Solving For Books How Many Did Hugh Purchase

Hugh's recent trip to the bookstore presents an interesting mathematical problem that we can solve by analyzing the cost of his magazine and book purchases. This article will dive deep into the problem, explaining the steps to find the solution while focusing on clarity and understanding, ensuring that readers can follow along and grasp the underlying concepts. Let's break down this problem step by step to determine how many books Hugh bought.

Setting up the Equation: Modeling the Problem

The core of solving this problem lies in understanding the equation provided: 3.95m + 8.95b = 47.65. This equation is a mathematical model representing Hugh's total expenditure, where:

• 3.95 represents the cost of each magazine.
• m represents the number of magazines Hugh bought.
• 8.95 represents the cost of each book.
• b represents the number of books Hugh bought.
• 47.65 represents the total amount Hugh spent.

This equation is a linear equation in two variables, a fundamental concept in algebra. It essentially states that the total cost of magazines (3.95 multiplied by the number of magazines) plus the total cost of books (8.95 multiplied by the number of books) equals the total amount spent, which is $47.65. The beauty of this equation is that it encapsulates the entire scenario in a concise mathematical form, making it easier to analyze and solve. To solve for the unknown, which in this case is the number of books (b), we will need to use the information given about the number of magazines Hugh bought.

Substituting the Known Value: Hugh's Magazine Purchase

The problem states that Hugh bought 3 magazines. This is crucial information because it allows us to substitute the value of m (number of magazines) in our equation. By replacing m with 3, we transform the equation into one with a single unknown (b), which we can then solve. Substituting m = 3 into the equation 3.95m + 8.95b = 47.65 gives us:

3. 95 * 3 + 8.95b = 47.65

Now, the equation simplifies to:

11. 85 + 8.95b = 47.65

This simplified equation is much easier to work with. We have effectively reduced the problem to a single-variable equation, a standard type of problem in algebra. The next step involves isolating the term with b to solve for the number of books.

Isolating the Variable: Preparing to Solve for Books

To find the value of b (the number of books), we need to isolate the term 8.95b on one side of the equation. This is achieved by performing the same operation on both sides of the equation to maintain the balance. In this case, we need to subtract 11.85 from both sides of the equation 11.85 + 8.95b = 47.65. This gives us:

11. 85 + 8.95b - 11.85 = 47.65 - 11.85

Simplifying both sides, we get:

8. 95b = 35.80

Now, we have a much simpler equation where the term containing b is isolated. This step is crucial in solving for any variable in an equation. By isolating the variable, we set the stage for the final step, which involves dividing both sides by the coefficient of the variable to find its value.

Solving for the Number of Books: The Final Calculation

With the equation now in the form 8.95b = 35.80, the final step to find the number of books (b) is to divide both sides of the equation by 8.95. This will isolate b on one side, giving us its value. Performing the division, we get:

b = 35.80 / 8.95

Calculating this gives us:

b = 4

Therefore, Hugh bought 4 books. This is the solution to the problem. We arrived at this answer by systematically working through the equation, substituting known values, isolating the variable, and performing the necessary calculations. This methodical approach is the cornerstone of solving mathematical problems, especially in algebra.

Verifying the Solution: Ensuring Accuracy

To ensure the accuracy of our solution, it's always a good practice to verify the answer. We can do this by plugging the value of b (which we found to be 4) back into the original equation along with the known value of m (which is 3) and checking if the equation holds true.

The original equation is:

3. 95m + 8.95b = 47.65

Substituting m = 3 and b = 4, we get:

3. 95 * 3 + 8.95 * 4 = 47.65

Calculating the left side:

11. 85 + 35.80 = 47.65

Adding the numbers, we get:

47. 65 = 47.65

Since the left side of the equation equals the right side, our solution is correct. This verification step is crucial in problem-solving as it confirms that our calculations and reasoning are accurate. It provides confidence in the answer and ensures that no errors were made during the process.

Conclusion: The Power of Mathematical Modeling

In conclusion, by using the given equation and systematically solving for the unknown variable, we determined that Hugh bought 4 books. This problem demonstrates the power of mathematical modeling in representing real-world scenarios. The equation 3.95m + 8.95b = 47.65 effectively captured the relationship between the number of magazines and books bought, their respective costs, and the total amount spent. By substituting the known value (number of magazines) and using algebraic techniques, we were able to isolate the variable representing the number of books and find its value.

This exercise highlights the importance of understanding algebraic concepts such as linear equations, variable substitution, and equation solving. These skills are not only essential in mathematics but also in various fields that require analytical thinking and problem-solving. Moreover, the verification step underscores the importance of checking one's work to ensure accuracy. The ability to translate real-world problems into mathematical equations and solve them is a valuable skill that empowers us to make informed decisions and understand the world around us in a more structured and logical way.

How many books did Hugh buy if he bought some magazines costing $3.95 each and some books costing $8.95 each, spending a total of $47.65, and he bought 3 magazines?