($3.99) by the number of books (b) to get the total price (p). This fundamental concept is used extensively in pricing and sales calculations, where understanding the cost-volume-profit relationship is essential for businesses to determine pricing strategies and profitability. The price per book acts as a constant multiplier, determining how much the total cost increases with each additional book purchased. This concept extends to other areas of mathematics, such as calculating total earnings based on an hourly wage or the total distance traveled based on speed and time. Recognizing these patterns is crucial for building a strong foundation in algebra and its applications.

The equation that represents this relationship is:

p = 3.99 b

This equation tells us that the total price (p) is equal to $3.99 multiplied by the number of books (b). This is a direct proportion equation, where p varies directly with b, and the constant of proportionality is 3.99. Understanding this type of equation is fundamental in algebra as it represents many real-world relationships where one quantity changes proportionally with another. For instance, the total cost of gasoline is directly proportional to the number of gallons purchased, or the total distance traveled at a constant speed is directly proportional to the time spent traveling. Recognizing and formulating such equations is a key skill in mathematical modeling and problem-solving. The equation also highlights the concept of variables and constants, where p and b are variables that can change, while 3.99 remains constant, representing the price of each book. This distinction is crucial in understanding and manipulating algebraic expressions and equations. The ability to correctly identify and represent relationships between variables using equations is a cornerstone of mathematical literacy.

Calculating the Cost of 13 Books

Now that we have the equation, we can use it to find the total cost of 13 books. To do this, we simply substitute b with 13 in our equation:

p = 3.99 * 13

Performing the multiplication:

p = 51.87

Therefore, the total cost of 13 books is $51.87. This calculation demonstrates a practical application of the equation we derived earlier. By substituting a specific value for the number of books, we can easily determine the corresponding total price. This process highlights the power of algebraic equations in solving real-world problems. The ability to substitute values into equations and solve for unknown variables is a fundamental skill in algebra and is used extensively in various fields, including finance, engineering, and science. This calculation also reinforces the concept of multiplication as repeated addition. We are essentially adding $3.99 thirteen times, which gives us the total cost. Visualizing the problem in this way can help solidify understanding, especially for those new to algebraic concepts. Moreover, this exercise emphasizes the importance of accurate arithmetic and attention to detail when performing calculations. A small error in multiplication can lead to a significant difference in the final result, which underscores the need for careful and methodical problem-solving.

Analyzing the Answer Choices

Let's examine the given answer choices to understand why the correct answer is what it is and why the others are incorrect.

A) b = 3.99 p; $51.87: This equation is incorrect because it represents the number of books as a function of the total price, rather than the total price as a function of the number of books. The equation implies that the number of books depends on the total price, which is the reverse of the actual relationship. While the total cost calculation provided ($51.87) is correct, the equation itself is flawed. This highlights the importance of understanding the relationship between variables and ensuring that the equation accurately reflects that relationship. The correct equation should express the dependent variable (total price) in terms of the independent variable (number of books). This misconception can arise if the relationship between the variables is not thoroughly understood or if the problem is misinterpreted. It's crucial to carefully analyze the problem statement and identify which variable depends on the other to formulate the correct equation. The correct calculation of $51.87, however, indicates an understanding of the arithmetic involved but a misunderstanding of the algebraic representation of the problem.
B) p = 3.99 b; $0.31: This equation is correct as it represents the total price (p) as a function of the number of books (b). However, the total cost calculation of $0.31 is incorrect. This indicates a correct understanding of the algebraic relationship but an error in the arithmetic calculation. The error likely stems from dividing 3.99 by 13 instead of multiplying. This type of mistake underscores the importance of carefully reviewing calculations to ensure accuracy. While the equation demonstrates the correct functional relationship, the incorrect calculation invalidates the answer choice as a whole. This highlights the two key aspects of solving such problems: correctly formulating the equation and accurately performing the necessary calculations. Both steps are equally important, and an error in either one will lead to an incorrect answer. This example emphasizes the need for a thorough and systematic approach to problem-solving, checking both the underlying logic and the arithmetic execution.
C) p = 3.99 b; $51.87: This answer choice is the correct one. The equation p = 3.99 b correctly represents the relationship between the total price and the number of books, and the total cost calculation of $51.87 is also accurate. This option demonstrates a complete understanding of the problem, both in terms of formulating the correct equation and performing the necessary calculations. This is the only option that accurately captures both the algebraic relationship and the numerical result. The consistency between the equation and the calculation reinforces the importance of a coherent and logical approach to problem-solving. Choosing this answer demonstrates not only the ability to perform the mathematical operations but also the understanding of the underlying principles and the ability to apply them correctly. This choice represents a comprehensive mastery of the concepts involved in the problem.
D) b = 3.99 c; $51.87: This equation is incorrect because it uses the variable c instead of p for the total price, which is a minor error in notation, and it also incorrectly represents the relationship between the number of books and the total price. While the total cost calculation ($51.87) is correct, the equation itself is flawed. The use of the incorrect variable c instead of p indicates a potential lack of attention to detail or a misunderstanding of the variables defined in the problem statement. However, the more significant error is the incorrect equation, which suggests a misunderstanding of the relationship between the number of books and the total price. This equation, similar to option A, implies that the number of books depends on the total price, which is the reverse of the actual relationship. Despite the correct calculation of the total cost, the incorrect equation makes this answer choice wrong. This highlights the importance of using correct notation and accurately representing the relationships between variables when formulating algebraic equations.

Conclusion

In conclusion, the correct equation that represents the total price (p) paid when buying books (b) that cost $3.99 each is p = 3.99 b, and the total cost of 13 books is $51.87. This problem demonstrates the application of basic algebraic principles in a real-world scenario. Understanding how to formulate equations and solve for unknown variables is crucial for various mathematical and practical applications. By carefully analyzing the problem, deriving the correct equation, and performing accurate calculations, we can confidently solve such problems. This exercise also highlights the importance of attention to detail and a thorough understanding of the underlying mathematical concepts. The ability to translate real-world situations into mathematical models and solve them effectively is a valuable skill that extends beyond the classroom and into everyday life.