In A Theater With 110 Seats, There Are Two Types Of Seats: 1000F Seats And 2500F Seats. When The Theater Is Full, The Total Revenue Is 1,700,000F. How Many Seats Of Each Type Are There?
Introduction
This article delves into a classic mathematical problem involving theater seating and revenue calculation. We'll explore how to determine the number of seats in each price category given the total capacity, ticket prices, and total revenue when the theater is full. This problem is a great example of how systems of equations can be used to solve real-world scenarios. Understanding the concepts of linear equations and system of equations is crucial for grasping the solution. We will break down the problem step-by-step, making it easy to follow and understand. The core of this problem lies in translating the word problem into mathematical equations. We need to identify the unknowns, which in this case are the number of seats in each price category. Once we have the equations, we can use various methods like substitution or elimination to solve for the unknowns. This article will not only provide the solution to the specific problem but also equip you with the skills to tackle similar problems in the future. We'll also discuss the practical applications of these types of problems, such as in budgeting, resource allocation, and pricing strategies. By the end of this article, you'll have a solid understanding of how mathematics can be applied to solve everyday challenges. Let's embark on this mathematical journey together and unravel the mysteries of theater seating!
Problem Statement
A theater has a seating capacity of 110 seats. There are two types of seats: seats priced at 1000F and seats priced at 2500F. When the theater is full, the total revenue is 1,700,000F. The question is: How many seats of each type are there? This is a typical problem that can be solved using a system of linear equations. To solve this problem effectively, we first need to define our variables. Let's assign variables to represent the unknowns, which are the number of seats in each price category. Let 'x' represent the number of seats priced at 1000F, and 'y' represent the number of seats priced at 2500F. The next step is to translate the given information into mathematical equations. We have two key pieces of information: the total number of seats and the total revenue. The total number of seats is 110, which can be expressed as an equation: x + y = 110. The total revenue is 1,700,000F, which can be expressed as another equation: 1000x + 2500y = 1,700,000. These two equations form our system of equations. Now we have a clear mathematical representation of the problem, making it easier to solve. The following sections will explore different methods to solve this system of equations and find the values of x and y, which will give us the number of seats in each price category. Understanding how to set up the equations correctly is half the battle in solving these types of problems. Once you have the equations, you can use various algebraic techniques to find the solution.
Setting up the Equations
To solve this problem, we need to translate the given information into mathematical equations. Let's define our variables: Let 'x' be the number of seats priced at 1000F, and let 'y' be the number of seats priced at 2500F. We have two key pieces of information: the total number of seats and the total revenue. The total number of seats is 110, which gives us our first equation:
x + y = 110
The total revenue when the theater is full is 1,700,000F. This can be represented by the equation:
1000x + 2500y = 1,700,000
We now have a system of two linear equations with two variables:
- x + y = 110
- 1000x + 2500y = 1,700,000
This system of equations represents the problem mathematically. The first equation represents the total number of seats, while the second equation represents the total revenue generated when all seats are occupied. Solving this system will give us the values of 'x' and 'y', which correspond to the number of seats in each price category. There are several methods to solve such systems, including substitution, elimination, and graphing. In the following sections, we will explore the substitution method to find the solution. The process of setting up the equations is crucial in solving mathematical problems. It requires careful reading and understanding of the problem statement, identifying the unknowns, and translating the information into mathematical expressions. Once the equations are set up correctly, the rest of the solution process becomes much easier. It's also important to check if the equations make sense in the context of the problem. For example, the sum of the number of seats in each category should equal the total number of seats, and the total revenue should be the sum of the revenue from each category. By carefully setting up and checking the equations, we can ensure that we are on the right track to solving the problem.
Solving the System of Equations
Now that we have our system of equations:
- x + y = 110
- 1000x + 2500y = 1,700,000
We can use the substitution method to solve for x and y. First, let's solve the first equation for x:
x = 110 - y
Now, substitute this expression for x into the second equation:
1000(110 - y) + 2500y = 1,700,000
Expand and simplify the equation:
110,000 - 1000y + 2500y = 1,700,000 1500y = 1,590,000
Now, solve for y:
y = 1,590,000 / 1500 y = 1060
Now that we have the value of y, we can substitute it back into the equation x = 110 - y:
x = 110 - 1060 x = -950
However, this solution doesn't make sense in the context of the problem because we cannot have a negative number of seats. Let's re-examine our calculations to identify any potential errors. It appears there was an error in the calculation of y. The correct calculation should be:
1500y = 1,700,000 - 110,000 1500y = 1,590,000 y = 1,590,000 / 1500 y = 1060
There seems to be an issue with the problem statement, as the number of seats (110) and the total revenue (1,700,000F) do not yield a realistic solution. The value of y (1060) is already greater than the total number of seats (110), indicating an inconsistency in the given information. It's important to note that in real-world problems, the accuracy of the data is crucial for obtaining meaningful solutions. If the data is incorrect or inconsistent, the results may not be valid. In this case, either the total number of seats or the total revenue needs to be adjusted to make the problem solvable and yield a realistic solution. Let's assume there was a typo in the total revenue, and it should have been a lower value. Alternatively, the number of seats could be higher. Without correcting the initial information, we cannot proceed to a valid solution. It's a good reminder that problem-solving often involves critical thinking and the ability to identify inconsistencies in the given data. Before blindly applying formulas and equations, it's essential to ensure that the problem is well-defined and the data is reasonable. In the next section, we will discuss the importance of verifying the solution and how to interpret the results in the context of the problem.
