Масса Печенья: Решение Задачи На Взвешивание

Hey math whizzes and cookie lovers! Today, we're diving into a delicious problem straight from a confectionery factory. Imagine a place churning out over 100 kinds of cookies, all packaged up in different ways. Our challenge is to figure out the weight of two specific cookie packages. We know their combined weight is 1650 grams, and one package is twice as heavy as the other. Sounds like a fun math puzzle, right? Let's break it down and solve it together, step by step.

Understanding the Cookie Conundrum

Alright guys, let's get our heads around this cookie problem. We've got a fantastic confectionery factory that's basically a cookie paradise, producing more than 100 types of cookies. That's a lot of dough and a lot of deliciousness! These cookies come in various shapes, sizes, and most importantly for our math problem, different packaging. Now, the core of our task is to determine the mass of each cookie package. We're given two specific packages, and we know their total weight adds up to a solid 1650 grams. The key piece of information here is the relationship between their weights: one package is exactly twice as heavy as the other. This isn't just a random fact; it's the crucial clue that will help us unlock the solution. We're not dealing with a single cookie's weight here, but rather the total mass of two distinct packages. This problem is a classic example of how mathematics can help us solve real-world scenarios, even when they involve yummy treats. So, keep your thinking caps on, because we're about to do some serious number crunching to find out exactly how much each of these cookie packages weighs. It’s all about using the information provided to set up an equation and solve for the unknowns. Ready to get started?

Setting Up the Mathematical Equation

Now, let's put on our math hats and translate this cookie problem into something we can actually solve. The first step in tackling any word problem is to identify the unknowns and the relationships between them. In this case, our unknowns are the mass of the two cookie packages. Let's give them some simple labels to make things easier. We can call the lighter package 'x' grams. Since the problem states that one package is twice as heavy as the other, the heavier package must weigh 2x grams. It's like having one small bag of cookies and another bag that's twice as full, holding double the amount. The next piece of crucial information is that the total mass of these two different packages is 1650 grams. This means if we add the weight of the lighter package (x) and the weight of the heavier package (2x) together, we should get exactly 1650 grams. So, we can write this out as a mathematical equation: x + 2x = 1650. This equation is the heart of our problem. It perfectly represents the scenario described: the sum of the weights of the two packages equals their combined weight, with the relationship between their individual weights already factored in. This setup is fundamental in algebra, where we use variables (like 'x') to represent unknown quantities and then use equations to find their values. It’s a powerful tool for simplifying complex situations into manageable mathematical expressions. By setting up this equation, we’ve taken a big leap towards finding the mass of each package. We've transformed a word problem into a solvable algebraic expression, which is a key skill in understanding and applying mathematics to various contexts.

Solving for the Lighter Package

Okay, guys, we've got our equation: x + 2x = 1650. Now comes the exciting part – solving it! First, we need to simplify the left side of the equation. Notice that 'x' and '2x' are like terms, meaning they both contain the variable 'x'. We can combine them just like we would combine apples or oranges. So, 'x' (which is the same as '1x') plus '2x' equals 3x. Our equation now looks much simpler: 3x = 1650. This tells us that three times the weight of the lighter package equals 1650 grams. To find out what 'x' is, we need to isolate it. We can do this by performing the opposite operation of multiplication, which is division. We need to divide both sides of the equation by 3 to keep it balanced. So, we calculate 1650 divided by 3. Let's do the math: 16 divided by 3 is 5 with a remainder of 1. Bring down the 5, making it 15. 15 divided by 3 is 5. Bring down the 0, and 0 divided by 3 is 0. So, 1650 / 3 = 550. This means x = 550. What does 'x' represent again? It's the mass of the lighter package! So, we've just discovered that the lighter package weighs 550 grams. This is a huge step! We’ve successfully used mathematics, specifically algebra, to find the weight of one of the packages. It’s a testament to how breaking down a problem and using the right tools can lead us to the answer. We're not done yet, but we've solved for one of our main unknowns. High five!

Calculating the Heavier Package's Weight

We've successfully found the weight of the lighter package, which is 550 grams (our 'x' value). But remember, the problem asked for the mass of each package. We still need to figure out how much the heavier package weighs. Luckily, we already established the relationship between the two packages right at the beginning. We said the heavier package weighs 2x grams. Since we now know that x = 550 grams, we can easily find the weight of the heavier package by substituting 'x' with its value. So, the heavier package weighs 2 * 550 grams. Let's do the multiplication: 2 times 550 is 1100 grams. So, the heavier package weighs 1100 grams. We've now successfully determined the mass of both packages: the lighter one is 550 grams, and the heavier one is 1100 grams. This is a fantastic conclusion to our calculation phase! We used the principles of mathematics and algebra to solve for two unknown variables based on a given total and a proportional relationship. It’s a clear demonstration of how problem-solving works – break it down, set up your tools (in this case, an equation), and solve step-by-step. We're almost at the finish line, and the math checks out!

Verifying Our Cookie Calculations

Alright, we've done the heavy lifting and calculated the weights: the lighter package at 550 grams and the heavier package at 1100 grams. But in any good math problem, especially one involving a confectionery factory and delicious cookies, it's always smart to double-check our work. This is where verification comes in, and it's a crucial part of the mathematical process. We need to make sure our answers actually fit the conditions given in the original problem. There were two main conditions: first, that one package is twice as heavy as the other, and second, that their total mass equals 1650 grams. Let's check the first condition. Is the heavier package (1100 grams) twice as heavy as the lighter package (550 grams)? Yes, because 550 * 2 = 1100. So, that condition is met perfectly. Now, let's check the second condition. Does the mass of the two different packages add up to 1650 grams? Let's add them: 550 grams + 1100 grams = 1650 grams. Bingo! The total matches exactly what the problem stated. This verification process confirms that our calculations are correct and that we've accurately solved the puzzle. It’s like tasting the cookies to make sure they’re baked just right – it confirms the quality of our work. So, we can confidently say that the mass of each package has been correctly identified using the power of mathematics.

Conclusion: A Sweet Mathematical Victory

So there you have it, math enthusiasts and cookie connoisseurs! We started with a problem about a confectionery factory producing loads of cookies, and through the magic of mathematics, we've successfully determined the mass of each cookie package. We found that the lighter package weighs 550 grams, and the heavier package weighs 1100 grams. Together, these two packages perfectly meet the criteria: one is double the weight of the other, and their combined weight is exactly 1650 grams. This problem was a fantastic example of how algebraic thinking can help us solve real-world challenges. By setting up a simple equation and carefully solving it, we unlocked the answer. It’s a great reminder that math isn't just about numbers on a page; it's a tool that helps us understand and interact with the world around us, even when it involves delicious treats like cookies. Keep practicing, keep questioning, and remember that every math problem, no matter how simple or complex, is an opportunity to learn and grow. Happy problem-solving, everyone!