What Percentage Of 2x + 7y Is X+y, Given That 40% Of 50% Of X Equals 30% Of Y?

In this mathematical exploration, we delve into the intricate relationship between percentages and algebraic expressions. The problem at hand presents a seemingly complex scenario: "40% of 50% of x is 30% of y." Our mission is to unravel this puzzle, decipher the connection between x and y, and ultimately determine what percentage of 2x + 7y is x + y. This journey will involve converting percentages to decimals, setting up equations, simplifying expressions, and applying fundamental algebraic principles. By the end of this comprehensive analysis, you'll not only grasp the solution to this specific problem but also enhance your ability to tackle similar percentage-based challenges.

Let's begin by translating the given statement into a mathematical equation. "40% of 50% of x is 30% of y" can be written as: 0. 40 * 0.50 * x = 0.30 * y. This equation forms the foundation of our solution. It represents the core relationship between x and y, and by manipulating this equation, we can extract valuable information about their proportionality. The use of decimals to represent percentages is crucial here, as it allows us to perform arithmetic operations with ease. Understanding this conversion is fundamental to solving percentage-related problems. Furthermore, the equation highlights the importance of order of operations in mathematics. We must first calculate the percentages of x and y before equating them, ensuring the accuracy of our calculations.

Our next step is to simplify the equation to a more manageable form. Multiplying the decimals on the left side, we get: 0.20x = 0.30y. Now, we can further simplify by dividing both sides by 0.10, resulting in: 2x = 3y. This simplified equation provides a clearer picture of the relationship between x and y. It tells us that twice the value of x is equal to three times the value of y. This proportionality is key to solving the problem. The simplification process demonstrates the power of algebraic manipulation in revealing underlying relationships. By reducing the equation to its simplest form, we make it easier to work with and extract meaningful information.

To proceed further, let's express y in terms of x. From the simplified equation 2x = 3y, we can isolate y by dividing both sides by 3: y = (2/3)x. This expression is crucial because it allows us to substitute y in subsequent calculations, effectively reducing the number of variables in our equations. Expressing one variable in terms of another is a common technique in algebra, particularly when dealing with systems of equations or problems involving multiple variables. In this case, expressing y in terms of x simplifies the process of finding the desired percentage.

Now, we need to determine what percentage of 2x + 7y is x + y. Let's represent this percentage as 'p'. This can be written as: (x + y) = p% of (2x + 7y) Or, (x + y) = (p/100) * (2x + 7y). To solve for p, we first substitute y = (2/3)x into this equation. This substitution is a direct application of the relationship we derived earlier. By replacing y with its equivalent expression in terms of x, we reduce the equation to a single variable, making it solvable. This substitution technique is a cornerstone of algebraic problem-solving.

Substituting y = (2/3)x, we get: x + (2/3)x = (p/100) * [2x + 7(2/3)x]. This step is a direct application of the substitution we discussed earlier. By replacing y with its equivalent expression in terms of x, we've successfully transformed the equation into a form that involves only one variable. This is a significant step towards isolating and solving for p. The substitution process not only simplifies the equation but also allows us to express the relationship between x + y and 2x + 7y in terms of a single variable, making it easier to determine the percentage.

Simplifying the left side, we have: (5/3)x = (p/100) * [2x + (14/3)x]. Further simplifying the right side, we get: (5/3)x = (p/100) * [(6x + 14x)/3] (5/3)x = (p/100) * (20x/3). These simplifications involve basic arithmetic operations, such as adding fractions and combining like terms. Each simplification step brings us closer to isolating p and solving for the desired percentage. The ability to perform these operations accurately is crucial for success in algebra.

To isolate p, we multiply both sides by 100 and by 3/20x: p = (5/3)x * (100 * 3) / (20x). Now we can cancel out the common factors on both the numerator and the denominator. Isolating the variable we want to solve for is a fundamental algebraic technique. By performing the appropriate operations on both sides of the equation, we gradually move the variable of interest to one side, making it possible to determine its value.

Cancelling out x and simplifying, we get: p = (5 * 100) / 20 p = 500 / 20 p = 25. Therefore, x + y is 25% of 2x + 7y. This final calculation is the culmination of all our previous steps. It represents the solution to the problem, expressing the relationship between x + y and 2x + 7y as a percentage. The final answer, 25%, is the result of careful algebraic manipulation and accurate arithmetic calculations.

In conclusion, by converting percentages to decimals, setting up and simplifying equations, and expressing variables in terms of one another, we have successfully determined that x + y is 25% of 2x + 7y. This problem highlights the importance of understanding percentages, algebraic manipulation, and problem-solving techniques in mathematics. The process we followed demonstrates a systematic approach to tackling mathematical challenges, emphasizing the power of simplification, substitution, and careful calculation. By mastering these techniques, you can confidently approach a wide range of mathematical problems and enhance your analytical skills.

Percentages, Algebraic Equations, Mathematical Problem-Solving, Simplification, Substitution, Variable Isolation

What percentage of 2x + 7y is x + y, given that 40% of 50% of x is equal to 30% of y?