Mixing Fuels: How To Calculate Ethanol Price?
Hey guys! Let's dive into a common problem faced by drivers: mixing gasoline and ethanol. This is a super practical scenario, especially with fluctuating fuel prices. We're going to break down a problem where a driver wants to mix these fuels, and we need to figure out the price of ethanol. It might sound a bit like a math puzzle, but trust me, it’s something you might actually use in real life! This guide provides a detailed, SEO-optimized explanation of how to calculate the price of ethanol when mixing it with gasoline, designed to help students prepare for exams and understand real-world applications of math.
Understanding the Problem
So, here’s the situation: A driver decides to mix gasoline and ethanol in their car's fuel tank. They pour in 30 liters of gasoline and 20 liters of ethanol. The total mixture, all 50 liters of it, ends up costing an average of R$ 5.80 per liter. Now, we also know that the gasoline costs R$ 6.50 per liter. The big question is: what’s the price of each liter of ethanol? To solve this, we need to use some basic math principles and a little bit of algebra. Don't worry, we'll go through it step by step so it's super clear.
This problem is a classic example of a weighted average question, often encountered in math and physics. It involves understanding how different quantities contribute to an overall average. In this case, the quantities are the volumes of gasoline and ethanol, and their respective prices per liter. The key to solving this problem lies in setting up the right equation that represents the total cost of the mixture in terms of the individual costs and volumes of its components. We'll need to carefully consider each piece of information provided and how they relate to each other. This kind of problem is not just about numbers; it's about applying mathematical concepts to a real-life scenario, which is why it's a great example for exam preparation and everyday problem-solving skills.
Setting Up the Equation
Okay, let’s get down to business. The first thing we need to do is translate the word problem into a mathematical equation. This might sound intimidating, but it’s really just about breaking down the information into smaller, manageable parts. Remember, the goal is to find the price of ethanol per liter, so let's call that x. This is our unknown, the variable we're trying to solve for. Now, think about the total cost of the mixture. We know the driver has 30 liters of gasoline at R$ 6.50 per liter and 20 liters of ethanol at x per liter. The total cost of the gasoline is simply the volume (30 liters) multiplied by the price per liter (R$ 6.50), and the total cost of the ethanol is the volume (20 liters) multiplied by its price per liter (x). We add these two costs together to get the total cost of the mixture. But we also know that the total mixture (50 liters) costs R$ 5.80 per liter. So, the total cost can also be calculated by multiplying the total volume (50 liters) by the average price per liter (R$ 5.80). This gives us two ways to express the total cost, which we can then set equal to each other to form our equation. This step-by-step approach of converting word problems into mathematical expressions is crucial for success in math and related fields. It allows us to systematically tackle complex problems by breaking them down into smaller, more manageable steps.
The equation we will be working with is this: (30 * 6.50) + (20 * x) = 50 * 5.80. This equation represents the total cost of the mixture in two different ways, allowing us to solve for the unknown variable, x, which is the price of ethanol per liter. Each part of the equation corresponds to a specific component of the problem. The term (30 * 6.50) represents the total cost of the gasoline, calculated by multiplying the volume of gasoline (30 liters) by its price per liter (R$ 6.50). The term (20 * x) represents the total cost of the ethanol, calculated by multiplying the volume of ethanol (20 liters) by its unknown price per liter (x). The sum of these two terms gives the total cost of the mixture based on the individual costs of its components. On the other side of the equation, (50 * 5.80) represents the total cost of the mixture calculated by multiplying the total volume of the mixture (50 liters) by its average price per liter (R$ 5.80). By setting these two expressions for the total cost equal to each other, we create an equation that allows us to solve for the unknown price of ethanol. Understanding how each term in the equation relates to the original problem is crucial for solving it correctly.
Solving for X: The Price of Ethanol
Now comes the fun part – solving for x! We've got our equation: (30 * 6.50) + (20 * x) = 50 * 5.80. The first step is to simplify both sides of the equation. Let's start by multiplying the numbers we know. 30 multiplied by 6.50 equals 195. So, the cost of the gasoline is R$ 195. Then, 50 multiplied by 5.80 equals 290. So, the total cost of the mixture is R$ 290. Now our equation looks like this: 195 + (20 * x) = 290. We're getting closer! The next step is to isolate the term with x on one side of the equation. To do this, we need to subtract 195 from both sides. This keeps the equation balanced, which is super important in algebra. So, 195 + (20 * x) - 195 = 290 - 195. This simplifies to 20 * x = 95. We're almost there! Now, to find x, we need to divide both sides of the equation by 20. This will give us the value of x, which is the price of ethanol per liter. So, (20 * x) / 20 = 95 / 20. This simplifies to x = 4.75. And there you have it! We've solved for x. This step-by-step approach to solving algebraic equations is a fundamental skill in mathematics and is crucial for tackling more complex problems.
