Solving For Movie Rentals How Many Movies Make The Costs Equal?

Introduction

Hey guys! Ever found yourself juggling between different online movie rental services, trying to figure out which one gives you the best bang for your buck? It's a common dilemma, especially with so many options available at our fingertips. In this article, we're going to break down a real-world problem using a simple algebraic equation. We'll tackle the equation 1.5r + 15 = 2.25r, which represents the number of movies, denoted as 'r', you need to rent to spend the same amount at two different internet movie services. So, grab your popcorn, and let's dive in!

Understanding the equation is the first step in solving the puzzle. The equation 1.5r + 15 = 2.25r might look a bit intimidating at first glance, but it's actually quite straightforward once you break it down. Let's dissect each part to understand what it represents in the context of our movie rental dilemma. On one side of the equation, we have 1.5r + 15. Here, 'r' stands for the number of movies you rent. The term 1.5r suggests that one movie rental service charges $1.5 per movie. The '+ 15' part indicates that this service also has an additional fixed cost, perhaps a membership fee or a one-time charge, of $15. On the other side of the equation, we have 2.25r. This implies that the second movie rental service charges $2.25 per movie, and there doesn't seem to be any additional fixed cost involved. The equation as a whole sets these two scenarios equal to each other, meaning we're trying to find the point where the total cost of renting movies from the first service (1.5r + 15) is exactly the same as the total cost of renting from the second service (2.25r). Solving this equation will give us the value of 'r', which is the number of movies you need to rent for the total spending to be the same across both services. This is incredibly useful information for anyone trying to budget their entertainment expenses or decide which service offers the best value for their movie-watching habits. So, now that we have a solid understanding of what the equation represents, let's move on to the next step: actually solving it!

Solving the Equation: Step-by-Step

Okay, let's get down to business and solve this equation! Solving equations might seem like a daunting task, but trust me, it's like piecing together a puzzle. Each step gets you closer to the final solution. We're going to take a step-by-step approach to make it super easy to follow.

Step 1: Isolate the Variable Terms

The first step in solving the equation 1.5r + 15 = 2.25r is to gather all the terms that contain our variable, 'r', on one side of the equation. This makes it easier to work with the equation and eventually isolate 'r' to find its value. Currently, we have '1.5r' on the left side and '2.25r' on the right side. To bring them together, we can subtract 1.5r from both sides of the equation. Remember, whatever we do to one side of an equation, we must do to the other to keep it balanced. So, subtracting 1.5r from both sides gives us: 1.5r + 15 - 1.5r = 2.25r - 1.5r. On the left side, the 1.5r terms cancel each other out, leaving us with just 15. On the right side, we subtract 1.5r from 2.25r, which gives us 0.75r. This simplifies our equation to 15 = 0.75r. Now, we have all our 'r' terms neatly on one side, and the constant term on the other. This is a significant step forward because it brings us closer to isolating 'r' and finding its value. The key here is to maintain balance by performing the same operation on both sides. By subtracting 1.5r, we've managed to streamline the equation and set the stage for the next step, which will involve isolating 'r' completely.

Step 2: Isolate the Variable

Now that we've got the equation in the form 15 = 0.75r, the next goal is to isolate 'r' completely. This means we want to get 'r' all by itself on one side of the equation. Currently, 'r' is being multiplied by 0.75. To undo this multiplication and isolate 'r', we need to perform the opposite operation, which is division. We'll divide both sides of the equation by 0.75. Again, it's crucial to remember that whatever we do to one side, we must do to the other to maintain the equation's balance. So, we divide both sides by 0.75: 15 / 0.75 = (0.75r) / 0.75. On the right side, 0.75 divided by 0.75 equals 1, so we're left with just 'r'. On the left side, 15 divided by 0.75 gives us 20. This means our equation simplifies to 20 = r. Voila! We've successfully isolated 'r' and found its value. This step is a classic example of using inverse operations to solve equations. By dividing both sides by 0.75, we effectively undid the multiplication and revealed the value of 'r'. So, what does this value mean in the context of our movie rental problem? Let's find out in the next section.

Step 3: State the Solution

We've crunched the numbers, and we've arrived at the solution: r = 20. But what does this actually mean in the real world? In the context of our original problem, 'r' represents the number of movies you need to rent for the total cost to be the same at both internet movie services. So, r = 20 tells us that you need to rent 20 movies for the total amount spent at the first service (which charges $1.5 per movie plus a $15 fee) to be equal to the total amount spent at the second service (which charges $2.25 per movie). This is a crucial piece of information for making informed decisions about which service to use. If you plan to rent fewer than 20 movies, one service might be more cost-effective, while if you plan to rent more, the other service might be the better deal. Stating the solution clearly, like this, is super important because it connects the mathematical result back to the original problem. It's not enough just to find a numerical answer; we need to understand what that answer means in the real-world scenario we're trying to solve. This step ensures that the math we've done is actually useful and can help us make practical decisions. So, with r = 20, we've not only solved the equation but also gained a valuable insight into our movie rental dilemma. Let's recap our findings and see how we can use this information.

Conclusion: Making Sense of the Solution

Alright, guys, we've journeyed through the equation 1.5r + 15 = 2.25r and successfully found that r = 20. That means you need to rent 20 movies for the total cost to be the same at both internet movie services. But what's the big picture here? Why is this solution so useful?

This result is super practical because it gives you a clear benchmark. If you're someone who only watches a few movies a month, say 5 or 10, the service with the flat fee of $15 might actually end up costing you more in the long run, even though their per-movie rental price is lower. On the other hand, if you're a movie buff who rents 25 or 30 movies a month, the service with the higher per-movie price but no flat fee might be the more economical choice. The solution r = 20 is the tipping point – the number of movies where the two services cost the same. Knowing this, you can make an informed decision based on your viewing habits. Moreover, this exercise highlights the power of algebra in everyday life. Equations aren't just abstract concepts; they're tools that can help us analyze situations, compare options, and make smarter choices. Whether it's budgeting for entertainment, planning a trip, or even deciding which phone plan to choose, the ability to set up and solve equations is a valuable skill. By breaking down the problem into smaller parts, understanding what each term in the equation represents, and systematically solving for the unknown, we've not only found the answer but also gained a deeper understanding of the underlying principles. So, the next time you're faced with a similar decision, remember the steps we took here – you might be surprised at how helpful a little algebra can be!

In conclusion, understanding and solving equations like 1.5r + 15 = 2.25r is more than just a mathematical exercise; it's a practical skill that empowers us to make informed decisions in various real-life scenarios. By systematically breaking down the equation, isolating the variable, and interpreting the solution in context, we've not only found the answer but also gained a valuable insight into the power of algebra in everyday problem-solving. So, keep those algebraic skills sharp, and you'll be well-equipped to tackle any mathematical challenge that comes your way!