Motel Room Tax: Calculate Proportion And Amount

Have you ever wondered how taxes are calculated on your hotel or motel bill? It might seem a bit mysterious, but it's actually quite straightforward using proportions! Let's break down a real-world example to understand this better. We'll use a scenario where Rita stayed at the Shoreline Motel, and we need to figure out the tax she paid. So, let’s dive in and explore how to calculate tax using proportions, making it easy and clear for everyone. Understanding how taxes are calculated is not only useful for your travels but also a great way to brush up on your math skills.

(a) Setting Up the Proportion for Tax Calculation

In this section, we'll focus on creating a proportion to find the amount of tax Rita paid on her motel room. Proportions are a fantastic tool for solving problems where two ratios are equal. In our case, we'll set up a proportion to relate the cost of the room to the tax amount. This involves understanding the relationship between the original price, the tax rate, and the actual tax amount. Let’s define our variables clearly: let x be the amount of tax in dollars. The room cost $45, and the tax rate is 16%. Our goal is to create an equation that accurately represents this scenario, allowing us to easily solve for the unknown tax amount. This foundational step is crucial for understanding not just this specific problem but also for tackling various percentage-related calculations in everyday life. By mastering this concept, you'll be able to confidently calculate taxes, discounts, and other percentage-based figures. Remember, setting up the proportion correctly is half the battle! A well-structured proportion makes the subsequent calculation much simpler and less prone to errors. So, let's get started and construct the proportion that will help us find the tax Rita paid. This skill is invaluable not only for academic purposes but also for real-world applications, such as budgeting and financial planning. Understanding proportions empowers you to make informed decisions and manage your finances effectively. Now, let’s see how we can put this into action and create our proportion.

Understanding Proportions

To understand proportions, think of them as two equal fractions. In our case, one fraction will represent the tax rate as a percentage, and the other will represent the actual tax amount relative to the original room cost. The key is to keep the units consistent on both sides of the equation. For instance, if we put the tax amount in the numerator of one fraction, we should also put the original cost in the denominator of that fraction, and mirror this structure on the other side with the percentage rate and its corresponding base (which is 100 since we're dealing with percentages). This consistency ensures that the proportion accurately reflects the relationship between the values. It’s like comparing apples to apples – you need to make sure you're comparing the same units to get a meaningful result. Proportions are not just useful for tax calculations; they can be applied to a wide range of problems, from scaling recipes to converting measurements. The ability to set up and solve proportions is a fundamental skill in mathematics and has practical applications in many different fields. By mastering proportions, you gain a powerful tool for problem-solving and critical thinking. So, let’s move on to applying this understanding to our specific problem and constructing the proportion for Rita's motel room tax.

Setting Up the Equation

Now, let's translate the given information into a mathematical proportion. We know the tax rate is 16%, which can be written as 16/100. This fraction represents the ratio of tax to the original cost. We also know the original cost of the room is $45, and we want to find the tax amount, which we've defined as x. So, we can set up the proportion as follows: x/$45 = 16/100. This equation states that the ratio of the tax amount (x) to the original cost ($45) is equal to the ratio of the tax rate (16) to 100 (since percentages are out of 100). This proportion is the foundation for solving our problem. It captures the relationship between the tax rate, the original cost, and the tax amount in a clear and concise way. Setting up the equation correctly is crucial because it ensures that the solution we find is accurate. A well-structured proportion simplifies the calculation process and reduces the risk of errors. This skill is not only valuable for academic purposes but also for practical situations where you need to calculate percentages, such as when shopping or managing your finances. By mastering the art of setting up proportions, you empower yourself to solve a wide range of mathematical problems with confidence. Now that we have our proportion, the next step is to solve for x and find the tax amount. Let’s move on to that in the next section.

(b) Calculating the Tax Amount

Now that we have established the proportion x/$45 = 16/100, our next step is to solve for x to find the tax amount. Solving a proportion involves cross-multiplication, a straightforward technique that allows us to isolate the variable and determine its value. This is a fundamental skill in algebra and is incredibly useful in various real-world scenarios, not just in calculating taxes. Understanding how to solve proportions can help you with everything from adjusting recipes to understanding scale models. Let’s walk through the process step by step to make sure it’s crystal clear. By the end of this section, you'll not only know how to calculate the tax amount for Rita's motel room but also have a solid understanding of how to solve proportions in general. This skill is a valuable addition to your mathematical toolkit and will serve you well in many different contexts. So, let’s dive in and learn how to solve our proportion and find the tax amount.