Interpreting the Results and Identifying the Issue
As we attempted to solve the system of equations, we encountered a situation where the calculated value for 'y' (the number of seats priced at 2500F) was 1060, which is significantly greater than the total number of seats in the theater (110). This result immediately indicates an issue with the problem statement or the given data. It's crucial to interpret the results in the context of the problem to determine if they make sense. In this case, a negative or excessively large number of seats is not a realistic solution. This discrepancy highlights the importance of verifying the solution. When solving mathematical problems, it's not enough to simply arrive at a numerical answer. You need to check if the answer fits the constraints and conditions of the problem. In this scenario, the constraints are the total number of seats and the total revenue. The calculated values must satisfy both equations and also be logical within the real-world context of a theater seating arrangement. To identify the issue, we need to re-examine the given information and the equations we set up. We have two equations:
- x + y = 110
- 1000x + 2500y = 1,700,000
The first equation represents the total number of seats, and it seems straightforward. The second equation represents the total revenue. Let's analyze the revenue equation more closely. If all 110 seats were priced at the higher rate of 2500F, the total revenue would be:
110 * 2500 = 275,000F
This is significantly less than the given total revenue of 1,700,000F. This suggests that the total revenue figure might be incorrect. The total revenue is far too high for the given number of seats and prices. This discrepancy is the root cause of the unrealistic solution. In problem-solving, it's important to be able to identify inconsistencies and errors in the given information. This often requires a critical and analytical approach. Before attempting to solve a problem, it's a good practice to quickly estimate a reasonable solution to see if the given data is plausible. In this case, a simple estimation would have revealed the inconsistency in the revenue figure. The key takeaway here is that mathematical solutions must align with the real-world context. If the results don't make sense, it's a signal to re-examine the problem statement and the solution process to identify any errors or inconsistencies.
Correcting the Problem and Finding a Valid Solution (Hypothetical)
Since we've identified that the original problem statement contains an inconsistency, let's hypothetically correct the total revenue to make the problem solvable and demonstrate the solution process. Let's assume the correct total revenue is 170,000F instead of 1,700,000F. With this corrected value, our system of equations becomes:
- x + y = 110
- 1000x + 2500y = 170,000
We can use the substitution method again. Solve the first equation for x:
x = 110 - y
Substitute this expression for x into the corrected second equation:
1000(110 - y) + 2500y = 170,000
Expand and simplify the equation:
110,000 - 1000y + 2500y = 170,000 1500y = 60,000
Now, solve for y:
y = 60,000 / 1500 y = 40
Now that we have the value of y, substitute it back into the equation x = 110 - y:
x = 110 - 40 x = 70
So, with the corrected total revenue, we find that there are 70 seats priced at 1000F and 40 seats priced at 2500F. This solution makes sense in the context of the problem. We have a positive number of seats for each category, and the total number of seats is 110. Let's verify the solution by calculating the total revenue:
(70 * 1000) + (40 * 2500) = 70,000 + 100,000 = 170,000F
The calculated total revenue matches the corrected total revenue, confirming our solution. This example demonstrates the importance of correcting inconsistencies in the problem statement to obtain a valid and meaningful solution. It also reinforces the idea that mathematical problem-solving is an iterative process. If the initial solution doesn't make sense, you need to go back, re-examine the problem, and identify any potential errors or inconsistencies. By making a hypothetical correction to the total revenue, we were able to successfully solve the problem and illustrate the solution process. This highlights the flexibility and adaptability required in problem-solving.
Conclusion
In conclusion, this article explored a problem involving theater seating and revenue calculation. We learned how to set up a system of equations to represent the problem mathematically and attempted to solve it using the substitution method. However, we encountered an issue where the given data (specifically, the total revenue) was inconsistent with the other information, leading to an unrealistic solution. This highlighted the importance of interpreting results in context and verifying the solution to ensure it makes sense. We then discussed the process of identifying inconsistencies and the need to re-examine the problem statement when the initial solution is not valid. By hypothetically correcting the total revenue, we demonstrated how to obtain a meaningful solution and illustrated the complete problem-solving process. The key takeaways from this exercise are:
- Setting up a system of equations is a powerful tool for solving real-world problems.
- It's crucial to interpret results in the context of the problem and verify the solution.
- Inconsistencies in the given data can lead to unrealistic solutions.
- Problem-solving is an iterative process that often requires critical thinking and analysis.
This type of problem-solving approach is applicable in various fields, including business, finance, and resource allocation. Understanding how to translate real-world scenarios into mathematical models and how to interpret the results is a valuable skill. By mastering these concepts, you can effectively tackle a wide range of problems and make informed decisions based on data and analysis. The ability to identify and correct errors in problem statements or data is also essential for successful problem-solving. This requires a flexible and adaptable mindset, as well as a willingness to challenge assumptions and re-evaluate the approach. Ultimately, problem-solving is not just about finding the right answer; it's about developing critical thinking skills and a deeper understanding of the underlying concepts. We hope this article has provided you with valuable insights and tools for tackling mathematical problems and applying them to real-world situations.