Calculate Total Revenue From Sales How To Solve The Problem
Hey guys! Let's dive into a super practical scenario today – calculating the total income you get from selling products. This is a fundamental concept in business and economics, and it's something that can help you make smart decisions whether you're running a lemonade stand or a big corporation. In this article, we'll break down a problem where we need to figure out the total revenue earned from selling a certain quantity of products at a specific price. We'll walk through each step, making sure it's crystal clear so you can tackle similar problems with confidence. So, let's jump right in and get those calculators ready!
Understanding the Problem
Before we crunch any numbers, let's make sure we understand what the problem is asking. Imagine you're selling some cool gadgets, and each one has a price tag of 5y + 3 pesos. That 'y' might seem a bit mysterious, but don't worry, we'll deal with it. Now, you manage to sell 3y + 1 of these awesome gadgets. The big question is: How much money did you make in total? In other words, what's the total income or revenue you earned? This is where our math skills come in handy. We need to figure out how to combine the price of each gadget with the number of gadgets sold. Think of it like this: if you sell one gadget, you get the price of that one gadget. If you sell two, you get twice the price, and so on. So, to get the total income, we're going to need to multiply the price of each item by the number of items sold. Let's get into the nitty-gritty and see how this works.
Breaking Down the Variables
Okay, let’s break down the variables we're working with. In our problem, the price of each product is given as 5y + 3 pesos. The variable 'y' here represents some unknown value, but don’t let that intimidate you. It’s just a placeholder that allows us to express the price in a general way. The number of products sold is given as 3y + 1. Again, 'y' is there, keeping things flexible. Now, to find the total income, we need to multiply these two expressions together. This is where our algebraic skills come into play. Think of it like this: each part of the price needs to be multiplied by each part of the quantity. We’re essentially distributing the multiplication across the terms. This might sound a bit complicated, but we'll take it step by step. By understanding what each variable represents, we can set up the equation correctly and solve for the total income. So, let's move on to the next step and see how we can multiply these expressions.
Setting Up the Equation
Alright, guys, it's time to set up the equation that will help us find the total income. Remember, the basic idea is that total income equals the price per product multiplied by the number of products sold. In our case, the price per product is 5y + 3 pesos, and the number of products sold is 3y + 1. So, the equation looks like this:
Total Income = (5y + 3) * (3y + 1)
This might seem a bit intimidating with all the parentheses and the 'y', but don't worry, we'll tackle it together. What we have here is essentially two binomials (expressions with two terms) that we need to multiply. To do this, we'll use a method called the distributive property, which is a fancy way of saying that we'll multiply each term in the first expression by each term in the second expression. Think of it like this: we're making sure that every part of the price gets multiplied by every part of the quantity. Once we've done that, we'll simplify the resulting expression by combining like terms. This will give us a nice, neat equation for the total income. So, let's roll up our sleeves and get multiplying!
Applying the Distributive Property
Okay, let's get down to the nitty-gritty of multiplying these expressions. We're going to use the distributive property, which, as we mentioned, means we'll multiply each term in the first expression by each term in the second expression. So, we have (5y + 3) * (3y + 1). Here’s how it breaks down:
First, we'll multiply 5y by both terms in the second expression:
5y * 3y = 15y² 5y * 1 = 5y
Next, we'll multiply 3 by both terms in the second expression:
3 * 3y = 9y 3 * 1 = 3
Now, we add all these terms together:
15y² + 5y + 9y + 3
See? It's like we're making sure each piece gets its turn. The distributive property helps us keep everything organized and ensures we don't miss any multiplications. But we're not done yet! We need to simplify this expression by combining like terms. This will make our equation cleaner and easier to work with. So, let’s move on to the next step and tidy things up!
Simplifying the Expression
Now that we've applied the distributive property, we've got a bit of a jumble of terms: 15y² + 5y + 9y + 3. Our next step is to simplify the expression by combining like terms. Like terms are terms that have the same variable raised to the same power. In our expression, we have two terms with 'y' to the power of 1: 5y and 9y. These are like terms, and we can combine them by simply adding their coefficients (the numbers in front of the 'y'). So, 5y + 9y equals 14y. Our 15y² term is the only term with y², so it stays as is. And the 3 is a constant term, meaning it doesn't have any variables, so it also stays as is.
Putting it all together, our simplified expression is:
15y² + 14y + 3
This is our final expression for the total income. It tells us that the total income is a quadratic expression in terms of 'y'. This means that the income will change based on the value of 'y', and it will change in a non-linear way. Depending on what 'y' represents, we could plug in different values to see how the income changes. But for now, we've successfully simplified the expression, and we have a clear formula for calculating the total income. So, let's recap what we've done and see how this can be applied in real-world scenarios.
Understanding the Result
So, we've arrived at our final expression for the total income: 15y² + 14y + 3. It's super important to understand what this result means. This expression tells us that the total income depends on the value of 'y'. Remember, 'y' is a variable, which means it can take on different values. Depending on what 'y' represents in our real-world scenario, the total income will change. For example, 'y' might represent the cost of materials, the number of hours worked, or some other factor that influences both the price and the quantity sold. The expression is a quadratic, meaning it has a y² term. This tells us that the relationship between 'y' and the total income isn't a straight line; it's a curve. This means that small changes in 'y' can sometimes lead to bigger changes in the total income. If we wanted to, we could plug in different values for 'y' and calculate the total income for each value. This could help us make decisions about pricing, production, and sales strategies. But for now, we've successfully found the expression for the total income, and we understand what it means in the context of our problem. Let’s wrap things up with a quick recap.
Conclusion
Alright, guys, we've reached the end of our journey into calculating total income from product sales! We started with a problem where the price of each product was given as 5y + 3 pesos, and the number of products sold was 3y + 1. Our goal was to find an expression for the total income. We broke down the problem, set up the equation, applied the distributive property, and simplified the expression. In the end, we found that the total income is given by the expression 15y² + 14y + 3. This expression tells us how the total income depends on the variable 'y'. We also discussed how understanding this expression can help in making business decisions. Whether you're selling gadgets, lemonade, or any other product, knowing how to calculate your total income is a crucial skill. It helps you understand your business better and make informed choices about pricing, sales, and overall strategy. So, next time you're faced with a similar problem, remember the steps we've covered, and you'll be able to tackle it with confidence. Keep practicing, and you'll become a pro at these calculations in no time!