Finding Maximum And Minimum Values Of 3x + 2y Given X * Y = 60

Hey guys! Ever stumbled upon a math problem that looks like a puzzle? Let's dive into one today! We're going to explore how to find the maximum and minimum values of an expression, specifically 3x + 2y, when x and y are natural numbers, and their product x * y equals 60. Sounds intriguing, right? Let's break it down step by step.

Understanding the Basics

Before we jump into solving this, let's make sure we're all on the same page with some basic concepts. First off, natural numbers are the positive whole numbers we use for counting (1, 2, 3, and so on). Now, the expression 3x + 2y is a linear combination of x and y, and we want to see how big or small it can get under a specific condition – that x * y must always be 60.

Our main goal here is to maximize and minimize the value of 3x + 2y. To do this effectively, we'll need to consider all the possible pairs of natural numbers x and y that multiply to 60. Think of it like this: we have a fixed area (60), and we want to play around with the dimensions (x and y) to see what makes our target expression (3x + 2y) the largest and smallest. This involves a bit of number sense and careful calculation, but don’t worry, we’ll tackle it together!

Listing the Factor Pairs of 60

The first crucial step in solving this problem is identifying all the pairs of natural numbers (x, y) that give us a product of 60. This means we need to find all the factor pairs of 60. Factor pairs are simply pairs of numbers that, when multiplied together, result in the given number – in our case, 60. Let's list them out systematically to make sure we don't miss any:

• 1 and 60 (1 * 60 = 60)
• 2 and 30 (2 * 30 = 60)
• 3 and 20 (3 * 20 = 60)
• 4 and 15 (4 * 15 = 60)
• 5 and 12 (5 * 12 = 60)
• 6 and 10 (6 * 10 = 60)

We've got six pairs of natural numbers that satisfy the condition x * y = 60. Now, the fun part begins! We'll use these pairs to figure out which ones give us the maximum and minimum values for the expression 3x + 2y.

Calculating 3x + 2y for Each Pair

Now that we have our factor pairs, let's plug each pair of x and y values into the expression 3x + 2y and see what we get. This will give us a set of values that we can then compare to find the largest and smallest. Grab your calculators, guys – let's crunch some numbers!

• Pair (1, 60): 3(1) + 2(60) = 3 + 120 = 123
• Pair (2, 30): 3(2) + 2(30) = 6 + 60 = 66
• Pair (3, 20): 3(3) + 2(20) = 9 + 40 = 49
• Pair (4, 15): 3(4) + 2(15) = 12 + 30 = 42
• Pair (5, 12): 3(5) + 2(12) = 15 + 24 = 39
• Pair (6, 10): 3(6) + 2(10) = 18 + 20 = 38

Alright, we've done the calculations! We now have a list of values for 3x + 2y corresponding to each factor pair of 60. The next step is super straightforward: we just need to pick out the highest and lowest values from this list.

Determining the Maximum and Minimum Values

After calculating 3x + 2y for each factor pair of 60, we have a set of values that show us how the expression changes as we change the values of x and y. Now, let's pinpoint the maximum and minimum values from our results. Remember, we're looking for the largest and smallest numbers in the list we just generated.

Looking back at our calculations, we have the following values:

• 123
• 66
• 49
• 42
• 39
• 38

It's pretty clear that the largest value is 123, which corresponds to the pair (1, 60). This means that when x = 1 and y = 60, the expression 3x + 2y reaches its maximum value.

On the other end of the spectrum, the smallest value in our list is 38, which comes from the pair (6, 10). So, when x = 6 and y = 10, the expression 3x + 2y hits its minimum value.

Summarizing the Results

Let's quickly recap what we've found. We were given the condition that x * y = 60, and we wanted to find the maximum and minimum values of the expression 3x + 2y, where x and y are natural numbers. We systematically found all factor pairs of 60, plugged them into the expression, and compared the results. Here’s what we discovered:

• Maximum Value: The maximum value of 3x + 2y is 123, which occurs when x = 1 and y = 60.
• Minimum Value: The minimum value of 3x + 2y is 38, which occurs when x = 6 and y = 10.

So, we've successfully solved the problem! We found both the maximum and minimum values of the expression under the given condition. This type of problem is a great example of how we can use basic algebraic concepts and number theory to solve interesting mathematical puzzles.

Tips and Tricks for Similar Problems

Now that we've tackled this problem together, let's chat about some general tips and tricks that you can use when you encounter similar problems in the future. These strategies will help you approach these questions methodically and increase your chances of finding the correct solution. Trust me, guys, these tips can be real game-changers!

1. Systematic Listing of Factor Pairs:

When you're dealing with problems that involve finding pairs of numbers that multiply to a specific value, like our x * y = 60 scenario, the first step should always be to systematically list out all the factor pairs. This ensures you don't miss any potential solutions. Start with 1 and the number itself, then move to 2, 3, and so on, checking for divisibility. This organized approach is key to accuracy.

2. Understanding the Impact of Coefficients:

Notice how the coefficients in the expression 3x + 2y (the 3 and the 2) play a crucial role. Since the coefficient of x is larger than that of y, increasing the value of x will have a more significant impact on the overall value of the expression than increasing y. This is why the maximum value occurred when x was small and y was large, and the minimum value occurred when x was larger and y was smaller. Keeping this in mind can help you make educated guesses and focus your efforts more efficiently.

3. Recognizing the Importance of Natural Number Constraints:

The fact that x and y were defined as natural numbers was essential. If they could have been any real numbers, the problem would have been much different (and trickier!). The natural number constraint limits our options to whole numbers, making it possible to list out all the pairs and test them. Always pay close attention to these types of constraints in problem statements.

4. Using Trial and Error Strategically:

Sometimes, the best way to get a feel for a problem is to simply start plugging in numbers and seeing what happens. This can help you develop intuition about how the expression behaves as the variables change. However, it's important to do this strategically, not randomly. Use your understanding of the coefficients and the factor pairs to guide your trials. For example, in our problem, you might start by trying extreme values for x and y to get a sense of the range of possible values for 3x + 2y.

5. Double-Check Your Calculations:

This might seem obvious, but it's worth emphasizing: always double-check your calculations! It's easy to make a small arithmetic error, especially when you're working under pressure. A simple mistake can throw off your entire solution. So, take a moment to review your work and make sure everything adds up correctly.

By keeping these tips in mind, you'll be well-equipped to tackle similar optimization problems with confidence. Remember, practice makes perfect, so keep working at it, and you'll become a pro in no time!

Wrapping Up

So, there you have it! We've successfully navigated through a problem that involved finding the maximum and minimum values of an expression under specific conditions. We started by understanding the problem, systematically listed factor pairs, calculated the values of the expression for each pair, and finally, determined the maximum and minimum values. We also picked up some handy tips and tricks along the way.

Problems like these are not just about finding the right answer; they're about developing your problem-solving skills and your understanding of mathematical concepts. By breaking down complex problems into smaller, manageable steps, we can tackle even the trickiest questions with confidence. Keep practicing, keep exploring, and most importantly, keep enjoying the world of math! You've got this, guys!