Let's dive into this math problem together, guys! We're given that the mean of two numbers, x and y, is 4. Our mission, should we choose to accept it (and we do!), is to find the mean of four numbers: x, 2x, y, and 2y. Buckle up, because we're about to embark on a mathematical adventure!
Understanding the Basics: What is the Mean?
Before we jump into solving the problem, let's make sure we're all on the same page about what the mean actually is. Simply put, the mean (also known as the average) of a set of numbers is the sum of those numbers divided by the total count of numbers. Think of it as evenly distributing a total amount among a group. For example, if you have three test scores – 80, 90, and 100 – the mean score is (80 + 90 + 100) / 3 = 90. This means that, on average, you scored a 90 on your tests.
In mathematical terms, the mean of n numbers (a1, a2, ..., an) is calculated as:
Mean = (a1 + a2 + ... + an) / n
This formula is the key to unlocking our problem. It's the foundation upon which we'll build our solution. Keep this formula in mind as we move forward, because we'll be using it extensively.
Applying the Mean to Our Problem
Now that we've refreshed our understanding of the mean, let's apply it to the problem at hand. We're told that the mean of x and y is 4. This means that if we add x and y together and divide by 2 (since there are two numbers), we should get 4. We can express this mathematically as:
(x + y) / 2 = 4
This equation is our first crucial piece of information. It establishes a relationship between x and y, which we'll need to leverage to solve the main problem. Let's take a moment to really understand what this equation tells us. It says that the sum of x and y is twice their mean. This is a handy insight that will simplify our calculations later on.
To make things even clearer, let's multiply both sides of the equation by 2. This will eliminate the fraction and give us a cleaner equation to work with:
x + y = 8
Ah, much better! This equation, x + y = 8, is our new best friend. It tells us that the sum of x and y is 8. This is a vital piece of the puzzle, and we'll be using it shortly to find the mean of the four numbers. Remember, the goal is to find the mean of x, 2x, y, and 2y, and we've already made significant progress by understanding the relationship between x and y. Let's keep going!
Finding the Mean of x, 2x, y, and 2y
Alright, guys, we've laid the groundwork, and now it's time to tackle the main question: what is the mean of x, 2x, y, and 2y? Remember the mean formula? We need to add these four numbers together and divide by 4 (since there are four numbers). So, the mean we're looking for can be expressed as:
Mean = (x + 2x + y + 2y) / 4
Notice anything interesting about the expression in the numerator? We have both x terms and y terms. This is a perfect opportunity to simplify things by combining like terms. Let's rewrite the expression:
Mean = (3x + 3y) / 4
Now we're getting somewhere! We've simplified the numerator by combining the x and y terms. But we're not quite done yet. Is there anything else we can do to make this expression even simpler? Hmmm...
The Power of Factoring
Take a closer look at the numerator: 3x + 3y. Do you see a common factor? That's right, both terms have a factor of 3! This means we can factor out the 3, which will make our expression even more manageable. Factoring is a powerful technique in algebra that allows us to rewrite expressions in a more convenient form. Let's factor out the 3:
Mean = 3(x + y) / 4
Now we're talking! This looks much better. We've successfully factored out the 3, and the expression is looking quite elegant. But why did we do this? What's the big advantage of factoring out the 3? The answer lies in the equation we derived earlier: x + y = 8. Remember that equation? It's about to come to our rescue!
Using x + y = 8 to Solve the Problem
Do you see it? The expression x + y appears in our equation for the mean! This is fantastic news, because we know the value of x + y. We determined earlier that x + y = 8. This means we can substitute 8 for x + y in our mean equation. Let's do it:
Mean = 3(8) / 4
Boom! We've made a major breakthrough. We've replaced x + y with its value, 8, and now we have a simple arithmetic expression to evaluate. The problem is practically solved at this point. All that's left is to do the calculation.
The Final Calculation
Let's finish this strong, guys! We have:
Mean = 3(8) / 4
First, let's multiply 3 by 8:
Mean = 24 / 4
And now, let's divide 24 by 4:
Mean = 6
There you have it! The mean of x, 2x, y, and 2y is 6. We've successfully navigated the problem, step by step, and arrived at the solution. Give yourselves a pat on the back!
Summarizing Our Journey
Let's take a moment to recap the steps we took to solve this problem. This is a great way to reinforce our understanding and make sure we haven't missed anything. Here's a summary of our mathematical adventure:
- Understanding the Mean: We started by defining the mean as the sum of the numbers divided by the count of numbers.
- Using the Given Information: We used the information that the mean of x and y is 4 to derive the equation x + y = 8.
- Setting Up the Mean Equation: We expressed the mean of x, 2x, y, and 2y as (x + 2x + y + 2y) / 4.
- Simplifying the Expression: We combined like terms and factored out a 3 to get Mean = 3(x + y) / 4.
- Substituting x + y = 8: We substituted 8 for x + y in the mean equation.
- Calculating the Result: We evaluated the expression to find that the mean is 6.
By following these steps, we were able to systematically solve the problem and arrive at the correct answer. This is the power of a structured approach to problem-solving. Always break down complex problems into smaller, manageable steps, and you'll be amazed at what you can achieve.
Key Takeaways and Problem-Solving Strategies
This problem wasn't just about finding a numerical answer; it was also about honing our problem-solving skills. Let's highlight some key takeaways and strategies that we can apply to other math problems (and even real-life situations!):
- Understand the Definitions: Make sure you have a solid grasp of the definitions of the concepts involved. In this case, understanding the definition of the mean was crucial.
- Use Given Information Wisely: The problem gave us the information that the mean of x and y is 4. We used this information to derive a crucial equation, x + y = 8, which was key to solving the problem.
- Simplify Expressions: Simplifying expressions by combining like terms and factoring can make problems much easier to manage.
- Look for Connections: Notice how the expression x + y appeared both in the equation we derived and in the mean equation. Recognizing these connections is a powerful problem-solving skill.
- Break It Down: Complex problems can seem daunting, but if you break them down into smaller steps, they become much more manageable.
- Review Your Work: Always take a moment to review your work and make sure your solution makes sense in the context of the problem.
These strategies are valuable tools in your mathematical arsenal. Practice applying them, and you'll become a more confident and effective problem solver.
Practice Makes Perfect
So, there you have it! We've successfully found the mean of x, 2x, y, and 2y. But don't stop here! The best way to solidify your understanding is to practice more problems. Try tackling similar problems with different numbers or slightly different conditions. The more you practice, the more comfortable and confident you'll become.
Maybe you could try finding the mean of x, 3x, y, and 3y, or what if the mean of x and y was 5 instead of 4? Experiment and challenge yourself! The world of mathematics is full of exciting problems to solve, and you're now well-equipped to tackle them. Keep up the great work, guys, and happy problem-solving!