A.M. And G.M. Problem: Find X² + Y²

Hey there, math enthusiasts! Today, we're going to tackle a fascinating problem that involves the Arithmetic Mean (A.M.) and the Geometric Mean (G.M.). These two concepts are fundamental in mathematics and pop up in various applications, from statistics to finance. We'll break down the problem step-by-step, making sure everyone understands the underlying principles. So, grab your thinking caps, and let's get started!

In this mathematical journey, we're presented with two positive numbers, let's call them x and y. The problem states that the Arithmetic Mean (A.M.) of these numbers is 6.5, and their Geometric Mean (G.M.) is 6. Our mission, should we choose to accept it, is to find the value of x² + y². Sounds intriguing, right? Well, fret not, because we're about to unravel this mathematical puzzle together. Remember, the key to solving any math problem is understanding the definitions and applying the correct formulas. So, let's refresh our memory on what A.M. and G.M. actually mean. The Arithmetic Mean, as the name suggests, is simply the average of a set of numbers. For two numbers, x and y, the A.M. is calculated by adding the numbers and dividing the sum by 2. Mathematically, we can express it as (x + y) / 2. On the other hand, the Geometric Mean is a different kind of average that is particularly useful when dealing with rates of change or multiplicative relationships. For two numbers, x and y, the G.M. is calculated by taking the square root of the product of the numbers. Mathematically, it's represented as √(xy). Now that we have a solid grasp of these definitions, let's see how we can apply them to solve our problem. We know that the A.M. of x and y is 6.5, so we can write the equation (x + y) / 2 = 6.5. Similarly, the G.M. is 6, which gives us the equation √(xy) = 6. Our goal is to find x² + y², but we currently have two equations involving x and y. The trick here is to manipulate these equations in a way that allows us to express x² + y² in terms of the known values. One common strategy is to square the equations, as this can help eliminate square roots and reveal hidden relationships. Let's try squaring both sides of the G.M. equation: (√(xy))² = 6². This simplifies to xy = 36. Now we have a new equation that tells us the product of x and y. This is a crucial piece of the puzzle. Next, let's work with the A.M. equation. We have (x + y) / 2 = 6.5. To get rid of the fraction, we can multiply both sides by 2, which gives us x + y = 13. Now we have two important pieces of information: x + y = 13 and xy = 36. But how do we connect these to x² + y²? This is where a clever algebraic identity comes into play. Remember the formula (x + y)² = x² + 2xy + y²? This identity relates the square of the sum of two numbers to the sum of their squares and their product. Notice that x² + y² appears in this formula, which is exactly what we want to find! Let's rearrange the formula to isolate x² + y²: x² + y² = (x + y)² - 2xy. Now we have an expression for x² + y² in terms of (x + y) and xy, both of which we know the values of! We can simply plug in the values we found earlier: x + y = 13 and xy = 36. So, x² + y² = (13)² - 2(36) = 169 - 72 = 97. And there you have it! We've successfully found the value of x² + y². The answer is 97.

