Calculating Costs Using Proportionality: Solving A National Exam Question

Hey guys! Ever get stumped by a math problem that seems simple but has you scratching your head? Well, let's dive into a classic proportionality question that many students face. This is a typical problem that you might encounter in national exams, and understanding how to solve it can really boost your confidence. We're going to break down the problem step-by-step, so you’ll not only get the answer but also understand the method behind it. So, let's get started!

Understanding Proportionality

Before we tackle the main question, it's super important to understand the concept of proportionality. Proportionality, at its core, is about how two quantities relate to each other. If two quantities are proportional, it means that when one changes, the other changes in a predictable way. Think of it like this: if you buy more of something, you expect to pay more, right? That’s proportionality in action! There are two main types of proportionality: direct and inverse.

Direct Proportionality

Direct proportionality is what we're dealing with in our problem today. In direct proportionality, as one quantity increases, the other quantity increases as well, and vice versa. The key here is that the ratio between the two quantities remains constant. For example, if you double the amount of material you buy, you expect to pay double the price. This direct relationship makes calculations straightforward once you grasp the fundamental principle. Understanding direct proportionality is essential not only for math problems but also for real-life situations, such as calculating grocery costs, understanding recipe ratios, or figuring out travel distances based on speed and time. The beauty of direct proportionality lies in its predictability and simplicity. If you can identify that two quantities are directly proportional, you can set up a simple proportion equation to find any unknown value.

Inverse Proportionality

On the flip side, there's inverse proportionality. In inverse proportionality, as one quantity increases, the other quantity decreases. Imagine you're planning a road trip. The faster you drive, the less time it takes to reach your destination. This is a classic example of inverse proportionality. While it's not the focus of our current problem, it’s still useful to know. Recognizing the difference between direct and inverse proportionality is crucial for tackling a variety of math and real-world problems. Inverse proportionality often involves scenarios where a resource is being shared or a task is being completed. For instance, the more workers you have on a project, the less time it will take to finish, assuming everyone is contributing effectively. Understanding these relationships helps you make informed decisions and solve problems more efficiently.

The Problem: 2.5 Meters Cost 15 Cents, What About 20 Meters?

Now, let's get to the heart of our problem. We know that 2.5 meters of something costs 15 cents. The question is: how much will 20 meters of the same thing cost? This is a classic proportionality problem, specifically a direct proportionality problem. Why? Because the more meters you buy, the more it's going to cost. Makes sense, right? The key to solving these problems is setting up a proportion. A proportion is just a statement that two ratios are equal. In this case, the ratio of meters to cost should remain the same, no matter how much we buy. Think of it like this: if you understand the cost per meter, you can easily calculate the cost for any number of meters.

The challenge, however, often lies in setting up the proportion correctly. You need to ensure that you're comparing the right quantities in the right order. Getting the setup wrong can lead to an incorrect answer, even if your calculations are perfect. So, before you start crunching numbers, take a moment to think about what you're comparing and how the quantities relate to each other. Once you've got the proportion set up correctly, the rest is usually just simple arithmetic. But remember, the devil is in the details, so pay close attention to each step.

Setting Up the Proportion

Okay, let's get down to business and set up the proportion. This is where we put our understanding of direct proportionality into action. Remember, a proportion is an equation stating that two ratios are equal. We're comparing meters to cost, so our ratios will look like this: (meters / cost). We know that 2.5 meters costs 15 cents, so one ratio is 2.5 / 15. We want to find out the cost for 20 meters, so we'll call the unknown cost 'x'. Our second ratio is 20 / x. Now, we set these two ratios equal to each other, forming our proportion equation: 2.5 / 15 = 20 / x. This equation is the key to solving our problem. It represents the relationship between the known quantities (2.5 meters and 15 cents) and the unknown quantity (the cost of 20 meters).

