Given That The Ratio Of Women To Men At A Party Of 90 Attendees Is 5:4, How Many Men Attended?

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In this article, we will delve into a classic mathematical problem involving ratios and proportions. We'll dissect the information provided, apply the principles of ratios, and arrive at the solution in a step-by-step manner. This problem is not just an exercise in numbers; it's a testament to how mathematical concepts can help us decipher real-world scenarios. Our journey will involve understanding the relationship between the number of women and men at a party, and using this relationship to determine the exact count of male attendees.

H2: Understanding the Problem: Setting the Stage for Solution

The core of the problem lies in the statement: "Out of 90 attendees at a party, the ratio of women to men is 5:4." This seemingly simple statement is a goldmine of information, providing us with the key to unlock the answer. To truly grasp the problem, let's break it down into its fundamental components. First, we have the total number of attendees, which is 90. This is our whole, the entire group we are working with. Second, we have the ratio of women to men, which is 5:4. This ratio tells us the proportional relationship between the number of women and the number of men. For every 5 women, there are 4 men. This doesn't mean there are exactly 5 women and 4 men, but rather that the number of women is a multiple of 5, and the number of men is the same multiple of 4. Understanding this proportional relationship is crucial for solving the problem. We need to figure out what that common multiple is so that we can determine the actual number of men who attended the party. The problem essentially asks us to divide the total attendees (90) into two groups (women and men) based on the given ratio. This involves concepts of proportions and fractions, where we treat the ratio as parts of a whole. By carefully analyzing the given information and translating it into mathematical terms, we can pave the way for a clear and concise solution. The next step involves translating this understanding into a mathematical framework, which will allow us to apply the necessary formulas and calculations.

H2: Deconstructing the Ratio: The Key to Unlocking the Solution

The ratio of women to men, given as 5:4, is the cornerstone of our problem-solving approach. This ratio doesn't directly tell us the exact number of women or men, but it establishes a proportional relationship between them. Think of it as a recipe: for every 5 parts of one ingredient (women), you need 4 parts of another ingredient (men). To convert this ratio into actual numbers, we need to introduce a common multiplier. Let's represent this multiplier with the variable 'x'. This means that the number of women can be represented as 5x, and the number of men as 4x. The total number of attendees is then the sum of the number of women and the number of men, which can be expressed as 5x + 4x. This is a crucial step because it allows us to create an equation that relates the ratio to the total number of attendees, which we know is 90. By introducing the variable 'x', we've transformed the ratio into a tangible algebraic expression. This expression, 5x + 4x, represents the total number of people at the party in terms of our unknown multiplier 'x'. Our goal now is to find the value of 'x', which will allow us to calculate the actual number of men and women. The beauty of using ratios lies in their ability to simplify complex problems. Instead of dealing with two separate unknowns (the number of women and the number of men), we've effectively reduced the problem to solving for a single unknown ('x'). This is a powerful technique that is widely used in mathematics and other fields. The next step involves setting up an equation and solving for this crucial variable, bringing us closer to our final answer.

H2: Setting Up the Equation: Translating Ratios into Algebra

Having deconstructed the ratio, the next logical step is to translate the information into a mathematical equation. We know that the total number of attendees is 90. We also know that the number of women can be represented as 5x, and the number of men as 4x. Therefore, the sum of 5x and 4x must equal the total number of attendees, which is 90. This can be written as the equation: 5x + 4x = 90. This equation is the bridge between the abstract ratio and the concrete numbers we are trying to find. It encapsulates the essence of the problem in a concise mathematical statement. The left side of the equation, 5x + 4x, represents the total number of attendees in terms of our unknown variable 'x'. The right side of the equation, 90, represents the actual total number of attendees. By setting these two expressions equal to each other, we create an equation that we can solve for 'x'. Solving this equation will reveal the value of 'x', which will then allow us to calculate the number of men and women at the party. The process of translating word problems into mathematical equations is a fundamental skill in algebra. It requires careful reading and comprehension of the problem, followed by the ability to identify the key relationships and express them in mathematical symbols. Once we have the equation, we can use algebraic techniques to solve for the unknown variable, unlocking the solution to the problem. Now, let's move on to solving the equation and finding the value of 'x'.

