Truck Capacity Problem: Solving For Trips
Hey guys! Ever wondered how to solve a real-world problem involving truck capacities and trips? Let's dive into a classic mathematical puzzle that's not only intriguing but also super practical. We're going to break down a problem where we need to figure out how many trips two trucks make, given their capacities and some extra information about their performance. So, buckle up and let's get started!
Understanding the Problem
Okay, so here's the scenario: We have two trucks, a smaller one and a larger one. The smaller truck has a capacity of 3 tons, while the larger truck can carry 4 tons. Now, here’s the twist: the smaller truck makes 18 more trips than the larger one. And because of these extra trips, it ends up delivering 12 more tons of freight than the larger truck. Our mission, should we choose to accept it, is to figure out how many trips each truck actually makes.
This type of problem might seem a bit daunting at first, but trust me, we can tackle it step by step. The key here is to translate the words into mathematical expressions. This involves using variables to represent the unknowns (in this case, the number of trips each truck makes) and setting up equations that reflect the relationships described in the problem. We'll use algebra to solve these equations, and by the end, we'll have our answers. So, let’s break down each part of the problem and see how it fits into our mathematical framework. Remember, the goal is not just to find the numbers but also to understand the process. That way, you'll be able to apply these problem-solving skills to all sorts of situations in the future!
Breaking Down the Information
To kick things off, let’s dissect the information we already have. The smaller truck’s 3-ton capacity and the larger truck’s 4-ton capacity are our foundational facts. These figures are constants, meaning they don't change throughout the problem. They're the bedrock upon which we’ll build our equations.
Next up, we have the relational data: the smaller truck makes 18 more trips than its larger counterpart. This is crucial because it sets up a direct comparison between the two. If we let 'x' represent the number of trips the larger truck makes, then the smaller truck makes 'x + 18' trips. See how we're already turning words into algebra? This is the magic of mathematical problem-solving!
Finally, we have the freight difference: the smaller truck delivers 12 more tons of freight. This gives us another comparative data point, this time focusing on the total weight delivered. We'll need to consider both the number of trips and the capacity of each truck to fully utilize this piece of information. It's like putting together a puzzle; each piece of data helps us see the bigger picture.
By carefully extracting and organizing this information, we’ve laid the groundwork for a clear, step-by-step solution. We're not just throwing numbers around; we're crafting a strategy. Each fact is a piece of the puzzle, and now we're ready to start assembling them into a coherent mathematical model.
Setting Up the Equations
Alright, let's get our hands dirty and translate this word problem into the language of mathematics! Setting up the equations is a crucial step; it's like creating the blueprint for our solution. If the equations are off, the whole house might crumble, so let's take our time and do it right.
First things first, we need to define our variables. In this case, the most logical choice is to let 'x' represent the number of trips the larger truck makes. Why 'x'? Well, it's the classic algebraic unknown, and it's a great starting point. Now, because the smaller truck makes 18 more trips than the larger one, we can express the number of trips it makes as 'x + 18'. Simple, right?
Now comes the trickier part: forming the equations. We know that the amount of freight a truck delivers is the product of its capacity and the number of trips it makes. So, the larger truck delivers 4x tons of freight (4 tons/trip * x trips), and the smaller truck delivers 3(x + 18) tons of freight (3 tons/trip * (x + 18) trips).
We also know that the smaller truck delivers 12 more tons than the larger truck. This gives us our golden equation: 3(x + 18) = 4x + 12. This equation beautifully encapsulates the relationship between the freight delivered by the two trucks. It's like a mathematical sentence that tells the story of our problem. With this equation in hand, we're more than halfway to the solution. The hard work of translation is done, and now we get to put on our algebraic hats and solve for 'x'. Let’s do it!
Explaining the Variables and Their Relationships
Let’s take a moment to really understand the variables and the relationships we've established. It's not just about writing equations; it's about grasping what they mean in the context of the problem. So, let's break it down, nice and easy.
