Cars Vs. Motorcycles: Solving A Vehicle Wheel Puzzle

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Cars vs. Motorcycles: Solving a Vehicle Wheel Puzzle

Hey guys! Ever find yourself scratching your head over a seemingly simple math problem that turns out to be a real brain-bender? Well, let's dive into one of those right now. We've got a classic riddle on our hands: imagine an apartment complex with a mix of cars and motorcycles parked inside. We know there are 55 vehicles in total, and if you were to count all the wheels, you'd get 170. The big question is, how many cars and how many motorcycles are we talking about? This isn't just about numbers; it's a fun exercise in problem-solving and applying basic math concepts to everyday scenarios.

Breaking Down the Problem: Setting Up the Equations

Okay, so how do we even begin to untangle this wheelie complex situation? The key here is to translate the word problem into mathematical equations. This might sound intimidating, but trust me, it's like cracking a code, and it's super satisfying when you get it!

Let's start by assigning variables. Variables are just symbols (usually letters) that represent the unknown quantities we're trying to find. In our case, we have two unknowns: the number of cars and the number of motorcycles.

  • Let's use 'c' to represent the number of cars.
  • And let's use 'm' to represent the number of motorcycles.

Now, let’s think about the information we already have. We know two crucial things:

  1. The total number of vehicles is 55. This means the number of cars (c) plus the number of motorcycles (m) equals 55. We can write this as an equation:

    c + m = 55

  2. The total number of wheels is 170. This is where it gets a little trickier, but don't worry, we've got this! Each car has 4 wheels, and each motorcycle has 2 wheels. So, the total number of wheels can be represented as 4 times the number of cars (4c) plus 2 times the number of motorcycles (2m). This gives us our second equation:

    4c + 2m = 170

So now we've transformed our word problem into a system of two equations:

  • c + m = 55
  • 4c + 2m = 170

This is a classic setup for solving a system of equations, and there are a couple of ways we can tackle it. We'll explore those next, but for now, give yourself a pat on the back! You've already taken the most important step: turning a real-world scenario into a mathematical model. That's some serious problem-solving power right there!

Methods to Solve: Substitution and Elimination

Alright, now that we've got our equations locked and loaded, it's time to choose our weapon of math and actually solve for 'c' and 'm'. There are two main techniques we can use to solve a system of equations like this: substitution and elimination. Both methods will lead us to the same answer, so it's really a matter of personal preference. Let's break down each one.

Method 1: Substitution – The Art of Replacing

The substitution method is all about isolating one variable in one equation and then substituting that expression into the other equation. Sounds fancy, right? But it's actually pretty straightforward.

  1. Choose an equation and isolate a variable. Let's take our first equation, c + m = 55, and isolate 'c'. To do that, we simply subtract 'm' from both sides:

    c = 55 - m

    Now we have 'c' expressed in terms of 'm'.

  2. Substitute. This is the key step! We're going to take the expression we just found for 'c' (which is 55 - m) and substitute it into our second equation, 4c + 2m = 170. This means we replace 'c' in the second equation with 55 - m:

    4(55 - m) + 2m = 170

    Notice what we've done here? We've turned the second equation into an equation with only one variable, 'm'. This is a huge step because we can now solve for 'm' directly!

  3. Solve for the remaining variable. Let's simplify and solve for 'm':

    • 220 - 4m + 2m = 170
    • -2m = -50
    • m = 25

    Boom! We've found that there are 25 motorcycles.

  4. Substitute back to find the other variable. Now that we know 'm' is 25, we can plug it back into either of our original equations to find 'c'. Let's use the simpler one, c + m = 55:

    • c + 25 = 55
    • c = 30

    And there you have it! We've found that there are 30 cars.

Method 2: Elimination – The Art of Cancellation

The elimination method, also known as the addition method, is all about strategically manipulating the equations so that one of the variables cancels out when you add the equations together. It's like a mathematical magic trick!

  1. Manipulate the equations. Our goal is to make the coefficients (the numbers in front of the variables) of either 'c' or 'm' opposites in the two equations. Let's target 'm'. In our equations:

    • c + m = 55
    • 4c + 2m = 170

    We can multiply the first equation by -2. This will give us a '-2m' term, which is the opposite of the '+2m' in the second equation:

    -2(c + m) = -2(55)

    This simplifies to:

    -2c - 2m = -110

    Now our system of equations looks like this:

    • -2c - 2m = -110
    • 4c + 2m = 170

  2. Add the equations. Now we simply add the two equations together, term by term:

    • (-2c + 4c) + (-2m + 2m) = (-110 + 170)

    Notice what happens? The '-2m' and '+2m' terms cancel each other out! This is the magic of elimination. We're left with:

    • 2c = 60

  3. Solve for the remaining variable. Now we can easily solve for 'c':

    • c = 30

    Just like in the substitution method, we've found that there are 30 cars.

  4. Substitute back to find the other variable. We plug 'c = 30' back into either of our original equations. Let's use c + m = 55 again:

    • 30 + m = 55
    • m = 25

    And just like that, we confirm that there are 25 motorcycles.

The Solution: 30 Cars and 25 Motorcycles

Drumroll, please! After all that mathematical maneuvering, we've finally arrived at the answer. Whether you prefer the elegance of substitution or the cancellation power of elimination, we've proven that there are 30 cars and 25 motorcycles parked in that apartment complex.

But it's not just about the numbers, guys. This problem highlights the power of translating real-world scenarios into mathematical models. By using variables and equations, we can represent complex relationships and solve for unknowns. This is a fundamental skill in many areas of life, from science and engineering to finance and even everyday decision-making.

Why This Matters: Real-World Applications

So, okay, maybe you're thinking,