Let's begin by dissecting the problem statement. We're given two positive numbers, x and y. The Arithmetic Mean (A.M.) between them is 6.5, and the Geometric Mean (G.M.) is 6. The ultimate question is: what is the value of x² + y²? Before we jump into calculations, it's crucial to understand what A.M. and G.M. represent. The Arithmetic Mean is the familiar average – the sum of the numbers divided by the count of the numbers. In this case, A.M. = (x + y) / 2. The Geometric Mean, on the other hand, is the nth root of the product of n numbers. For two numbers, it's the square root of their product: G.M. = √(xy). With these definitions in mind, we can translate the given information into mathematical equations. The A.M. being 6.5 translates to (x + y) / 2 = 6.5. The G.M. being 6 translates to √(xy) = 6. Now we have two equations, and our goal is to find x² + y². The challenge lies in connecting these two equations to the expression we want to evaluate. We need a strategy to bridge the gap between the means and the sum of squares. One approach is to manipulate the equations to isolate relevant terms. For instance, we can multiply both sides of the A.M. equation by 2 to get x + y = 13. This gives us the sum of x and y. Similarly, we can square both sides of the G.M. equation to eliminate the square root: (√(xy))² = 6², which simplifies to xy = 36. Now we have the product of x and y. But how do we relate the sum and product of x and y to the sum of their squares? This is where algebraic identities come to the rescue. We need an identity that involves (x + y), xy, and x² + y². The identity that fits the bill is: (x + y)² = x² + 2xy + y². This identity connects the square of the sum to the sum of squares and the product. It's a powerful tool in our arsenal. Now, let's rearrange this identity to isolate the term we're interested in: x² + y². Subtracting 2xy from both sides, we get: x² + y² = (x + y)² - 2xy. Aha! We've expressed x² + y² in terms of (x + y) and xy, both of which we know the values of. This is a breakthrough. All that remains is to substitute the values we found earlier: x + y = 13 and xy = 36. Plugging these into our expression, we get: x² + y² = (13)² - 2(36). Now it's just a matter of arithmetic. Let's calculate: (13)² = 169, and 2(36) = 72. So, x² + y² = 169 - 72 = 97. Therefore, the value of x² + y² is 97. We've successfully solved the problem by carefully dissecting the problem statement, understanding the definitions of A.M. and G.M., and leveraging a crucial algebraic identity. This problem highlights the power of algebraic manipulation and the importance of recognizing key relationships between mathematical concepts.

Alright, guys, let's dive into the nitty-gritty of solving this problem algebraically. We've already established that the Arithmetic Mean (A.M.) of x and y is 6.5, which translates to the equation: (x + y) / 2 = 6.5. We also know that the Geometric Mean (G.M.) is 6, giving us the equation: √(xy) = 6. Our ultimate goal is to find the value of x² + y². To get there, we need to strategically manipulate these equations. Let's start by simplifying the A.M. equation. To get rid of the fraction, we can multiply both sides by 2: x + y = 13. This gives us a direct relationship between x and y. Next, let's tackle the G.M. equation. To eliminate the square root, we can square both sides: (√(xy))² = 6², which simplifies to xy = 36. Now we have two crucial pieces of information: the sum of x and y (x + y = 13) and the product of x and y (xy = 36). The key to connecting these to x² + y² lies in a well-known algebraic identity. The identity we need is: (x + y)² = x² + 2xy + y². This identity beautifully relates the square of the sum to the sum of squares and the product. It's like a mathematical bridge connecting the pieces of our puzzle. Notice that x² + y² is part of this identity, which is exactly what we're trying to find. To isolate x² + y², we can rearrange the identity: x² + y² = (x + y)² - 2xy. Now we have an expression for x² + y² in terms of (x + y) and xy, both of which we know the values of! This is a major step forward. All that remains is to substitute the values we found earlier: x + y = 13 and xy = 36. Plugging these into our expression, we get: x² + y² = (13)² - 2(36). Now it's just a matter of performing the arithmetic operations. First, let's calculate (13)²: 13 * 13 = 169. Next, let's calculate 2(36): 2 * 36 = 72. Now we can substitute these values back into our expression: x² + y² = 169 - 72. Finally, let's subtract: 169 - 72 = 97. Therefore, the value of x² + y² is 97. We've successfully solved the problem using algebraic manipulation and a key algebraic identity. This approach highlights the power of algebra in solving mathematical problems. By recognizing and applying the appropriate identities, we can often simplify complex expressions and arrive at the solution more easily. It's like having a set of mathematical tools that allow us to disassemble and reassemble equations to suit our needs. This problem also reinforces the importance of understanding the definitions of mathematical concepts like A.M. and G.M. A solid understanding of these concepts is crucial for translating word problems into mathematical equations and for choosing the right strategies to solve them.