Setting up the proportion correctly is crucial, and it's a skill that gets easier with practice. A common mistake is to mix up the units or the order of the quantities. For instance, you might accidentally set up the proportion as 15 / 2.5 = 20 / x, which would lead to an incorrect answer. Always double-check that you're comparing like with like: meters with meters, and cost with cost. Once you're confident that your proportion is set up correctly, the next step is to solve for the unknown variable, which we'll tackle in the next section.

Solving for the Unknown

Now that we've got our proportion set up correctly (2.5 / 15 = 20 / x), it's time to solve for x, which represents the cost of 20 meters. The most common method to solve a proportion is cross-multiplication. It's a handy trick that simplifies the equation and makes it easy to isolate the variable. Here’s how it works: you multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa. So, in our case, we multiply 2.5 by x and 15 by 20. This gives us the equation: 2.5x = 15 * 20. Now, we simplify the right side of the equation: 2.5x = 300. To isolate x, we need to get it by itself on one side of the equation. We can do this by dividing both sides of the equation by 2.5. So, we have x = 300 / 2.5. Performing the division, we find that x = 120. This means that 20 meters will cost 120 cents, or $1.20.

This step-by-step process of solving for the unknown is a fundamental skill in mathematics and is used in various contexts, not just in proportionality problems. Understanding how to manipulate equations and isolate variables is essential for problem-solving in algebra and beyond. It's also a great feeling when you successfully solve for the unknown and find the answer! The key is to take it one step at a time, ensuring that each operation you perform is mathematically sound and brings you closer to the solution.

The Answer and Practical Application

So, after all that math, we've arrived at our answer: 20 meters will cost 120 cents, which is the same as $1.20! Isn't it satisfying to solve a problem like this? But beyond just getting the right answer, it's crucial to understand the process we used. This type of problem isn't just an academic exercise; it has practical applications in everyday life. Imagine you're at the store, and you see that a certain length of fabric costs a specific amount. You can use proportionality to quickly calculate how much a different length will cost. Or, if you're cooking, you might need to adjust a recipe to make a larger or smaller batch. Understanding proportionality helps you do this accurately.

The ability to solve these kinds of problems also boosts your confidence in dealing with numbers and quantitative information. It's a skill that's valuable in many different fields, from finance and engineering to healthcare and the arts. Being able to think proportionally allows you to make informed decisions and solve problems effectively. Moreover, mastering these fundamental math concepts lays a strong foundation for more advanced topics in mathematics and science. So, the effort you put into understanding proportionality pays off in more ways than one.

Tips for Solving Proportionality Problems

Before we wrap up, let's go over a few tips that can help you tackle proportionality problems with confidence. First and foremost, always read the problem carefully. Make sure you understand what's being asked and what information you're given. Identify the quantities that are proportional and determine whether they are directly or inversely proportional. This initial step is crucial for setting up the problem correctly. Next, set up your proportion equation accurately. Remember to compare like quantities with like quantities. For example, if you're comparing meters to cost, make sure you have meters in the numerators and cost in the denominators (or vice versa) in both ratios. Double-check that you haven't mixed up any units or values.

Another useful tip is to estimate the answer before you start calculating. This can help you catch any major errors. For example, in our problem, we knew that 20 meters would cost more than 2.5 meters, so we expected the answer to be greater than 15 cents. If we had gotten an answer less than 15 cents, we would have known something was wrong. Finally, practice, practice, practice! The more you work with proportionality problems, the more comfortable you'll become with them. Look for real-world examples and try to solve them using proportions. With consistent effort, you'll master this important math skill and be able to tackle any proportionality problem that comes your way.

Conclusion

So, guys, we've successfully solved a proportionality problem and learned some valuable tips along the way! Remember, the key to these problems is understanding the relationship between the quantities and setting up the proportion correctly. With a little practice, you'll be solving these problems like a pro. Keep practicing, keep learning, and you'll ace those exams in no time! And remember, math isn't just about numbers; it's about understanding the world around us. Keep exploring and keep asking questions!