Solving Ratio And Average Problems A Step By Step Guide
In this article, we will delve into a fascinating mathematical problem that involves ratios and averages. Our primary focus will be on deciphering the relationship between variables x, y, and z, and then using this knowledge to solve a complex equation. The problem is structured in such a way that it requires a deep understanding of how ratios work, how averages are calculated, and how percentages play a role in determining the final answer. Let's embark on this mathematical journey together to unravel the mystery behind this problem and equip ourselves with the skills to tackle similar challenges in the future.
The core of our discussion lies in the following mathematical puzzle: Given that x : y = 11 : 9 and y : z = 3 : 4, we are tasked with finding the value of the expression (x + y) - 1.5z. To add a layer of complexity, we are presented with two additional statements:
I. The average of all three variables (x, y, and z) is two more than the average of x and y. II. The sum of 9 1/11% of x and 11 1/9% of y is equal to √36.
Our mission is to utilize these statements, in conjunction with the given ratios, to determine the numerical value of (x + y) - 1.5z. This problem is a classic example of how different mathematical concepts can be interwoven to create a challenging yet rewarding exercise. To solve this, we will systematically break down each piece of information, establish relationships between the variables, and then use these relationships to arrive at the final answer. This problem not only tests our ability to perform calculations but also our logical reasoning and problem-solving skills. The journey to the solution is just as important as the answer itself, as it teaches us how to approach complex problems with a structured and methodical approach. So, let's put on our thinking caps and dive into the world of ratios, averages, and percentages!
Part 1: Ratio Analysis Establishing the Foundation
The initial step in solving this mathematical puzzle is to analyze the given ratios. We are provided with two key pieces of information: x : y = 11 : 9 and y : z = 3 : 4. These ratios establish a direct relationship between the variables x, y, and z. However, to effectively use these ratios, we need to find a common term that links all three variables together. The variable 'y' serves as the bridge between the two ratios. To create a unified ratio, we need to make the 'y' component consistent across both ratios.
Currently, the ratio of x to y is 11 : 9, and the ratio of y to z is 3 : 4. To equate the 'y' values, we can find the least common multiple (LCM) of 9 and 3, which is 9. We need to adjust the second ratio (y : z = 3 : 4) so that the 'y' component becomes 9. To achieve this, we multiply both sides of the second ratio by 3. This gives us a new ratio of y : z = 9 : 12.
Now that we have a common 'y' value, we can combine the two ratios into a single, unified ratio. The ratio x : y is 11 : 9, and the adjusted ratio y : z is 9 : 12. Combining these, we get the comprehensive ratio x : y : z = 11 : 9 : 12. This unified ratio is a crucial stepping stone in solving the problem, as it allows us to express all three variables in terms of a common factor. Let's denote this common factor as 'k'. Therefore, we can express x as 11k, y as 9k, and z as 12k. These expressions will be instrumental in utilizing the information provided in the subsequent statements.
The power of ratios lies in their ability to simplify complex relationships into manageable proportions. By understanding and manipulating ratios effectively, we can solve a wide range of problems, from simple comparisons to intricate mathematical puzzles. In this case, establishing the unified ratio x : y : z = 11 : 9 : 12 is the foundation upon which the rest of the solution will be built. This methodical approach of breaking down the problem into smaller, more manageable parts is a key strategy in problem-solving. With the variables now expressed in terms of 'k', we are well-equipped to tackle the additional information provided in the problem statement.
Part 2: Leveraging Statement I The Average Connection
Now, let's shift our focus to the first additional statement provided in the problem: "The average of all three variables (x, y, and z) is two more than the average of x and y." This statement introduces the concept of averages and establishes a direct relationship between the average of all three variables and the average of only two of them. To effectively use this information, we need to translate this verbal statement into a mathematical equation. Recall that the average of a set of numbers is calculated by summing the numbers and then dividing by the count of numbers.
The average of x, y, and z is given by (x + y + z) / 3. Similarly, the average of x and y is (x + y) / 2. According to the statement, the average of all three variables is two more than the average of x and y. Therefore, we can write the equation as:
(x + y + z) / 3 = (x + y) / 2 + 2
This equation is a powerful tool that links the variables x, y, and z together. However, we already have expressions for x, y, and z in terms of 'k' from our ratio analysis in Part 1. Recall that x = 11k, y = 9k, and z = 12k. Substituting these expressions into the equation, we get:
(11k + 9k + 12k) / 3 = (11k + 9k) / 2 + 2
This substitution transforms the equation from one involving three variables into one involving a single variable, 'k'. This simplifies the problem significantly and allows us to solve for 'k'. Now, we can simplify the equation further by performing the arithmetic operations. The left side of the equation becomes (32k) / 3, and the right side becomes (20k) / 2 + 2, which simplifies to 10k + 2. Therefore, the equation now reads:
(32k) / 3 = 10k + 2
This equation is a linear equation in one variable, which we can solve using standard algebraic techniques. Our next step is to isolate 'k' and determine its value. Solving for 'k' will provide us with the key to unlocking the values of x, y, and z, and ultimately help us find the value of the expression (x + y) - 1.5z. The power of this statement lies in its ability to provide a concrete equation that links the variables together, allowing us to progress towards the final solution. The strategic use of averages, combined with our previous ratio analysis, is a testament to the importance of a structured approach in problem-solving.