We started with 'x', which represents the number of trips made by the larger truck. It's our baseline, the foundation upon which everything else is built. Think of it as the starting point in our journey to solve the problem. Everything else is defined in relation to 'x'.
The expression 'x + 18' then comes into play, representing the number of trips the smaller truck makes. This shows us the direct relationship between the two trucks’ trips – the smaller truck always makes 18 more than the larger. This is a key piece of the puzzle because it links the two unknowns together.
Next, we consider the amount of freight each truck delivers. The larger truck, with its 4-ton capacity, delivers 4x tons in total. This is a straightforward multiplication: capacity times trips. It’s a direct reflection of how much the truck can carry in a single journey multiplied by the number of journeys it makes.
Similarly, the smaller truck, with its 3-ton capacity, delivers 3(x + 18) tons. Notice how this expression combines both the capacity and the relationship between the trips? It's a bit more complex, but it perfectly captures the total freight delivered by the smaller truck.
Finally, the equation 3(x + 18) = 4x + 12 ties everything together. It states that the smaller truck delivers 12 more tons than the larger truck. This equation is the heart of our problem, the key to unlocking the solution. By understanding each variable and how they relate to each other, we’ve built a solid foundation for the next step: solving the equation.
Solving the Equation
Here comes the fun part – solving the equation we’ve set up! This is where we put our algebra skills to the test and find the value of 'x'. Think of it like cracking a code; we have the equation, and now we need to decipher it to reveal the hidden number of trips.
Our equation is 3(x + 18) = 4x + 12. The first thing we need to do is simplify. Let's distribute that 3 on the left side: 3 * x + 3 * 18 = 3x + 54. So, our equation now looks like this: 3x + 54 = 4x + 12.
Now, we want to get all the 'x' terms on one side and the constants on the other. A good move here is to subtract 3x from both sides: 3x + 54 - 3x = 4x + 12 - 3x, which simplifies to 54 = x + 12. We're getting closer!
To isolate 'x', we need to get rid of that pesky 12. So, we subtract 12 from both sides: 54 - 12 = x + 12 - 12, which gives us 42 = x. Boom! We’ve cracked the code. We now know that x = 42. But hold on, we're not quite done yet.
Remember, 'x' represents the number of trips the larger truck makes. We still need to find the number of trips the smaller truck makes. But we already know that the smaller truck makes 'x + 18' trips. So, we just add 18 to 42: 42 + 18 = 60. Voila! The smaller truck makes 60 trips.
We’ve solved the equation and found the number of trips for both trucks. It’s like reaching the summit of a mountain after a challenging climb. But before we celebrate, let’s make sure our solution actually makes sense. We'll need to verify our answers to ensure they fit the original problem conditions. Let's move on to the verification step!
Step-by-Step Solution Breakdown
Let’s recap the step-by-step solution we just walked through. It’s like retracing our steps on a map to make sure we took the right path. This breakdown will not only solidify our understanding but also provide a clear process we can use for similar problems in the future.
- Distribute: We started with the equation 3(x + 18) = 4x + 12. The first step was to distribute the 3 across the parentheses: 3 * x + 3 * 18, which gave us 3x + 54. So, the equation became 3x + 54 = 4x + 12. Distribution is a fundamental algebraic technique, and it's crucial for simplifying equations like this.
- Isolate x terms: Next, we wanted to gather all the terms with 'x' on one side of the equation. We subtracted 3x from both sides: 3x + 54 - 3x = 4x + 12 - 3x. This simplified to 54 = x + 12. Isolating the variable is a key strategy in solving equations; it’s like sorting ingredients before you start cooking.
- Isolate the variable: To get 'x' all by itself, we needed to eliminate the 12. We subtracted 12 from both sides: 54 - 12 = x + 12 - 12. This resulted in 42 = x. We’ve isolated 'x', which means we’ve found its value. This is the moment of triumph in our algebraic journey!
- Find the other unknown: We found that x = 42, which is the number of trips the larger truck makes. But we still needed to find the number of trips the smaller truck makes. We knew the smaller truck makes 'x + 18' trips, so we added 18 to 42: 42 + 18 = 60. We've now found both unknowns, but we’re not quite done yet.
