Olive Oil And Palm Heart Cans: A Budget-Friendly Breakdown

Hey there, math enthusiasts! Ever find yourself staring at a grocery bill and wondering where all your money went? Let's break down a fun little problem involving olive oil and palm heart cans. This isn't just about numbers; it's about understanding how to use those numbers to solve real-world scenarios. We'll be using some basic algebra, so don't worry if you're not a math whiz. The goal here is to determine the exact number of olive oil and palm heart cans a person purchased, given their spending habits and the prices of each item. This problem provides a practical application of mathematical principles, demonstrating how algebra can be used to solve everyday financial problems. By understanding the relationships between the quantities and prices of the cans, we can determine the exact quantities purchased. This exercise not only strengthens your problem-solving skills but also highlights the usefulness of algebra in real-world situations, like budgeting and managing expenses.

Let's get started. We're given a scenario where a person buys a certain number of olive oil cans and a different number of palm heart cans. The total amount spent is known, along with the individual prices of each item. Our mission, should we choose to accept it, is to figure out how many of each type of can were purchased. We'll approach this by setting up equations based on the given information. The key is to translate the words into mathematical expressions that we can then manipulate to find the solution. Each step of the way, we'll explain the 'why' behind the 'how', making sure you understand the logic. This is not just about getting to the answer; it's about understanding the process of how to get there. It's like a treasure hunt, and algebra is our map and compass. So, grab your calculators (or your brains), and let's go on an adventure to find those hidden numbers! We'll start by defining our variables, which is a crucial first step in any algebraic problem.

Setting Up the Problem: Defining Variables and Equations

Alright, let's dive into this problem, shall we? First things first, we need to translate the word problem into a set of mathematical equations. This involves assigning variables to the unknown quantities and then formulating equations based on the information provided. It's like building a bridge; you need to start with the foundations. So, what do we know? We know the person bought x cans of olive oil at five reais each. This means the total cost for the olive oil cans is 5x. We also know that the person bought x + 4 cans of palm heart at seven reais each. The total cost for the palm heart cans is 7*(x+4). And finally, we know the total amount spent was 172 reais.

So, let's define our variables: Let x represent the number of olive oil cans. Therefore, the number of palm heart cans is x + 4. Now, we can write our equation. The total cost is the sum of the costs of the olive oil cans and the palm heart cans. In mathematical terms, this is 5x + 7*(x+4) = 172. This equation is the heart of the problem. It encapsulates all the information given and sets us up to solve for x. Remember, x is the number of olive oil cans. Once we find x, we can easily find the number of palm heart cans by adding 4. We can use algebraic techniques to solve for x, which includes simplifying expressions, isolating variables, and solving linear equations. Let's simplify and solve this equation step by step, which we'll break down into manageable parts. Doing so helps us keep track of our work and ensures that we don't miss any critical steps. Each step we take will get us closer to our goal: determining the number of olive oil and palm heart cans.

Solving for x: Unveiling the Number of Olive Oil Cans

Now, let's get down to the fun part: solving the equation! We have our equation: 5x + 7*(x+4) = 172. Our goal is to isolate x to find out how many olive oil cans were purchased. First, we need to simplify the equation. This means getting rid of those parentheses and combining like terms. Let's start by distributing the 7 across the terms inside the parentheses. So, 7*(x+4) becomes 7x + 28. Now our equation looks like this: 5x + 7x + 28 = 172. The next step is to combine like terms. On the left side of the equation, we have 5x and 7x. Combine them, and we get 12x. Our equation now is: 12x + 28 = 172. Next, we want to isolate the term with x. To do this, we need to get rid of that +28. We do this by subtracting 28 from both sides of the equation. Why both sides? Because whatever we do to one side of the equation, we must do to the other to keep it balanced.

So, 172 - 28 = 144. Our equation now becomes 12x = 144. Almost there! Now we just need to solve for x. To do this, we divide both sides of the equation by 12. So, 144 / 12 = 12. Therefore, x = 12. This means the person bought 12 cans of olive oil! We have successfully determined the value of x. The series of steps that we've taken show that solving algebraic equations involves a systematic approach, using principles of arithmetic to isolate the variable. This means each step we take must be based on mathematical rules, ensuring that our solution is accurate and reliable. Once you understand the process, you'll be able to solve increasingly complex equations, which is a valuable skill in math and many other areas. With the value of x in hand, we have the first piece of our puzzle.

Finding the Number of Palm Heart Cans and the Final Answer

We're in the home stretch, folks! We've found that x, the number of olive oil cans, is 12. Remember, the problem stated that the person bought x + 4 cans of palm heart. So, to find the number of palm heart cans, we simply substitute x with 12. Therefore, the number of palm heart cans is 12 + 4, which equals 16. So, the person bought 16 cans of palm heart. And there you have it! We've solved the problem.

So, here's our final answer: The person bought 12 cans of olive oil and 16 cans of palm heart. We successfully used algebra to solve a practical, real-world problem. By setting up the equations correctly and systematically solving for the unknown variables, we were able to determine the exact quantities of each type of can purchased. This exercise shows you that mathematics is more than just abstract concepts; it's a powerful tool that can be used to understand and solve problems we encounter every day. In summary, we started with a word problem, translated it into a mathematical equation, solved for our unknown variables, and arrived at a precise answer. This process reinforces the importance of using mathematical concepts in our daily lives. So next time you're at the grocery store, maybe you'll start calculating your purchases in your head. Well done, everyone!