Solving For A: A Tricky Math Problem!

Hey guys! Ever stumbled upon a math problem that looks simple but has a sneaky twist? Well, today we're diving into one of those! We've got the equation A+B+C+D+E = 240, and here's the kicker: there's a difference of 3 between each of these numbers. Our mission? To find out what A is. Sounds like a fun challenge, right? Let's break it down step by step and crack this mathematical puzzle together!

Understanding the Problem

Okay, let's really understand what we're dealing with here. The core of the problem is this equation: A + B + C + D + E = 240. This tells us that we have five variables (A, B, C, D, and E) which, when added together, give us a sum of 240. That’s our foundation. But here’s the extra layer of complexity, the thing that makes this more than just a simple addition problem: there's a difference of 3 between each consecutive number. This is super important because it tells us that the numbers aren't just random; they follow a specific pattern. This pattern is the key to unlocking the solution.

To put it simply, imagine if A was, say, 10. Then B would be 13, C would be 16, D would be 19, and E would be 22. See how that works? Each number is 3 more than the one before it. Now, we don't know what A actually is yet, but understanding this relationship is crucial. It allows us to express all the other variables (B, C, D, and E) in terms of A. This is a common strategy in algebra – simplifying things by relating them to each other. So, before we start crunching numbers, let’s make sure we've got this concept down pat. This difference of 3 is the secret ingredient that will help us turn this seemingly complex problem into something much more manageable. We are going to manipulate the variables to rewrite the equation in terms of A. Once we have the value of A, we will find the values of all the other variables. Then, we can also double-check our answer to make sure all the conditions are met.

Expressing Variables in Terms of A

Alright, guys, now for a bit of algebraic magic! We're going to take that understanding of the difference of 3 and use it to rewrite our variables. This is a crucial step because it's going to allow us to simplify our equation and solve for A. Remember, the goal here is to express B, C, D, and E in terms of A. This means we want to write equations that show how each of these variables relates directly to A. Let's start with B. We know that B is 3 more than A, right? So, we can write that as: B = A + 3. Easy peasy! Now, let's move on to C. C is 3 more than B, which means it's 6 more than A (3 more than B, which is already 3 more than A). So, we get: C = A + 6. Notice the pattern here? We're just adding multiples of 3 to A. Following this pattern, D is 3 more than C (or 9 more than A), giving us: D = A + 9. And finally, E is 3 more than D (or 12 more than A), so: E = A + 12. Boom! We've done it! We've successfully expressed all our variables in terms of A. This is a major victory because now we can substitute these expressions back into our original equation. Instead of having five different variables, we'll have just one (A), which makes the equation much easier to solve. Think of it like translating a sentence from a complicated language into something you understand perfectly. We've taken our original equation and re-written it in a way that's much more friendly to our mathematical brains. This is the power of algebra, folks! Now, let's use these expressions to simplify our main equation.

Substituting into the Main Equation

Okay, team, we've done the prep work, and now it's time for the main event! We're going to take those expressions we just figured out (B = A + 3, C = A + 6, D = A + 9, and E = A + 12) and plug them into our original equation: A + B + C + D + E = 240. This is where things start to get really exciting because we're about to turn this seemingly complex problem into a straightforward one. So, let's do it! We'll replace B, C, D, and E with their equivalent expressions in terms of A. This gives us: A + (A + 3) + (A + 6) + (A + 9) + (A + 12) = 240. See what we did there? We've essentially rewritten the entire equation using only A as our variable. This is a huge step forward. Now, the equation might look a little long and intimidating, but don't worry, we're going to simplify it in the next step. The key thing to remember here is that we're just substituting equivalent values. We haven't changed the underlying math; we've just re-expressed it in a more useful way. Think of it like swapping out different Lego bricks to build the same structure. We're using different pieces (our expressions in terms of A), but we're still building the same thing (the equation). This substitution technique is a fundamental tool in algebra, and it's something you'll use again and again. So, let’s move on and simplify this equation. We're on the home stretch now!

Simplifying and Solving for A

Alright, let's get down to business and simplify this equation! We've got: A + (A + 3) + (A + 6) + (A + 9) + (A + 12) = 240. The first thing we want to do is get rid of those parentheses. Luckily, they're just hanging out there with plus signs in front, so we can simply remove them without changing anything: A + A + 3 + A + 6 + A + 9 + A + 12 = 240. Now, things are looking much cleaner! Our next step is to combine like terms. This means we're going to group all the 'A's together and all the numbers together. How many 'A's do we have? We've got five of them! So, we can write that as 5A. Now, let's add up the numbers: 3 + 6 + 9 + 12. That gives us 30. So, our equation now looks like this: 5A + 30 = 240. We're getting super close to solving for A! We've transformed our complicated equation into a much simpler one. Now, we need to isolate the term with A (the 5A). To do that, we'll subtract 30 from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep things balanced. So, we get: 5A = 240 - 30, which simplifies to 5A = 210. We're almost there! Now, to get A by itself, we need to divide both sides of the equation by 5: A = 210 / 5. And finally, drumroll please... A = 42! We did it! We've successfully solved for A. But hold on, we're not quite done yet. It's always a good idea to double-check our work to make sure we haven't made any mistakes along the way. So, let's verify our answer and make sure it makes sense in the context of the original problem.

Verifying the Solution

Fantastic job, everyone! We've found that A = 42, but before we declare victory, let's verify our solution to be absolutely sure. This is a crucial step in any math problem because it helps us catch any sneaky errors we might have made along the way. Remember, our original equation is A + B + C + D + E = 240, and we know that there's a difference of 3 between each number. We've already expressed B, C, D, and E in terms of A, so let's use those expressions and our value for A to find the values of the other variables. We have: B = A + 3 = 42 + 3 = 45 C = A + 6 = 42 + 6 = 48 D = A + 9 = 42 + 9 = 51 E = A + 12 = 42 + 12 = 54 Now we have values for all our variables: A = 42, B = 45, C = 48, D = 51, and E = 54. Let's check if these values satisfy our original equation. We'll add them all together: 42 + 45 + 48 + 51 + 54. What do we get? If you add those up, you'll find that they indeed equal 240! Awesome! But we're not done yet. We also need to make sure that there's a difference of 3 between each consecutive number. Let's check: 45 - 42 = 3 48 - 45 = 3 51 - 48 = 3 54 - 51 = 3 Yep, everything checks out! The difference between each number is 3, just like the problem stated. So, we can confidently say that our solution is correct. We've not only found the value of A, but we've also verified that it fits all the conditions of the problem. This is how you become a math superstar, guys! You don't just find the answer; you make sure it's the right answer.

Conclusion

So there you have it, mathletes! We've successfully navigated this tricky problem and found that A = 42. We took a seemingly complex equation, broke it down into smaller, manageable steps, and used the power of algebra to solve it. We expressed variables in terms of each other, substituted them into the main equation, simplified, and then verified our solution. It's a fantastic journey of problem-solving, and you guys nailed it! Remember, the key to tackling these kinds of challenges is to stay organized, understand the relationships between the variables, and always double-check your work. Math isn't just about finding the right answer; it's about the process of getting there. And in this case, we not only got the answer, but we also learned some valuable problem-solving skills along the way. Keep practicing, keep challenging yourselves, and you'll be amazed at what you can achieve. Until next time, keep those brains buzzing!