C) A Hotel Has 200 Rooms. Rooms With Kitchen Facilities Rent For $\$200$ Per Night, And Those Without Rent For $\$160$ Per Night. On A Night When The Hotel Was Completely Booked, Total Revenues Were $\$34,000$. How Many Of

Introduction

In this problem, we are given information about a hotel with 200 rooms. The hotel has two types of rooms: those with kitchen facilities and those without. The rooms with kitchen facilities rent for $\$200$ per night, while those without rent for $\$160$ per night. On a night when the hotel was completely booked, the total revenues were $\$34,000$. Our goal is to determine how many of the rooms have kitchen facilities.

Let's Define the Variables

Let $x$ be the number of rooms with kitchen facilities and $y$ be the number of rooms without kitchen facilities. We know that the total number of rooms is 200, so we can write an equation:

$$x + y = 200$$

We also know that the total revenues were $\$34,000$, and that the rooms with kitchen facilities rent for $\$200$ per night and those without rent for $\$160$ per night. We can write an equation based on this information:

$$200x + 160y = 34000$$

Solving the System of Equations

We have a system of two equations with two variables. We can solve this system using substitution or elimination. Let's use substitution.

First, we can solve the first equation for $y$:

$$y = 200 - x$$

Now, we can substitute this expression for $y$ into the second equation:

$$200x + 160(200 - x) = 34000$$

Expanding and simplifying, we get:

$$200x + 32000 - 160x = 34000$$

Combine like terms:

$$40x = 2000$$

Divide by 40:

$$x = 50$$

Now that we have found the value of $x$, we can find the value of $y$ by substituting $x$ into one of the original equations. Let's use the first equation:

$$x + y = 200$$

$$50 + y = 200$$

Subtract 50 from both sides:

$$y = 150$$

Conclusion

We have found that the hotel has 50 rooms with kitchen facilities and 150 rooms without kitchen facilities. This solution makes sense, as the total number of rooms is 200, and the total revenues were $\$34,000$.

What We Learned

In this problem, we learned how to solve a system of linear equations using substitution. We also learned how to use variables to represent unknown values and how to write equations based on given information.

Real-World Applications

This problem has real-world applications in the hospitality industry. Hotels and resorts need to manage their room inventory and pricing to maximize revenue. By understanding how to solve systems of linear equations, hotel managers can make informed decisions about room pricing and inventory.

Additional Practice Problems

If you want to practice solving systems of linear equations, try the following problems:

• A bakery sells two types of bread: whole wheat and white. The whole wheat bread sells for $\$2$ per loaf, and the white bread sells for $\$3$ per loaf. If the bakery sells 100 loaves of bread per day and makes a total of $\$250$ per day, how many loaves of whole wheat bread does the bakery sell?
• A company produces two types of products: A and B. Product A sells for $\$10$ per unit, and product B sells for $\$15$ per unit. If the company produces 100 units per day and makes a total of $\$1200$ per day, how many units of product A does the company produce?

Answer Key

• A bakery sells 60 loaves of whole wheat bread per day.
• A company produces 80 units of product A per day.
  A Hotel Revenue Problem: Solving for the Number of Rooms with Kitchen Facilities ===========================================================

Q&A: A Hotel Revenue Problem

Q: What is the problem about?

A: The problem is about a hotel with 200 rooms. The hotel has two types of rooms: those with kitchen facilities and those without. The rooms with kitchen facilities rent for $\$200$ per night, while those without rent for $\$160$ per night. On a night when the hotel was completely booked, the total revenues were $\$34,000$. We need to determine how many of the rooms have kitchen facilities.

Q: What are the variables in the problem?

A: The variables in the problem are $x$ and $y$. $x$ represents the number of rooms with kitchen facilities, and $y$ represents the number of rooms without kitchen facilities.

Q: What are the equations in the problem?

A: The two equations in the problem are:

$$x + y = 200$$

$$200x + 160y = 34000$$

Q: How do we solve the system of equations?

A: We can solve the system of equations using substitution or elimination. In this case, we used substitution.

Q: What is the solution to the problem?

A: The solution to the problem is that the hotel has 50 rooms with kitchen facilities and 150 rooms without kitchen facilities.

Q: Why is this solution correct?

A: This solution is correct because it satisfies both equations. When we substitute $x = 50$ and $y = 150$ into the first equation, we get:

$$50 + 150 = 200$$

Which is true. When we substitute $x = 50$ and $y = 150$ into the second equation, we get:

$$200(50) + 160(150) = 34000$$

Which is also true.

Q: What are some real-world applications of this problem?

A: This problem has real-world applications in the hospitality industry. Hotels and resorts need to manage their room inventory and pricing to maximize revenue. By understanding how to solve systems of linear equations, hotel managers can make informed decisions about room pricing and inventory.

Q: Can you give me some additional practice problems?

A: Yes, here are a few additional practice problems:

• A bakery sells two types of bread: whole wheat and white. The whole wheat bread sells for $\$2$ per loaf, and the white bread sells for $\$3$ per loaf. If the bakery sells 100 loaves of bread per day and makes a total of $\$250$ per day, how many loaves of whole wheat bread does the bakery sell?
• A company produces two types of products: A and B. Product A sells for $\$10$ per unit, and product B sells for $\$15$ per unit. If the company produces 100 units per day and makes a total of $\$1200$ per day, how many units of product A does the company produce?

Q: Can you give me the answer key for the practice problems?

A: Yes, here is the answer key for the practice problems:

• A bakery sells 60 loaves of whole wheat bread per day.
• A company produces 80 units of product A per day.

Conclusion

We have solved the problem of determining how many rooms with kitchen facilities a hotel has based on the total revenues. We used substitution to solve the system of linear equations and found that the hotel has 50 rooms with kitchen facilities and 150 rooms without kitchen facilities. This solution has real-world applications in the hospitality industry and can be used to make informed decisions about room pricing and inventory.