Maximize Card Sum: A Tricky Math Problem!

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Okay, guys, let's dive into this math problem that involves picking cards of different colors and trying to maximize their sum. It's like a mini-game of strategy and arithmetic all rolled into one! We've got a selection of cards, some blue and some purple, and our mission is to figure out the highest possible total we can get by picking one blue card and two purple cards. Sounds fun, right? Let's break it down step by step so we can nail this question.

Understanding the Problem

So, the main goal is to maximize the sum. This means we need to think carefully about which numbers on the cards will give us the biggest possible result. When you see a question like this, always remember that positive numbers are your friends! We want to pick the largest positive numbers available, but there's a catch: we might also have to deal with negative numbers. Negative numbers can bring our sum down, so we need to be strategic about how we include them (or avoid them altogether!).

Also, pay attention to the specifics. We must pick one blue card and two purple cards. This constraint is super important because it tells us exactly how many of each color we need to select. Don't accidentally pick too many blue cards or not enough purple ones!

Now, let's consider the answer options: A) -27, B) -17, C) -11, and D) -9. Notice that all the options are negative. This tells us that the cards likely contain some negative numbers that we can't completely avoid. Our job is to minimize the impact of these negative numbers while maximizing the impact of the positive ones. Keep these options in mind as we solve the problem; they'll help us check if our answer makes sense.

Strategy for Solving

Here’s the plan we're going to follow to crack this problem:

  1. Identify the Cards: First, we need to know the numbers on all the blue and purple cards. Since the actual numbers aren't provided in the prompt, let’s assume we have a set of numbers for each color. For example, let's say the blue cards have the numbers [-10, -5, 2] and the purple cards have the numbers [-15, -8, 3, 7]. (We'll use these example numbers to illustrate the solution, but remember, the exact numbers will be given in the actual problem).
  2. Maximize the Positives: Look for the largest positive numbers among the blue and purple cards. We definitely want to include these in our sum.
  3. Minimize the Negatives: If we have to include negative numbers, we want to pick the ones that will reduce our sum the least. In other words, pick the negative numbers that are closest to zero.
  4. Calculate the Sum: Add the numbers on the selected cards together to find the total sum.
  5. Check the Options: Compare our calculated sum with the answer options provided. If our sum matches one of the options, that's likely the correct answer. If not, we need to double-check our work and make sure we haven't made any mistakes.

Step-by-Step Solution with Example Numbers

Alright, let's apply our strategy to the example numbers we came up with:

Blue Cards: [-10, -5, 2] Purple Cards: [-15, -8, 3, 7]

We need to pick one blue card and two purple cards.

  1. Best Blue Card: To maximize the sum, we want to pick the largest positive number from the blue cards. In this case, that's 2.
  2. Best Purple Cards: Next, we want to pick the two largest numbers from the purple cards. These are 7 and 3.
  3. Calculate the Sum: Now, let's add these numbers together: 2 (from the blue card) + 7 (from a purple card) + 3 (from the other purple card) = 12.

So, based on these example numbers, the maximum sum we can get is 12. But remember, this is just an example! The actual numbers on the cards in the problem will be different.

Dealing with Negative Numbers

What if the cards have only negative numbers or a mix of positive and negative numbers? Here’s how to handle those situations:

  • All Negative Numbers: If all the cards have negative numbers, we want to pick the ones that are closest to zero (i.e., the least negative numbers). For example, if the blue cards are [-10, -5, -2] and the purple cards are [-15, -8, -3, -1], we would pick -2 from the blue cards and -1 and -3 from the purple cards. The sum would be -2 + (-1) + (-3) = -6.
  • Mix of Positive and Negative Numbers: If we have both positive and negative numbers, we still want to pick the largest positive numbers possible. If we have to include a negative number, we pick the one that is closest to zero. For example, if the blue cards are [-10, -5, 2] and the purple cards are [-15, -8, 3, 7], and we had to pick a negative blue card, we would pick -5 instead of -10 because -5 will reduce the sum less.

Back to the Original Problem

Now that we understand the strategy, let's revisit the original problem and the answer options:

  • Options: A) -27, B) -17, C) -11, D) -9

Without knowing the exact numbers on the cards, it's hard to give a definitive answer. However, we can think about what the options tell us. The fact that all the options are negative suggests that the cards probably have a lot of negative numbers.

To get the maximum sum (which is still negative), we need to pick the least negative numbers. Looking at the options, -9 is the largest (least negative) number. So, the correct answer is likely D) -9, assuming that the cards have numbers that, when added strategically, result in a sum close to -9.

Final Tips and Tricks

Before we wrap up, here are a few final tips and tricks to help you ace this type of problem:

  • Read Carefully: Always read the problem carefully and make sure you understand exactly what you need to find. Pay attention to any constraints or conditions (like the number of cards you need to pick from each color).
  • Write it Down: Write down all the numbers and conditions given in the problem. This will help you keep track of everything and avoid making mistakes.
  • Think Strategically: Plan your approach before you start calculating. Think about how to maximize the positives and minimize the negatives.
  • Double-Check: After you find an answer, double-check your work to make sure you haven't made any calculation errors.
  • Practice: The more you practice these types of problems, the better you'll become at solving them. Look for similar problems online or in textbooks and work through them.

Conclusion

So, there you have it! That’s how you can tackle a math problem that involves picking cards and maximizing their sum. Remember, the key is to think strategically, pay attention to the details, and double-check your work. With a little practice, you'll be solving these problems like a pro in no time! Keep practicing, and you'll get the hang of it. Good luck, and have fun with your math adventures! This kind of analytical thinking is super useful in real life too, so you're not just learning math – you're building valuable problem-solving skills.