Hey everyone! Let's delve deeper into the magic of algebraic identities, those nifty formulas that make complex math problems a whole lot easier. In the problem we're tackling today, the identity (x + y)² = x² + 2xy + y² played a starring role. But why are these identities so powerful? Well, algebraic identities are essentially pre-packaged shortcuts. They're equations that are always true, regardless of the values of the variables involved. This means we can use them to transform expressions, simplify equations, and solve problems more efficiently. In our case, the identity (x + y)² = x² + 2xy + y² allowed us to connect the sum of two numbers (x + y) and their product (xy) to the sum of their squares (x² + y²). Without this identity, we'd be stuck trying to find the individual values of x and y, which would be much more complicated. The identity acts like a bridge, allowing us to jump directly from the known information (A.M. and G.M.) to the desired result (x² + y²). Think of it like having a secret code that unlocks the solution. But the power of algebraic identities doesn't stop there. There are many other identities that can be used to solve a wide variety of problems. For example, the identity (x - y)² = x² - 2xy + y² is useful when dealing with the difference of two numbers. The identity (x + y)(x - y) = x² - y² is a powerful tool for factoring and simplifying expressions. And the identities for the sum and difference of cubes, x³ + y³ and x³ - y³, can be used to solve problems involving higher powers. Learning and mastering these identities is like expanding your mathematical toolkit. The more identities you know, the better equipped you'll be to tackle challenging problems. It's not just about memorizing the formulas, though. It's about understanding how they work and when to apply them. This requires practice and a willingness to explore different approaches. When faced with a math problem, try to identify the key relationships between the given information and the desired result. Then, think about which algebraic identities might help you bridge the gap. Sometimes, it's not immediately obvious which identity to use. You might need to try a few different approaches before finding the one that works. But don't be discouraged! The more you practice, the better you'll become at recognizing patterns and applying the appropriate identities. In our A.M. and G.M. problem, the identity (x + y)² = x² + 2xy + y² was the key to unlocking the solution. By rearranging this identity, we were able to express x² + y² in terms of (x + y) and xy, which we already knew from the given information. This allowed us to solve the problem quickly and efficiently. So, the next time you encounter a math problem that seems daunting, remember the power of algebraic identities. They might just be the secret weapon you need to succeed. Keep practicing, keep exploring, and keep expanding your mathematical toolkit! You'll be amazed at what you can accomplish.

Alright, let's recap our journey to conquering this math question. We started with the problem statement: given two positive numbers x and y, the Arithmetic Mean (A.M.) is 6.5 and the Geometric Mean (G.M.) is 6. Our mission was to find the value of x² + y². The first step was to translate the given information into mathematical equations. We knew that A.M. = (x + y) / 2, so (x + y) / 2 = 6.5. Similarly, G.M. = √(xy), so √(xy) = 6. Next, we simplified these equations. Multiplying both sides of the A.M. equation by 2 gave us x + y = 13. Squaring both sides of the G.M. equation gave us xy = 36. Now we had two key pieces of information: the sum of x and y and their product. The challenge was to connect these to x² + y². This is where the algebraic identity (x + y)² = x² + 2xy + y² came to the rescue. By rearranging this identity, we isolated x² + y²: x² + y² = (x + y)² - 2xy. This was a crucial step, as it expressed what we wanted to find in terms of what we already knew. We then substituted the values we found earlier: x + y = 13 and xy = 36. This gave us x² + y² = (13)² - 2(36). Finally, we performed the arithmetic calculations. (13)² = 169 and 2(36) = 72, so x² + y² = 169 - 72 = 97. And there you have it! The value of x² + y² is 97. We successfully solved the problem by following a systematic approach: understanding the definitions, translating the information into equations, simplifying the equations, recognizing and applying the appropriate algebraic identity, and performing the calculations. This step-by-step recap highlights the key strategies we used to solve the problem. It emphasizes the importance of: 1. Understanding the Definitions: Knowing what A.M. and G.M. mean was crucial for setting up the initial equations. 2. Translating into Equations: Converting the word problem into mathematical expressions allowed us to manipulate the information effectively. 3. Simplifying Equations: Getting rid of fractions and square roots made the equations easier to work with. 4. Recognizing Key Identities: The identity (x + y)² = x² + 2xy + y² was the key to connecting the sum and product to the sum of squares. 5. Systematic Calculation: Performing the arithmetic operations carefully ensured we arrived at the correct answer. By mastering these strategies, you'll be well-equipped to tackle a wide range of math problems. Remember, practice makes perfect! The more you solve problems, the better you'll become at recognizing patterns and applying the appropriate techniques. So, keep practicing, keep exploring, and keep challenging yourself. You've got this!

Therefore, the correct answer is D. 97.