Brown Sugar Recipe: Scaling Servings With Math

Hey guys! Let's dive into this fun math problem based on a comic strip (tirinha in Portuguese!). We've got a recipe, some brown sugar, and a scaling challenge. It’s like we’re chefs in a mathematical kitchen today! So, let’s break it down and see how many portions we can whip up.

Understanding the Brown Sugar Ratio

First off, let’s understand the core of the problem. The comic strip snippet gives us a crucial piece of information: “E então, adicionam-se duas colheres de açúcar mascavo e mexe-se a calda por dois minutos” which translates to “Then, add two tablespoons of brown sugar and stir the syrup for two minutes.” This sets our base ratio. However, the main question revolves around scaling this recipe. The question states, “Suponha que a receita da tirinha rendesse três porções. Se fossem utilizadas 8 colheres de açúcar mascavo, a receita renderia quantas porções?” This means: Suppose the comic strip recipe yields three servings. If 8 tablespoons of brown sugar were used, how many servings would the recipe yield? This is where the fun begins – we need to figure out how brown sugar relates to the number of portions. This is a classic proportionality problem, and we'll need to set up a ratio to solve it. We know the initial recipe uses a certain amount of brown sugar (let's call it the base amount) and yields three portions. The question asks what happens when we use eight tablespoons. Think of it like this: if we double the brown sugar, will we double the portions? Probably! But let's get the exact math right. This type of problem is super common in everyday life, whether you're baking cookies, mixing drinks, or even figuring out how much paint you need for a room. The key is to identify the relationship between the ingredients and the final output. So, before we jump into calculations, let's make sure we've really wrapped our heads around what the question is asking. We're not just looking for a number; we're looking for a proportional relationship. This understanding will guide us to the correct solution and help us apply this concept to other scenarios in the future.

Setting Up the Proportion

Alright, let’s set up the proportion like true math whizzes! We know that the original recipe (let’s call it Recipe 1) yields three portions, and we are told (implicitly) that this recipe uses 2 tablespoons of brown sugar (from the first part of the comic strip description). Now, we have a new scenario (Recipe 2) where we're using 8 tablespoons of brown sugar, and we want to figure out how many portions this will yield. This is where setting up a proportion comes in handy. A proportion is just a statement that two ratios are equal. In our case, we can set up the following proportion: (tablespoons of brown sugar in Recipe 1) / (portions yielded by Recipe 1) = (tablespoons of brown sugar in Recipe 2) / (portions yielded by Recipe 2). Let's plug in the numbers we know: 2 / 3 = 8 / x. Here, 'x' is the number of portions we're trying to find. It's our unknown variable, the star of our mathematical show! Now, to solve for 'x', we'll use a technique called cross-multiplication. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. So, we get 2 * x = 3 * 8. This simplifies to 2x = 24. Now, we're just one step away from finding our answer. We need to isolate 'x', which means getting it all by itself on one side of the equation. To do this, we'll divide both sides of the equation by 2. This gives us x = 24 / 2, which simplifies to x = 12. So, what does this '12' mean? It means that if we use 8 tablespoons of brown sugar, the recipe will yield 12 portions. We've successfully used proportions to scale up our recipe! Proportions are such a powerful tool in math because they allow us to compare different quantities and find unknown values. Whether you're adjusting a recipe, converting measurements, or even calculating distances on a map, proportions can help you solve all sorts of problems. The key is to set up the ratios correctly and then use cross-multiplication to find the missing piece of the puzzle. So, next time you encounter a scaling problem, remember our brown sugar adventure and think: proportions are the way to go!

Solving for the Number of Portions

Okay, we've solved for the number of portions using the proportion, but let’s really break down what we did and why it works. We established the proportion 2/3 = 8/x. This equation is saying that the ratio of brown sugar to portions should remain constant, whether we're making the original small batch or a larger batch. Think of it like a seesaw – if we add more brown sugar, we need to add more portions to keep the balance. Cross-multiplication is a nifty trick that helps us solve proportions. When we multiply 2 * x and 3 * 8, we're essentially creating equivalent fractions. It's like finding a common denominator, but instead of working with fractions, we're working with an equation. So, 2x = 24 is just a way of rewriting our original proportion in a more manageable form. Now, dividing both sides by 2 might seem like a simple step, but it's super important. We're using the principle of equality, which says that if we do the same thing to both sides of an equation, the equation remains true. By dividing by 2, we're isolating 'x' and figuring out its value. The result, x = 12, tells us that 8 tablespoons of brown sugar will yield 12 portions. This makes intuitive sense, right? We used four times as much brown sugar (8 tablespoons compared to 2 tablespoons), and we ended up with four times as many portions (12 portions compared to 3 portions). This confirms that our proportion was set up correctly and that our answer is reasonable. Now, let's think about why this is useful in the real world. Imagine you're baking a cake for a party, and the recipe is for a small cake. You need to scale it up to feed a crowd. By using proportions, you can easily calculate how much of each ingredient you need. Or, let's say you're working on a science experiment, and you need to mix chemicals in a specific ratio. Proportions can help you ensure that you're using the right amounts. The beauty of proportions is that they apply to so many different situations. They're a fundamental concept in math and science, and they're also incredibly practical in everyday life. So, mastering proportions is like unlocking a superpower – you'll be able to solve all sorts of scaling and ratio problems with confidence.

Conclusion: The Sweet Taste of Math Victory!

So, guys, we've reached the conclusion of our tirinha-inspired math adventure! We started with a simple recipe snippet, a question about scaling, and a whole lot of mathematical curiosity. By understanding the problem, setting up a proportion, and solving for the unknown, we discovered that using 8 tablespoons of brown sugar would yield a delicious 12 portions. This wasn't just about finding the right answer; it was about understanding the process, the logic, and the power of proportions. We saw how proportions can help us scale recipes, adjust measurements, and solve real-world problems. Math isn't just about numbers and equations; it's about problem-solving, critical thinking, and making sense of the world around us. This little tirinha question turned into a fantastic opportunity to explore these concepts. And the best part? We did it together! We broke down the problem step by step, explained the reasoning behind each step, and celebrated our mathematical victory. So, next time you encounter a scaling problem, remember our brown sugar recipe. Remember the proportion, the cross-multiplication, and the joy of finding the solution. And most importantly, remember that math can be fun, engaging, and even a little bit delicious!

Final Answer

Therefore, if 8 tablespoons of brown sugar were used, the recipe would yield 12 portions.