Part 3: Decoding Statement II Percentages and Square Roots
Now, let's turn our attention to the second statement provided: "The sum of 9 1/11% of x and 11 1/9% of y is equal to √36." This statement introduces the concept of percentages and square roots, adding another layer of complexity to our puzzle. Similar to the previous statement, our goal is to translate this verbal statement into a mathematical equation. To do this effectively, we need to understand how to work with mixed percentages and square roots.
First, let's convert the mixed percentages into fractions. 9 1/11% can be written as 9 + 1/11, which is equal to 100/11%. To convert this percentage to a fraction, we divide by 100, giving us (100/11) / 100 = 1/11. Similarly, 11 1/9% can be written as 11 + 1/9, which is equal to 100/9%. Converting this percentage to a fraction, we get (100/9) / 100 = 1/9.
Now that we have the percentages as fractions, we can express the statement in terms of x and y. 9 1/11% of x is (1/11)x, and 11 1/9% of y is (1/9)y. The statement says that the sum of these two quantities is equal to √36. The square root of 36 is 6. Therefore, we can write the equation as:
(1/11)x + (1/9)y = 6
This equation provides another relationship between x and y. Recall that we have already expressed x and y in terms of 'k' from our ratio analysis in Part 1. Substituting x = 11k and y = 9k into the equation, we get:
(1/11)(11k) + (1/9)(9k) = 6
This substitution simplifies the equation significantly. The (1/11)(11k) term simplifies to k, and the (1/9)(9k) term also simplifies to k. Therefore, the equation now reads:
k + k = 6
This equation is a simple linear equation in one variable, 'k'. Combining the terms on the left side, we get 2k = 6. Dividing both sides by 2, we find that k = 3. This is a significant breakthrough in our problem-solving journey. We have successfully determined the value of 'k', which is the common factor that links x, y, and z together. With the value of 'k' in hand, we can now easily calculate the values of x, y, and z. The strategic use of percentages and square roots, combined with our previous ratio analysis, has led us to a crucial piece of the puzzle. The power of this statement lies in its ability to provide a direct equation that allows us to solve for 'k'.
Part 4: Solving for Variables and the Final Calculation
With the value of 'k' determined to be 3, we are now in a position to calculate the values of x, y, and z. Recall that from our ratio analysis in Part 1, we expressed x, y, and z in terms of 'k' as follows: x = 11k, y = 9k, and z = 12k. Substituting k = 3 into these expressions, we get:
x = 11 * 3 = 33 y = 9 * 3 = 27 z = 12 * 3 = 36
We have successfully determined the numerical values of x, y, and z. Now that we know the values of these variables, we can finally calculate the value of the expression (x + y) - 1.5z, which is the ultimate goal of our problem. Substituting the values of x, y, and z into the expression, we get:
(x + y) - 1.5z = (33 + 27) - 1.5 * 36
Now, we can simplify the expression by performing the arithmetic operations. First, we add 33 and 27, which gives us 60. Next, we multiply 1.5 by 36, which gives us 54. Therefore, the expression becomes:
60 - 54
Subtracting 54 from 60, we get 6. Therefore, the final value of the expression (x + y) - 1.5z is 6. This is the solution to our mathematical puzzle. We have successfully navigated through the complexities of ratios, averages, percentages, and square roots to arrive at the answer. The journey to the solution has been a testament to the power of a structured and methodical approach in problem-solving. By breaking down the problem into smaller, more manageable parts, we were able to systematically utilize the information provided and establish relationships between the variables. The strategic use of each statement, combined with our initial ratio analysis, has led us to the final answer. The ability to solve such problems is not just about performing calculations; it's about developing logical reasoning and critical thinking skills, which are essential in various aspects of life.
In conclusion, we have successfully solved a challenging mathematical problem involving ratios, averages, and percentages. The problem required us to decipher the relationships between variables x, y, and z, and then use this knowledge to calculate the value of the expression (x + y) - 1.5z. We approached the problem with a structured and methodical approach, breaking it down into smaller, more manageable parts. We started by analyzing the given ratios and establishing a unified ratio between the variables. Then, we strategically utilized the additional statements provided, translating them into mathematical equations. By solving these equations, we were able to determine the values of the variables and ultimately calculate the value of the expression.
The key to solving this problem was the ability to connect different mathematical concepts and apply them in a logical and systematic manner. The problem not only tested our computational skills but also our ability to think critically and reason logically. The journey to the solution has been a valuable learning experience, highlighting the importance of a structured approach in problem-solving. The ability to solve such problems is not just about finding the right answer; it's about developing the skills and mindset necessary to tackle complex challenges in any field. Mathematical reasoning is a powerful tool that can be applied in various aspects of life, from making informed decisions to solving real-world problems. By mastering these skills, we can empower ourselves to navigate the complexities of the world around us.
The problem we have solved is a testament to the beauty and elegance of mathematics. It demonstrates how seemingly disparate concepts can be interwoven to create a challenging yet rewarding exercise. The satisfaction of arriving at the solution is a testament to the power of human intellect and the joy of intellectual exploration. Mathematics is not just a subject to be studied; it is a way of thinking, a way of understanding the world. By embracing the challenges that mathematics presents, we can unlock our potential and develop the skills necessary to succeed in an increasingly complex world. The journey of solving this problem has been a journey of intellectual growth, and the skills we have acquired will serve us well in future endeavors. The ability to approach problems with a structured and methodical approach, combined with a strong understanding of mathematical concepts, is a recipe for success in any field.