H2: Solving for 'x': Unveiling the Multiplier

Now that we have the equation 5x + 4x = 90, the next crucial step is to solve for 'x'. This involves using basic algebraic principles to isolate 'x' on one side of the equation. First, we can simplify the left side of the equation by combining the like terms: 5x and 4x. Adding these terms together, we get 9x. So, our equation now becomes 9x = 90. To isolate 'x', we need to undo the multiplication by 9. We can do this by dividing both sides of the equation by 9. Dividing both sides by 9, we get: (9x) / 9 = 90 / 9. This simplifies to x = 10. Therefore, the value of 'x' is 10. This value is the key to unlocking the solution. Remember that 'x' represents the common multiplier that relates the ratio to the actual number of attendees. Now that we know 'x', we can use it to calculate the number of women and men at the party. Solving for 'x' is a fundamental skill in algebra, and it's a technique that is used in a wide variety of mathematical problems. The process involves manipulating the equation using algebraic operations, such as addition, subtraction, multiplication, and division, until the unknown variable is isolated. Once we have solved for 'x', we can substitute its value back into the original expressions to find the values of other unknowns. In this case, knowing 'x' allows us to determine the number of men who attended the party, which is the ultimate goal of the problem. With the value of 'x' in hand, we can now proceed to calculate the number of men who attended the party.

H2: Calculating the Number of Men: The Final Revelation

With the value of 'x' determined to be 10, we are now in the final stretch to answering the problem. Recall that the number of men is represented by 4x. To find the actual number of men, we simply need to substitute the value of 'x' into this expression. So, the number of men is 4 * 10, which equals 40. Therefore, there were 40 men at the party. This is the solution to our problem. We have successfully used the given ratio and the total number of attendees to determine the number of male attendees. This final calculation demonstrates the power of ratios and proportions in solving real-world problems. By understanding the proportional relationship between the number of women and men, and by using algebra to translate this relationship into an equation, we were able to arrive at a precise answer. The process of solving this problem highlights the importance of breaking down complex problems into smaller, more manageable steps. By carefully analyzing the given information, identifying the key relationships, and applying the appropriate mathematical techniques, we can successfully tackle a wide range of problems. This problem also serves as a reminder that mathematics is not just about abstract numbers and formulas; it's a powerful tool for understanding and solving problems in the real world. We have now successfully answered the question: there were 40 men at the party. Let's move on to summarizing the steps we took to reach this solution.

H2: Summary and Conclusion: Reflecting on the Solution Process

In conclusion, we successfully solved the problem by systematically breaking it down into smaller, manageable steps. First, we understood the problem, identifying the key information: the total number of attendees (90) and the ratio of women to men (5:4). Then, we deconstructed the ratio, representing the number of women as 5x and the number of men as 4x, where 'x' is a common multiplier. This allowed us to translate the ratio into algebraic expressions. Next, we set up an equation, recognizing that the sum of the number of women (5x) and the number of men (4x) must equal the total number of attendees (90). This gave us the equation 5x + 4x = 90. We then solved for 'x' by simplifying the equation and isolating 'x', finding that x = 10. Finally, we calculated the number of men by substituting the value of 'x' into the expression 4x, which gave us 4 * 10 = 40 men. Therefore, there were 40 men at the party. This problem demonstrates the power of using ratios and proportions to solve real-world problems. By carefully analyzing the given information, translating it into mathematical terms, and applying algebraic techniques, we can arrive at accurate solutions. The process of problem-solving involves a combination of understanding the problem, developing a strategy, executing the strategy, and verifying the solution. In this case, we successfully navigated each of these steps to arrive at the correct answer. This problem also highlights the importance of mathematical literacy in everyday life. The ability to understand and apply mathematical concepts is essential for making informed decisions and solving problems in various aspects of life. We hope this detailed explanation has provided a clear understanding of the problem and its solution, and has demonstrated the power and elegance of mathematical reasoning.

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