Cross-Multiplication

To solve the proportion x/$45 = 16/100, we use the method of cross-multiplication. This involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the two products equal to each other. In our case, we multiply x by 100 and $45 by 16. This gives us the equation 100x = 45 * 16. Cross-multiplication is a powerful technique because it eliminates the fractions, making the equation easier to solve. It’s like turning a complex problem into a simpler one by rearranging the terms. This method is based on the fundamental principle that if two ratios are equal, their cross-products are also equal. Understanding this principle is key to mastering proportions. Cross-multiplication is not just a mathematical trick; it’s a logical way of manipulating equations to isolate the variable we want to solve for. This skill is essential for algebra and beyond, and it can be applied to a wide range of problems involving ratios and proportions. By mastering cross-multiplication, you gain a valuable tool for solving mathematical problems efficiently and accurately. Now that we have our equation, 100x = 45 * 16, let’s proceed to simplify it and isolate x to find the tax amount.

Isolating the Variable

After cross-multiplying, we have the equation 100x = 45 * 16. The next step is to isolate x to find its value. First, we need to calculate the product of 45 and 16, which equals 720. So, our equation becomes 100x = 720. Now, to isolate x, we need to divide both sides of the equation by 100. This is a fundamental principle of algebra: whatever operation you perform on one side of the equation, you must perform on the other side to maintain the equality. Dividing both sides by 100 gives us x = 720/100. This step is crucial because it separates x from the other terms, allowing us to determine its value. Isolating the variable is a key skill in solving algebraic equations, and it's used in a wide range of mathematical problems. Whether you're solving for an unknown in a simple equation or tackling a more complex problem, the ability to isolate the variable is essential. This skill empowers you to manipulate equations and find solutions effectively. Now that we have x = 720/100, we can perform the division to find the tax amount. Let’s do that in the next step.

Finding the Solution

Now that we have the equation x = 720/100, we can easily find the value of x by performing the division. Dividing 720 by 100 gives us x = 7.2. Since x represents the amount of tax in dollars, we can conclude that the tax for the room is $7.20. This is the final answer to our problem! We have successfully used a proportion to calculate the tax on Rita's motel room. Finding the solution is the culmination of all the steps we've taken so far, from setting up the proportion to isolating the variable. It’s the moment when we see the result of our efforts and gain a concrete answer to the question we set out to solve. This process not only provides us with the specific answer to this problem but also reinforces our understanding of proportions and how they can be used in real-world situations. Calculating taxes is just one example of the many applications of proportions. They can also be used to calculate discounts, scale recipes, convert measurements, and much more. By mastering the art of solving proportions, you equip yourself with a versatile tool for problem-solving and critical thinking. Now that we've found the tax amount, let's take a moment to reflect on the entire process and appreciate the power of proportions in solving mathematical problems.

Conclusion

In conclusion, we've successfully calculated the tax on Rita's motel room by setting up and solving a proportion. We defined the unknown tax amount as x, established the proportion x/$45 = 16/100, and used cross-multiplication and division to find that x = $7.20. This example demonstrates the power and practicality of proportions in solving real-world problems. Proportions are a fundamental concept in mathematics with wide-ranging applications, from calculating taxes and discounts to scaling recipes and converting units. By mastering proportions, you gain a valuable tool for problem-solving and critical thinking. The ability to set up and solve proportions is not just useful in academic settings; it's also essential for everyday life. Whether you're managing your finances, shopping for groceries, or planning a trip, proportions can help you make informed decisions and solve problems efficiently. We hope this step-by-step guide has made the process of calculating taxes using proportions clear and accessible. Remember, practice makes perfect, so try applying this method to other scenarios and see how it works. The more you use proportions, the more confident and proficient you'll become in solving mathematical problems. Keep exploring the world of mathematics, and you'll discover many more fascinating concepts and techniques that can enrich your understanding and empower you to tackle challenges with confidence. For further learning and to solidify your understanding of proportions and percentages, you might find resources on websites like Khan Academy's Arithmetic and Pre-algebra section incredibly helpful. They offer a wealth of lessons and practice exercises that can help you master these essential mathematical skills.