Sweater Sales Equation: Find P(c) Based On Customers
Let's dive into figuring out how to write an equation that tells us the sweater sales, p(c), depending on how many customers we have (c). We know that the sales go up directly with the number of customers, and we've got a little clue: eight customers spent a total of $335 on sweaters. So, how do we crack this nut and find the right equation? Let's break it down, guys, into simple steps so we can understand it better. This problem is a classic example of direct variation, a concept that pops up frequently in mathematics and real-world scenarios. Understanding it thoroughly will not only help you solve this particular problem but also equip you to tackle similar situations with confidence. So, let's roll up our sleeves and get started!
Understanding Direct Variation
First, let's understand direct variation. In mathematical terms, direct variation means that two variables are related in such a way that when one variable increases, the other variable increases proportionally, and vice-versa. This relationship can be expressed by the equation y = kx, where y and x are the variables, and k is the constant of variation. This constant, k, represents the factor by which x must be multiplied to obtain y. In our sweater sales scenario, the sales, p(c), vary directly with the number of customers, c. This means we can write a similar equation: p(c) = kc, where k is the constant of proportionality that we need to determine. Finding k is the key to unlocking the specific equation for this problem. To find the constant of variation, we need a set of values for both variables. That's where the information about the eight customers and their $335 spending comes in handy. We'll use this data to calculate k and then plug it back into our equation, giving us the final formula to predict sales based on customer numbers. So, keep this concept of direct variation in mind as we move forward, because it's the foundation of our solution!
Setting up the Proportion
Now, let's set up a proportion to help us find the magic number, k. We know that the sales, p(c), are directly proportional to the number of customers, c. This means the ratio of sales to customers will always be the same. We're given that eight customers spent $335. So, we can express this as a ratio: . To find the general relationship, we can set this equal to the ratio of p(c) to c: p(c) / c = 335 / 8. This equation sets the foundation for solving our problem. It states that for any number of customers, the total sales will be proportional to the sales from the initial group of eight customers. By understanding this proportion, we can isolate the constant of variation and ultimately derive the equation that accurately predicts sweater sales based on customer count. The next step involves manipulating this equation to solve for p(c), which will give us the formula we're looking for. So, we're on the right track, guys! Let's keep going and see how we can simplify this proportion to get to our final answer.
Solving for p(c)
Alright, guys, let's get down to brass tacks and solve for p(c). We've got our proportion set up: p(c) / c = 335 / 8. To isolate p(c), we need to get rid of the division by c. How do we do that? Simple! We multiply both sides of the equation by c. This gives us: p(c) = (335 / 8) * c. Now, this looks much cleaner, doesn't it? This equation tells us that the total sales, p(c), is equal to the number of customers, c, multiplied by the constant ratio 335 / 8. This ratio represents the average amount spent per customer. By performing the division, we can find the decimal value of this constant, but leaving it as a fraction is perfectly fine and often more accurate. So, what we've done here is transformed the initial proportion into a clear, concise equation that directly relates sales to the number of customers. This is the core of our solution, and it's a fantastic demonstration of how algebraic manipulation can help us solve real-world problems. Now, let's take a closer look at what this equation actually means and how we can use it.
The Final Equation
So, after all that brain-bending work, we've arrived at our final equation: p(c) = (335 / 8) * c. This equation is the key to predicting sweater sales based on the number of customers. It's a neat little formula that tells us exactly how the sales, p(c), change as the number of customers, c, varies. Think of it like this: if we know how many customers come to the website, we can plug that number into c, do the math, and boom! We've got an estimate of the total sweater sales. The fraction 335 / 8 is super important here. It's the constant of proportionality, the k we talked about earlier. It essentially represents the average spending per customer. By multiplying the number of customers by this constant, we get the total sales, assuming that the spending habits of the customers remain consistent. This equation isn't just a bunch of symbols; it's a powerful tool that can help the website owner forecast sales, plan inventory, and make informed business decisions. It's a prime example of how math can be applied to real-world scenarios, making it not just an academic exercise but a practical skill.
In conclusion, the equation p(c) = (335 / 8) * c can be used to find the sales for c customers. This equation directly relates the total sales to the number of customers, using the constant of proportionality derived from the given information. Understanding direct variation and how to set up and solve proportions are key skills in tackling problems like this. So, next time you're faced with a similar challenge, remember the steps we've taken here, and you'll be well on your way to finding the solution! And remember, guys, math isn't just about numbers; it's about understanding relationships and using them to solve problems in the world around us. Keep practicing, and you'll become a math whiz in no time!