The solution, x = 4.75, tells us that the price of each liter of ethanol is R$ 4.75. This is the answer to our original question! Now, let’s recap how we got there. We started with a word problem, identified the unknown (the price of ethanol), and translated the problem into a mathematical equation. We then simplified the equation by performing multiplication and subtraction, step by step, always ensuring that we kept the equation balanced. Finally, we isolated x by dividing both sides of the equation by the coefficient of x. This process of breaking down a complex problem into smaller, manageable steps is a powerful problem-solving strategy that can be applied in many different contexts. It's not just about getting the right answer; it's about understanding the process and the logic behind it. This understanding will help you tackle similar problems in the future with confidence. Remember, practice makes perfect, so the more you work through problems like this, the more comfortable and proficient you'll become at solving them.
Checking Our Answer
It's always a good idea to double-check your work, especially in math! Let's make sure our answer makes sense in the context of the problem. We found that the price of ethanol is R$ 4.75 per liter. To check this, we can plug this value back into our original equation and see if both sides of the equation are equal. Our original equation was: (30 * 6.50) + (20 * x) = 50 * 5.80. Now, let’s substitute x with 4.75: (30 * 6.50) + (20 * 4.75) = 50 * 5.80. First, we calculate the products: 30 * 6.50 = 195 and 20 * 4.75 = 95. So, the left side of the equation becomes 195 + 95, which equals 290. Now, let's calculate the right side of the equation: 50 * 5.80 = 290. Both sides of the equation are equal to 290! This confirms that our answer is correct. Checking our answer in this way not only gives us confidence in our solution but also helps us reinforce our understanding of the problem and the steps we took to solve it. This practice is especially valuable in exam situations, where accuracy is crucial.
But why is checking the answer so crucial? It's more than just a formality; it’s a way to ensure that the solution you’ve found fits the initial conditions of the problem. In real-world scenarios, like managing fuel costs, an incorrect calculation can lead to financial errors. Similarly, in scientific and engineering applications, accuracy is paramount. By checking our answer, we're essentially validating our solution against the original problem, ensuring that we haven't made any logical or computational errors along the way. This step also helps to solidify the understanding of the concepts involved, as we revisit the problem from a different angle. The habit of verifying solutions is a hallmark of critical thinking and effective problem-solving, skills that are valuable not just in mathematics, but in all aspects of life. So, always remember to check your work – it’s a small step that can make a big difference!
Real-World Applications
Okay, so we've solved the problem, but why is this even important? Well, understanding how to calculate the cost of mixtures like this has tons of real-world applications! Think about it – mixing fuels is something drivers do all the time, especially with the rise of ethanol-blended gasoline. Knowing how to calculate the price per liter helps you make informed decisions at the pump and potentially save money. But it's not just about fuel. This type of calculation can be used in cooking, where you might need to mix ingredients in specific ratios to achieve a certain flavor or consistency. It's used in chemistry, where you might need to calculate the concentration of a solution. It's even used in finance, where you might need to calculate the weighted average cost of investments. The underlying principle of weighted averages is applicable across a wide range of fields, making this a valuable skill to have. So, by mastering this type of problem, you're not just learning math; you're learning a skill that can be applied in many different areas of your life. This is the beauty of mathematics – it's not just about numbers; it's about understanding the world around us and making informed decisions.
Key Takeaways for Exam Preparation
So, what are the main things to remember when tackling a problem like this on an exam? First, read the problem carefully. Make sure you understand exactly what's being asked and what information you're given. Identify the unknown variable – in this case, the price of ethanol. Second, translate the word problem into a mathematical equation. This is a crucial step, so take your time and make sure the equation accurately represents the situation. Third, simplify the equation step by step, showing your work clearly. This will help you avoid mistakes and also make it easier for the examiner to follow your reasoning. Fourth, solve for the unknown variable using algebraic techniques. Remember to keep the equation balanced by performing the same operations on both sides. Fifth, check your answer by plugging it back into the original equation or by thinking about whether it makes sense in the context of the problem. Finally, practice, practice, practice! The more problems you solve, the more comfortable you'll become with this type of calculation. And remember, it's not just about memorizing formulas; it's about understanding the concepts and being able to apply them in different situations. By following these steps and practicing regularly, you'll be well-prepared to tackle similar problems on your exam and in real life.
Conclusion
Alright guys, we've covered a lot in this article! We started with a practical problem about mixing gasoline and ethanol, and we walked through the steps to solve it. We learned how to translate a word problem into a mathematical equation, how to solve for an unknown variable, and how to check our answer. We also discussed the real-world applications of this type of calculation and the key takeaways for exam preparation. I hope this has been helpful and that you now feel more confident in your ability to tackle similar problems. Remember, math is not just about numbers; it's about problem-solving and critical thinking. So keep practicing, keep asking questions, and keep exploring the world of mathematics! You've got this!
