Calculating Movie Ticket Costs A Family Outing Problem

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Paige and her family embarked on a delightful movie night, a classic family outing filled with anticipation and shared excitement. They purchased four tickets to the latest blockbuster and, like any true moviegoers, indulged in a large bucket of popcorn, a quintessential movie snack. The total cost for this cinematic adventure amounted to $40, a sum that included the price of the tickets and the popcorn. The popcorn alone set them back $12, leaving us with a mathematical puzzle to solve: how much did each ticket cost?

This scenario presents a real-world problem that can be effectively tackled using basic algebraic principles. We are essentially dealing with a simple equation where the unknown variable is the cost of each ticket. By carefully dissecting the information provided, we can construct an equation that accurately represents the situation and leads us to the solution. This exercise not only helps us determine the ticket cost but also reinforces our understanding of how mathematical concepts can be applied to everyday situations. Understanding these types of problems helps solidify mathematical foundations for more complex problem-solving in the future. The ability to translate real-world scenarios into mathematical equations is a crucial skill, applicable in various fields beyond the classroom. From budgeting and finance to engineering and science, the power of mathematical modeling is undeniable.

Setting Up the Equation: A Step-by-Step Approach

To unravel the mystery of the ticket cost, we need to translate the word problem into a mathematical equation. Let's break down the information and assign variables to the unknown quantities. Let 'x' represent the cost of each ticket, which is the value we are trying to determine. Since they bought four tickets, the total cost of the tickets would be 4 * x, or simply 4x. We also know that they spent $12 on popcorn, and the overall expense for the movie night was $40. Putting these pieces together, we can formulate the following equation:

4x + 12 = 40

This equation accurately models the situation, capturing the relationship between the number of tickets, the cost per ticket, the popcorn expense, and the total expenditure. The equation states that the total cost of the tickets (4x) plus the cost of the popcorn ($12) equals the total amount spent ($40). Solving this equation will reveal the value of 'x', which represents the cost of each ticket. The process of translating word problems into equations is a fundamental skill in algebra. It requires careful reading and comprehension to identify the key information and the relationships between the variables. By mastering this skill, students can confidently approach a wide range of mathematical problems presented in real-world contexts.

Solving for the Ticket Cost: A Journey Through Algebra

Now that we have our equation, 4x + 12 = 40, we can embark on the journey of solving for 'x', the cost of each ticket. The goal is to isolate 'x' on one side of the equation, effectively revealing its value. To achieve this, we employ the principles of algebraic manipulation, performing the same operations on both sides of the equation to maintain balance. The first step is to eliminate the constant term, 12, from the left side of the equation. We can do this by subtracting 12 from both sides:

4x + 12 - 12 = 40 - 12

This simplifies to:

4x = 28

Now, we have 4x equal to 28. To isolate 'x', we need to get rid of the coefficient 4. We can accomplish this by dividing both sides of the equation by 4:

4x / 4 = 28 / 4

This leads us to the solution:

x = 7

Therefore, the cost of each ticket is $7. This step-by-step process of solving for 'x' demonstrates the power of algebraic techniques. By applying the principles of inverse operations and maintaining balance in the equation, we can systematically isolate the unknown variable and arrive at the correct answer. The ability to solve equations is a cornerstone of mathematical proficiency, essential for tackling more advanced concepts and applications.

Verifying the Solution: Ensuring Accuracy

With our solution in hand, it's always prudent to verify its accuracy. We can do this by plugging the value of 'x' back into the original equation and checking if it holds true. Our solution is x = 7, meaning each ticket cost $7. Substituting this value into the equation 4x + 12 = 40, we get:

4 * 7 + 12 = 40

Simplifying the left side:

28 + 12 = 40

40 = 40

The equation holds true! This confirms that our solution, x = 7, is indeed correct. Each ticket cost $7. Verifying the solution is a crucial step in the problem-solving process. It provides assurance that the answer is accurate and that no errors were made during the calculations. This practice reinforces the importance of precision and attention to detail in mathematics. Moreover, it fosters a deeper understanding of the relationship between the equation and the problem it represents.

The Answer and the Equation: Putting It All Together

In conclusion, after carefully analyzing the scenario and applying algebraic principles, we have successfully determined the cost of each movie ticket. The equation that models this situation is:

4x + 12 = 40

And the correct answer, representing the cost of each ticket, is:

$7

This problem highlights the practical application of mathematics in everyday life. By understanding how to translate real-world situations into mathematical equations and solve them, we can gain valuable insights and make informed decisions. From calculating expenses to managing budgets, the power of mathematical reasoning is undeniable. This exercise also underscores the importance of breaking down complex problems into smaller, manageable steps. By carefully dissecting the information, identifying the key variables, and applying the appropriate techniques, we can confidently tackle any mathematical challenge.

Key takeaways from this problem include:

  • The ability to translate word problems into algebraic equations.
  • The importance of understanding variables and their relationships.
  • The step-by-step process of solving equations using inverse operations.
  • The significance of verifying solutions to ensure accuracy.

By mastering these skills, students can develop a strong foundation in mathematics and apply their knowledge to a wide range of real-world scenarios. The confidence gained from successfully solving problems like this can empower individuals to approach future challenges with a positive and solution-oriented mindset.