Solving For Hotel Room Types: A Mathematical Approach
Introduction
In the realm of mathematical problem-solving, a classic scenario involves deciphering the composition of a hotel's room inventory. Imagine a hotel that accommodates guests in two primary room types: double rooms and single rooms. This hotel boasts a total of 50 rooms and 87 beds. The challenge lies in determining the exact number of each room type. This seemingly straightforward problem provides a fascinating glimpse into the world of linear equations and their practical applications. By carefully formulating and solving a system of equations, we can unravel the mystery and reveal the precise number of double and single rooms within the hotel.
To fully grasp the solution, we must first define the variables and translate the given information into mathematical expressions. Let's denote the number of double rooms as 'x' and the number of single rooms as 'y'. With this foundation, we can construct two equations that accurately represent the problem's constraints. The first equation will capture the total number of rooms, while the second equation will express the total number of beds. By employing the principles of algebra, we can then manipulate these equations to isolate the variables and arrive at the solution. This problem serves as an excellent illustration of how mathematics can be used to model and solve real-world scenarios, providing valuable insights into the inner workings of a hotel's operations. Furthermore, it highlights the importance of analytical thinking and problem-solving skills, which are essential in various fields beyond mathematics itself.
Setting up the Equations
To successfully solve this problem, the first crucial step is to translate the given information into a set of mathematical equations. This involves carefully identifying the variables and the relationships between them. As mentioned earlier, let's represent the number of double rooms as 'x' and the number of single rooms as 'y'. These variables will serve as the foundation for our equations.
The first piece of information we have is that the hotel has a total of 50 rooms. This can be directly translated into an equation: x + y = 50. This equation simply states that the sum of the number of double rooms (x) and the number of single rooms (y) must equal 50. It provides a fundamental constraint that our solution must satisfy.
The second piece of information pertains to the total number of beds. We know that the hotel has 87 beds in total. A double room, by definition, has 2 beds, while a single room has 1 bed. Therefore, the total number of beds can be expressed as 2x + y = 87. This equation reflects the fact that each double room contributes 2 beds to the total, while each single room contributes 1 bed. This equation is crucial as it introduces a second constraint, allowing us to solve for both variables.
Now, we have a system of two linear equations with two variables:
- x + y = 50
- 2x + y = 87
This system of equations provides a complete mathematical representation of the problem. The next step is to choose a suitable method for solving this system and finding the values of 'x' and 'y'. Several techniques can be employed, such as substitution, elimination, or matrix methods. The choice of method often depends on personal preference and the specific structure of the equations. In the following section, we will explore one such method to determine the solution.
Solving the System of Equations
Now that we have established the system of equations, the next step is to solve for the variables 'x' and 'y'. Several methods can be employed to solve such a system, including substitution, elimination, and graphical methods. For this particular problem, the elimination method proves to be an efficient and straightforward approach.
The elimination method involves manipulating the equations in a way that allows one of the variables to be eliminated when the equations are added or subtracted. In our case, we can observe that the coefficient of 'y' is the same in both equations (1). This makes the elimination process particularly simple.
To eliminate 'y', we can subtract the first equation (x + y = 50) from the second equation (2x + y = 87). This yields:
(2x + y) - (x + y) = 87 - 50
Simplifying the equation, we get:
2x + y - x - y = 37
x = 37
Therefore, we have found that the value of 'x' is 37. This means that the hotel has 37 double rooms.
Now that we have the value of 'x', we can substitute it back into either of the original equations to solve for 'y'. Let's use the first equation (x + y = 50):
37 + y = 50
Subtracting 37 from both sides, we get:
y = 50 - 37
y = 13
Hence, the value of 'y' is 13, indicating that the hotel has 13 single rooms.
We have successfully solved the system of equations using the elimination method. The solution reveals that the hotel has 37 double rooms and 13 single rooms. To ensure the accuracy of our solution, it's always a good practice to verify it by substituting the values of 'x' and 'y' back into the original equations. In the following section, we will perform this verification step.
Verifying the Solution
After obtaining a solution to a mathematical problem, it's crucial to verify its accuracy. This step ensures that the solution satisfies all the given conditions and constraints. In the case of our hotel room problem, we found that the hotel has 37 double rooms (x = 37) and 13 single rooms (y = 13). To verify this solution, we need to substitute these values back into the original equations and check if they hold true.
The first equation is x + y = 50. Substituting the values of x and y, we get:
37 + 13 = 50
This equation is satisfied, as 37 + 13 indeed equals 50. This confirms that our solution correctly accounts for the total number of rooms in the hotel.
The second equation is 2x + y = 87. Substituting the values of x and y, we get:
2(37) + 13 = 87
74 + 13 = 87
This equation is also satisfied, as 74 + 13 equals 87. This confirms that our solution accurately reflects the total number of beds in the hotel, considering that each double room has 2 beds and each single room has 1 bed.
Since both equations are satisfied by our solution (x = 37, y = 13), we can confidently conclude that our solution is correct. The hotel indeed has 37 double rooms and 13 single rooms. This verification step reinforces the importance of not only finding a solution but also ensuring its validity within the context of the problem.
Conclusion
In conclusion, the problem of determining the number of double and single rooms in a hotel, given the total number of rooms and beds, provides a practical application of linear equations. By carefully translating the problem's information into a system of equations and employing algebraic techniques, we successfully solved for the unknowns. The solution revealed that the hotel has 37 double rooms and 13 single rooms. This solution was rigorously verified by substituting the values back into the original equations, ensuring its accuracy.
This exercise highlights the power of mathematics in modeling real-world scenarios and providing quantitative solutions. The ability to formulate equations, solve systems of equations, and verify results is a valuable skill that extends far beyond the classroom. From resource allocation to financial planning, mathematical problem-solving plays a crucial role in various aspects of our lives.
Moreover, this problem underscores the importance of analytical thinking and attention to detail. The process of translating the word problem into mathematical expressions requires careful consideration of the relationships between the variables. Similarly, the solution process demands precision and accuracy in applying algebraic operations. The verification step serves as a final check, ensuring that no errors have been made along the way.
In essence, this hotel room problem exemplifies the beauty and utility of mathematics in addressing practical challenges. It demonstrates how abstract concepts can be applied to concrete situations, providing insights and solutions that can inform decision-making and enhance our understanding of the world around us. This problem serves as a testament to the enduring relevance of mathematics in a wide range of disciplines and its continued importance in our daily lives.