Average Cost Per Item: A Step-by-Step Calculation

Let's dive into how to figure out the average cost per item when someone buys a bunch of stuff! In this case, a person purchased $5k + 2$ items, and the total cost came out to be $35k^2 + 29k + 6$. Sounds a bit like algebra, right? Don't worry, we'll break it down step by step. Understanding average costs is crucial in many real-life scenarios, from managing personal finances to making informed business decisions. So, let’s get started and make sure you grasp the concept fully.

Understanding the Problem

Before we jump into calculations, let's make sure we understand the problem clearly. The core question here is: if you buy a certain number of items for a total price, how much does each item cost on average? This is a common scenario in everyday life, whether you're buying groceries or office supplies. We need to find the average cost per item, which means we'll need to divide the total cost by the number of items purchased. In our specific case, we have algebraic expressions representing these values, which adds a bit of complexity but also makes it a great exercise in applying algebraic principles to practical problems.

First, we identify the two key pieces of information we have: the total number of items ($5k + 2$) and the total cost ($35k^2 + 29k + 6$). The variable k might seem intimidating, but it’s just a placeholder for a number. Think of it as a way to generalize the problem – the same method will work no matter what value k has. The presence of k also means our answer will be an expression rather than a simple number, which is perfectly fine. This is a common situation in algebra, where we often deal with general solutions rather than specific numerical answers.

Our goal is to find a simplified expression that represents the cost per item. We will achieve this by dividing the total cost expression by the total number of items expression. This is where your knowledge of algebraic division, particularly polynomial division or factoring, will come in handy. Don't worry if those terms sound scary; we’ll walk through the process. Remember, the average cost is a fundamental concept, and being able to calculate it, even with algebraic expressions, is a valuable skill. It's like knowing how to decode a secret message – once you understand the rules, you can apply them in many different situations.

Step-by-Step Calculation

Now, let's get our hands dirty with the math. The key to solving this problem is recognizing that we need to divide the total cost ($35k^2 + 29k + 6$) by the number of items ($5k + 2$). This looks like a job for polynomial division, but before we jump into that, let’s see if we can simplify things by factoring. Factoring is a powerful tool in algebra that allows us to rewrite expressions in a more manageable form. If we can factor both the numerator (total cost) and the denominator (number of items), we might be able to cancel out common factors and simplify the division process.

Let's start by trying to factor the quadratic expression $35k^2 + 29k + 6$. We're looking for two binomials that multiply together to give us this expression. This often involves some trial and error, but there are techniques to make it easier. Think about the factors of the leading coefficient (35) and the constant term (6). We need to find a combination that, when multiplied and added in the right way, gives us the middle coefficient (29). After some experimentation, you might find that $35k^2 + 29k + 6$ can be factored into $(5k + 2)(7k + 3)$. Isn't that neat? One of the factors matches the denominator! This is a great sign because it means we're on the right track.

Now, let's rewrite our division problem using the factored form: $(5k + 2)(7k + 3)$ divided by $(5k + 2)$. Do you see it? We have a common factor of $(5k + 2)$ in both the numerator and the denominator. This means we can cancel them out, just like simplifying a regular fraction. When we cancel out the $(5k + 2)$ terms, we're left with simply $(7k + 3)$. And that's our answer! The average cost per item is $7k + 3$. This demonstrates the power of factoring in simplifying algebraic expressions and solving problems more efficiently.

Verifying the Solution

Great! We've found that the average cost per item is $7k + 3$. But how can we be sure that this is the correct answer? Verification is a crucial step in problem-solving, especially in mathematics. It helps us catch any potential errors and build confidence in our solution. There are a couple of ways we can verify our answer in this case. One way is to substitute a value for k and see if the numbers work out. Let's try k = 1. If k = 1, then the total number of items is $5(1) + 2 = 7$, and the total cost is $35(1)^2 + 29(1) + 6 = 70$. The average cost per item would then be $70 / 7 = 10$. Now let's plug k = 1 into our answer: $7(1) + 3 = 10$. It matches! This gives us some confidence, but it's not a foolproof method since it only tests one specific case.

Another way to verify our solution is to multiply the average cost per item by the number of items and see if we get the total cost. In other words, we should multiply $(7k + 3)$ by $(5k + 2)$ and see if it equals $35k^2 + 29k + 6$. Let's do that: $(7k + 3)(5k + 2) = 35k^2 + 14k + 15k + 6 = 35k^2 + 29k + 6$. It matches the original total cost! This is a much stronger verification method because it checks our solution in a general way, not just for a specific value of k. It confirms that our algebraic manipulation was correct and that our answer is indeed the average cost per item.

By verifying our solution, we not only ensure accuracy but also deepen our understanding of the problem and the mathematical concepts involved. It’s a good habit to develop for any mathematical problem, as it can save you from making mistakes and help you build a stronger foundation in mathematics.

Conclusion

So, to wrap it all up, the average cost per item for this purchase is $7k + 3$. Remember, the key to solving this problem was to divide the total cost by the number of items. We simplified this process by factoring the expressions and canceling out common terms. We also verified our solution to ensure its accuracy. This problem highlights how algebraic concepts can be applied to real-world situations, making math not just an abstract exercise but a useful tool for understanding the world around us.

Understanding average costs is essential in various aspects of life, from budgeting to business management. The ability to work with algebraic expressions allows us to generalize solutions and apply them to a wide range of scenarios. Keep practicing these skills, and you'll find yourself becoming more confident and capable in your mathematical abilities. Don't be afraid to tackle challenging problems – they're the best way to learn and grow!

For further learning and practice on similar algebraic problems, you might find helpful resources on websites like Khan Academy's Algebra I section.