Max Difference Of X+y When X*y=28? Math Solution!

Hey guys! Let's dive into a fun math problem where we explore the relationship between two integers and their sum. Our main goal here is to figure out the biggest possible difference between the maximum and minimum values of x + y, given that x and y are integers and their product, x * y, equals 28. Sounds interesting, right? Let's break it down step by step and make sure we get a solid grasp on how to tackle this kind of problem. We will explore different integer pairs that multiply to 28, and then we'll calculate their sums. By comparing these sums, we can easily pinpoint the maximum and minimum values and, ultimately, find the difference between them. So, grab your thinking caps, and let's get started on this mathematical journey!

Understanding the Problem

So, the core of our problem is this: we have two integers, let's call them x and y. The rule they follow is that when you multiply them together (x * y), you always get 28. Now, here's the tricky part: we need to find out what's the biggest possible value you can get if you add x and y together, and what's the smallest possible value. Once we know those, we just subtract the smallest from the biggest to find the difference. But why is this important, you might ask? Well, these types of problems are super helpful in understanding how numbers work and how they relate to each other. It's not just about finding an answer; it's about flexing those mathematical muscles and getting better at problem-solving. Plus, it's a great way to see how different factors of a number can lead to different sums. Okay, so with the challenge set, let's roll up our sleeves and start figuring out how to crack this!

Finding Integer Pairs

Okay, guys, let's get into the nitty-gritty of this problem! The first thing we need to do is figure out all the pairs of integers that multiply together to give us 28. Think of it like this: we're playing a detective game with numbers, trying to find all the combinations that fit our clue, which is the number 28. We can start with the obvious ones, like 1 and 28, because 1 multiplied by 28 is definitely 28. Then, we can think about 2. Can 2 be multiplied by another integer to get 28? Yep, 2 times 14 works perfectly! What about 3? Nope, 3 doesn't fit. But 4 does! 4 times 7 gives us 28. Now, here’s a little secret: once you reach a point where the numbers start repeating in reverse (like we've gone from 1 to 4, and now we're at 7), you’ve likely found all the positive pairs. But hold on, we're not just dealing with positive numbers here! Remember, we're talking about integers, which means negative numbers are in the mix too. So, we also have to consider the negative pairs, because a negative times a negative also gives a positive. This means we have pairs like -1 and -28, -2 and -14, and -4 and -7. Finding all these pairs is super important because each pair will give us a different sum, and we need to find the biggest and smallest of those sums. So, make sure you’ve got all these pairs jotted down – they're the key to cracking the rest of the problem!

Calculating the Sums

Alright, now that we've rounded up all our pairs of integers that multiply to 28, it's time for the next step: calculating the sum of each pair. This is where we add the two numbers in each pair together to see what we get. It’s like we’re putting these pairs through a little “summing machine” to get a new set of numbers. Let’s start with the positive pairs we found earlier. We had 1 and 28, so if we add those, we get 29. Easy peasy! Next up is 2 and 14, which adds up to 16. Then we have 4 and 7, giving us a sum of 11. Now, let's tackle the negative pairs. Remember, adding negative numbers can be a bit tricky, so let's take our time. -1 plus -28 gives us -29. -2 plus -14 equals -16. And last but not least, -4 plus -7 adds up to -11. So, we've got a whole bunch of sums now: 29, 16, 11, -29, -16, and -11. These sums are super important because they represent all the possible values of x + y when x * y is 28. Our next mission is to sift through these sums and find the biggest and the smallest. That's how we'll solve the last piece of the puzzle!

Identifying Maximum and Minimum Values

Okay, so we've got our list of sums, and now it's time to play a little game of "spot the extremes"! We need to figure out which sum is the biggest and which one is the smallest. Think of it like lining up all the sums on a number line – we're looking for the one furthest to the right (the largest) and the one furthest to the left (the smallest). Looking at our sums, which are 29, 16, 11, -29, -16, and -11, it's pretty clear that 29 is our big winner. It's the largest positive number on the list, so it definitely takes the crown for the maximum value. Now, let's find the smallest. Remember, with negative numbers, the bigger the number looks, the smaller it actually is. So, -29 is much smaller than -11, even though 29 looks bigger than 11. That means -29 is our smallest value. Identifying these extremes is crucial because they're the two numbers we need for the final step. We're in the home stretch now! We know the biggest possible sum is 29, and the smallest possible sum is -29. All that's left is to find the difference between them. Are you ready to wrap this up? Let’s do it!

Calculating the Difference

Alright guys, we've made it to the final showdown! We know the largest possible sum of x + y is 29, and the smallest possible sum is -29. Now, the question asks us to find out how much bigger the largest value is than the smallest value. In other words, we need to find the difference between these two numbers. So, how do we do that? Well, when we're finding the difference between two numbers, especially when one of them is negative, it's like we're measuring the distance between them on a number line. To calculate this, we subtract the smallest number from the largest number. So, in our case, that means we need to subtract -29 from 29. Now, here's a little math tip: subtracting a negative number is the same as adding its positive counterpart. So, 29 minus -29 is the same as 29 plus 29. And what does that give us? 58! So, the difference between the largest and smallest possible values of x + y is 58. This final calculation brings it all home, showing us the total range of possible sums when x * y equals 28. We've taken a deep dive into this problem, exploring integer pairs, calculating sums, and identifying extremes. Great job, guys! You've tackled this math challenge like true pros!

Therefore, the difference between the largest and smallest possible values of x + y is 58.