This step-by-step breakdown shows us how each action logically leads to the next, transforming a complex-looking equation into a simple solution. But, as any good mathematician knows, finding a solution is only half the battle. The other half is making sure our solution is correct!
Verifying the Solution
Okay, we've got our answers – the larger truck makes 42 trips, and the smaller truck makes 60 trips. But before we declare victory, let’s put on our detective hats and verify our solution. It's like double-checking your work on an exam; we want to make sure we didn't make any silly mistakes along the way.
To verify, we need to go back to the original problem and see if our numbers fit the conditions. First, let's check the number of trips. The smaller truck makes 18 more trips than the larger truck. Does 60 = 42 + 18? Yes, it does! So far, so good.
Now, let's check the freight delivered. The larger truck, with its 4-ton capacity, delivers 4 * 42 = 168 tons. The smaller truck, with its 3-ton capacity, delivers 3 * 60 = 180 tons. The problem stated that the smaller truck delivers 12 more tons than the larger truck. Does 180 = 168 + 12? Absolutely! Our solution checks out on all fronts.
By verifying our solution, we've not only confirmed that our calculations are correct but also deepened our understanding of the problem. It's like seeing the puzzle pieces fit perfectly together; there's a satisfying sense of completeness. Verification is a crucial step in any mathematical problem-solving process, and it's a habit worth developing. It gives us confidence in our answers and ensures that we're not just blindly following steps, but truly understanding what we're doing.
Checking Against the Original Problem Conditions
Let's dive a bit deeper into how we checked our solution against the original problem conditions. This isn't just about plugging numbers in; it's about ensuring our answers make logical sense in the real-world context of the problem.
The first condition we verified was the relationship between the number of trips. The problem stated that the smaller truck makes 18 more trips than the larger one. We found that the larger truck makes 42 trips and the smaller truck makes 60 trips. So, we checked if 60 is indeed 18 more than 42, which it is. This simple check ensures that we've correctly translated the trip relationship into our solution.
The second, and more complex, condition involved the freight delivered. The problem stated that the smaller truck delivers 12 more tons of freight than the larger truck. This required us to calculate the total freight for each truck, considering both their capacity and the number of trips.
For the larger truck, we multiplied its 4-ton capacity by its 42 trips, giving us 168 tons. For the smaller truck, we multiplied its 3-ton capacity by its 60 trips, resulting in 180 tons. Then, we checked if 180 tons is 12 more than 168 tons, which it is. This freight check is crucial because it incorporates both the capacity of the trucks and the number of trips, ensuring our solution aligns with all aspects of the problem.
By meticulously checking our answers against both conditions, we’ve demonstrated a thorough understanding of the problem and a commitment to accuracy. This process is not just a formality; it's a vital part of mathematical problem-solving. It's like proofreading a document before you submit it; you want to catch any errors and ensure your message is clear and correct. So, remember, always verify your solutions!
Conclusion
So, there you have it, guys! We've successfully navigated through this truck capacity problem, and we've not only found the answers but also understood the process along the way. We discovered that the larger truck makes 42 trips, and the smaller truck makes 60 trips. But more importantly, we've reinforced our problem-solving skills.
We started by carefully breaking down the problem, identifying the knowns and unknowns, and establishing the relationships between them. Then, we translated the word problem into mathematical equations, a crucial step in turning a real-world scenario into an algebraic puzzle. We solved the equation step by step, using basic algebraic techniques like distribution and isolating variables. And finally, we verified our solution, ensuring that our answers not only made mathematical sense but also fit the original problem conditions.
This problem is a great example of how math can be applied in everyday situations. It's not just about numbers and equations; it's about logical thinking and problem-solving strategies. The skills we've used here – translating words into equations, solving equations, and verifying solutions – are applicable in countless other contexts. So, next time you encounter a challenging problem, remember the steps we've taken today, and tackle it with confidence. Keep practicing, keep learning, and you'll become a problem-solving pro in no time!