Danu's Math Competition Score: How Many Correct Answers?

Hey guys! Let's break down this math problem together. We've got Danu, who's a whiz at math competitions, and we need to figure out how many questions he nailed to get a score of 148. The competition has a pretty straightforward scoring system: 4 points for a correct answer, -2 points for a wrong one, and 0 points for skipping a question. Sounds like a fun challenge, right? So, let's dive into the details and see if we can crack this code and figure out how many questions Danu answered correctly!

Understanding the Scoring System

First off, let's really get our heads around this scoring system. A correct answer is like hitting the jackpot, giving Danu a sweet 4 points boost. But, watch out! A wrong answer isn't just a zero; it's a penalty of -2 points, kinda like a little setback. And then there's the unanswered question, which is a neutral 0 points – no gain, no loss. Understanding this system is key because it helps us set up the equation we'll need to solve the problem. We need to consider how the number of correct answers, wrong answers, and unanswered questions all play a role in Danu's final score of 148. It's like balancing an equation, where each type of answer has its own weight. So, with this scoring system in mind, let's move on to setting up our problem-solving strategy!

Setting Up the Equation

Okay, time to put on our math hats and get a little algebraic! Let's use some letters to represent the unknowns. We'll let 'C' stand for the number of correct answers – that's what we're trying to find out. Then, 'W' will represent the number of wrong answers, and 'U' will stand for the number of unanswered questions. Now, how do we translate the scoring system into an equation? Each correct answer gives 4 points, so the total points from correct answers will be 4 * C. Similarly, each wrong answer takes away 2 points, so the total deduction from wrong answers will be -2 * W. Unanswered questions don't affect the score, so they contribute 0 points. Combining all these, we get the equation: 4C - 2W + 0U = 148. Or, simplifying it a bit, we have: 4C - 2W = 148. This equation is the heart of our problem, but here's the catch: we have one equation and two unknowns (C and W). That means we can't directly solve for C just yet. We need to think about what other information we might need or what assumptions we can make to help us out. Let's keep going and see what we can figure out!

Making Assumptions and Simplifying

Alright, so we've got our equation 4C - 2W = 148, but it's like trying to solve a puzzle with a piece missing. We have two unknowns (C and W) and only one equation. To make things easier, let's make a smart assumption. In most math competitions, participants try to answer as many questions as possible, even if they're not 100% sure. So, let's assume that Danu attempted every question, meaning there are no unanswered questions (U = 0). This simplifies our problem a lot! Now we just need to focus on the relationship between correct and wrong answers.

To make things even simpler, let's divide our equation by 2. This gives us 2C - W = 74. This new equation is much easier to work with. We still have two unknowns, but the numbers are smaller, and the relationship is clearer. We know that 2 times the number of correct answers minus the number of wrong answers equals 74. Now, we need to think about how the total number of questions in the competition fits into this. If we knew the total number of questions, we could express W in terms of C and solve for C. Let's dig deeper and see if we can find any more clues or make another logical assumption to help us crack this problem!

Considering the Total Number of Questions

Okay, guys, let's think about the bigger picture here. We know Danu took a math competition, but we don't know how many questions were on the test! This is a crucial piece of information we're missing. Let's say the competition had a total of 'T' questions. If Danu attempted every question (which we're assuming), then the number of correct answers (C) plus the number of wrong answers (W) must equal the total number of questions (T). So, we have another equation: C + W = T. Now we're getting somewhere! We've got two equations: 2C - W = 74 and C + W = T. It looks like we're setting up a system of equations, which is a fancy way of saying we have two equations that we can solve together.

However, we still have a little snag: we don't know the value of T. Without knowing the total number of questions, we can't find an exact solution for C. But, let's not give up just yet! We can use some logical reasoning to narrow down the possibilities. We know that C and W must be whole numbers (you can't answer half a question!). Also, we know that C must be greater than half of 74 (because 2C - W = 74, W can't be a negative number). So, C must be greater than 37. This gives us a starting point for thinking about possible values of C and T. Let's explore this a bit more and see if we can find a solution that makes sense in the context of a math competition.

Finding a Plausible Solution

Alright, let's put on our detective hats and try to narrow down the possibilities! We know that C (the number of correct answers) has to be a whole number greater than 37. We also have two equations: 2C - W = 74 and C + W = T. Let's play around with the first equation a bit. We can rearrange it to get W = 2C - 74. This tells us that W must be an even number because 2C and 74 are both even. Now, let's think about the total number of questions, T. Since C + W = T, and we know something about C and W, we can start to make some educated guesses. Let's try a value for C and see if it leads to a reasonable solution.

Let's start with C = 40. If C = 40, then W = 2 * 40 - 74 = 80 - 74 = 6. So, if Danu answered 40 questions correctly and 6 questions incorrectly, then the total number of questions would be T = 40 + 6 = 46. This sounds like a reasonable number of questions for a math competition. Let's check if these values fit our original score equation: 4 * 40 - 2 * 6 = 160 - 12 = 148. Bingo! It works! So, we found a solution that fits all the conditions. Danu answered 40 questions correctly, 6 questions incorrectly, and there were a total of 46 questions in the competition. Hooray!

Conclusion

So, after diving into the math competition problem, making some smart assumptions, and working through the equations, we've figured out that Danu answered 40 questions correctly. This problem was a fun mix of algebra and logical reasoning, and it shows how important it is to break down a problem into smaller parts and use all the information you have. Remember, guys, math isn't just about formulas; it's about problem-solving and thinking creatively! I hope you enjoyed this breakdown, and keep practicing those math skills! You'll be math whizzes in no time! And remember, if you ever get stuck on a problem, just take a deep breath, revisit the basics, and think it through step by step. You got this!