Tania's Tire Troubles Solving A Motorcycle And ATV Puzzle
Introduction
Hey guys! Ever find yourself puzzling over a word problem that seems like it’s straight out of a math textbook? Well, today we’re diving into one that involves Tania, her motorcycle and ATV rental business, and a whole lot of tires. It’s a classic scenario that combines basic math with a real-world situation, and we’re going to break it down step by step. So, buckle up and let’s get started!
This problem is all about figuring out how many motorcycles and ATVs Tania has in her rental fleet, given that she bought 110 tires to replace all the old ones on her 40 vehicles. It might sound tricky at first, but with a little bit of algebra and some logical thinking, we’ll crack this nut in no time. Word problems like these are not just academic exercises; they help us develop critical thinking skills that are useful in everyday life. Whether you’re calculating the cost of groceries or figuring out the best route to work, problem-solving is a skill we all use constantly. So, let’s sharpen those skills and see how many motorcycles and ATVs Tania has!
Understanding the Problem
So, the first step in tackling any word problem is to really understand what’s being asked. In this case, Tania has a rental business with a mix of motorcycles and ATVs. We know she has a total of 40 vehicles, and she bought 110 new tires to replace all the old ones. The question we need to answer is: how many motorcycles and how many ATVs does Tania have? It’s crucial to identify the key information here: the total number of vehicles and the total number of tires. These are our clues, and they’ll guide us toward the solution.
To make things even clearer, let’s break down what we know about the vehicles themselves. Motorcycles, as we all know, have two tires. ATVs, on the other hand, typically have four tires. This difference in the number of tires is what allows us to distinguish between the two types of vehicles in the problem. If all the vehicles had the same number of tires, we wouldn’t be able to solve it! So, we have a mix of two-wheeled and four-wheeled vehicles, and we need to figure out how many of each there are. This is where algebra comes in handy. By using variables to represent the unknown quantities, we can set up equations that describe the relationships between them. It might sound intimidating, but trust me, it’s just a matter of translating the words into math.
Setting Up the Equations
Alright, now comes the fun part – turning this word problem into math equations! This might sound intimidating, but don’t worry, we’ll take it slow. The key here is to use variables to represent the unknowns. Let’s say “m” represents the number of motorcycles Tania has, and “a” represents the number of ATVs. These are the two things we’re trying to figure out, so it makes sense to give them their own symbols.
Now, let’s think about the information we have. We know that Tania has a total of 40 vehicles. This means that the number of motorcycles plus the number of ATVs must equal 40. We can write this as our first equation:
m + a = 40
This equation is a simple way of saying, “The number of motorcycles and the number of ATVs add up to 40.” Pretty straightforward, right? But we’re not done yet. We also know that Tania bought 110 tires. Each motorcycle needs two tires, and each ATV needs four tires. So, the total number of tires can be expressed in terms of “m” and “a” as well. This gives us our second equation:
2m + 4a = 110
This equation says, “Two times the number of motorcycles plus four times the number of ATVs equals 110.” This is because each motorcycle contributes two tires, and each ATV contributes four tires. Now we have two equations with two unknowns, which means we can solve for “m” and “a”! This is a classic system of equations, and there are several ways to solve it. We could use substitution, elimination, or even graphing. But for this problem, let’s use substitution. It’s a method that involves solving one equation for one variable and then plugging that expression into the other equation. Sound complicated? Don’t worry, we’ll walk through it step by step.
Solving the System of Equations
Okay, we’ve got our two equations: m + a = 40 and 2m + 4a = 110. Now, let’s use the substitution method to solve for m and a. The first step is to solve one of the equations for one variable. It doesn’t matter which equation or which variable we choose, but it’s often easiest to pick the one that looks simplest. In this case, the first equation, m + a = 40, looks pretty straightforward. Let’s solve it for m.
To isolate m, we just need to subtract a from both sides of the equation:
m = 40 - a
Now we have an expression for m in terms of a. This is the key to the substitution method. We can now substitute this expression for m into the second equation. This might sound a little confusing, but it just means that wherever we see an “m” in the second equation, we’re going to replace it with “40 - a”. So, our second equation, 2m + 4a = 110, becomes:
2(40 - a) + 4a = 110
See what we did there? We replaced the “m” with “(40 - a)”. Now we have an equation with only one variable, a, which means we can solve for it! The next step is to simplify the equation by distributing the 2 and combining like terms:
80 - 2a + 4a = 110
Combine the “a” terms:
80 + 2a = 110
Now, subtract 80 from both sides to isolate the term with “a”:
2a = 30
Finally, divide both sides by 2 to solve for a:
a = 15
Great! We’ve found that a = 15. This means Tania has 15 ATVs. But we’re not done yet! We still need to find the number of motorcycles. Luckily, we already have an equation that relates m and a: m = 40 - a. We can simply plug in our value for a to find m:
m = 40 - 15
m = 25
So, Tania has 25 motorcycles. We’ve solved the system of equations! But before we celebrate, let’s make sure our answer makes sense.
Verifying the Solution
Alright, we’ve done the math and found that Tania has 25 motorcycles and 15 ATVs. But before we declare victory, it’s always a good idea to double-check our work. This is a crucial step in problem-solving because it helps us catch any mistakes and ensure that our answer is reasonable. So, how do we verify our solution?
Well, we have two pieces of information that we can use to check our answer: the total number of vehicles and the total number of tires. First, let’s make sure the number of vehicles adds up correctly. We found that Tania has 25 motorcycles and 15 ATVs. If we add these together, we get:
25 + 15 = 40
That’s exactly the total number of vehicles Tania has! So far, so good. Now, let’s check the number of tires. Each motorcycle has 2 tires, and each ATV has 4 tires. So, the total number of tires should be:
(25 motorcycles * 2 tires/motorcycle) + (15 ATVs * 4 tires/ATV) = 50 + 60 = 110
That’s also the correct number of tires! Our solution checks out. This gives us confidence that we’ve solved the problem correctly. Verifying your solution is a great habit to develop, not just in math class, but in any situation where you’re solving a problem. It’s like proofreading a paper or testing a new recipe. It’s a way to catch errors and make sure you’re on the right track.
Conclusion
Woohoo! We did it! We successfully navigated the world of word problems and figured out that Tania has 25 motorcycles and 15 ATVs in her rental fleet. This problem might have seemed a little daunting at first, but by breaking it down into smaller steps, setting up equations, and solving them systematically, we were able to find the answer.
This is a great example of how math can be used to solve real-world problems. Whether you’re running a rental business like Tania or just trying to figure out how many pizzas to order for a party, math is a powerful tool that can help you make informed decisions. But more than just finding the answer, this exercise has helped us sharpen our problem-solving skills. We learned how to translate a word problem into mathematical equations, how to solve a system of equations, and how to verify our solution. These are skills that will serve you well in all areas of life.
So, the next time you encounter a word problem, don’t panic! Remember the steps we used today: understand the problem, set up the equations, solve the equations, and verify your solution. With a little practice and a lot of perseverance, you’ll be a word problem whiz in no time! And who knows, maybe you’ll even start your own motorcycle and ATV rental business someday!
