Solve 4/n = 4/5: A Step-by-Step Guide

by Omar Yusuf 38 views

Hey guys! Let's dive into a math problem that might seem tricky at first glance but is actually super straightforward once you break it down. We're going to tackle the equation 4/n = 4/5 and figure out which of the given options is the correct value for n. So, buckle up, and let's get started!

Understanding the Problem

In this section, let's deeply understand the core of the problem we're tackling: solving for n in the equation 4/n = 4/5. This isn't just about finding the right answer; it's about grasping the underlying mathematical principles that allow us to manipulate equations and isolate variables. When we look at 4/n = 4/5, we're seeing a proportion—two ratios set equal to each other. Proportions are fundamental in mathematics and appear in various real-world scenarios, from scaling recipes to calculating distances on a map. Understanding proportions equips us with a versatile tool for problem-solving. The variable n represents an unknown quantity, and our goal is to pinpoint its exact value. Think of n as a missing piece in a puzzle; by solving the equation, we're finding the piece that makes the puzzle complete. This process of solving for a variable is a cornerstone of algebra, a branch of mathematics that uses symbols and letters to represent numbers and quantities. Algebra provides us with the framework to express relationships between quantities and solve complex problems. The beauty of algebra lies in its ability to generalize mathematical concepts. Instead of just dealing with specific numbers, we can use variables like n to represent a range of possibilities. This allows us to create equations that model real-world situations and find solutions that apply universally. Solving for n involves isolating n on one side of the equation. This means manipulating the equation in such a way that n is by itself, with its value clearly revealed on the other side. To do this, we employ various algebraic techniques, such as multiplying or dividing both sides of the equation by the same quantity. These techniques are based on the principle of maintaining balance. Just like a scale, an equation must remain balanced—whatever operation we perform on one side, we must perform on the other to keep the equation true. This principle ensures that we're not changing the fundamental relationship expressed by the equation. By understanding these core concepts, we're not just solving a single problem; we're building a foundation for more advanced mathematical explorations. The ability to solve for variables is crucial in fields ranging from engineering to economics, making it a valuable skill for anyone to develop. Remember, math isn't just about memorizing formulas; it's about understanding the logic and reasoning that underpin them. As we move forward, keep these principles in mind, and you'll find that even the most challenging equations become manageable.

Step-by-Step Solution

Now, let's break down how to solve the equation 4/n = 4/5 step by step. This will make it super clear how we arrive at the correct answer. This part is where the magic happens! We're going to take our equation and use some cool mathematical tricks to get n all by itself. Remember, the goal is to isolate n on one side of the equation. One common strategy for solving equations involving fractions is to eliminate the fractions. This makes the equation easier to work with and reduces the chance of making errors. In our case, we have a fraction with n in the denominator. To get rid of this fraction, we can multiply both sides of the equation by n. This is a valid operation because, as we discussed earlier, whatever we do to one side of the equation, we must do to the other to maintain balance. So, let's multiply both sides of 4/n = 4/5 by n. On the left side, the n in the numerator and the n in the denominator cancel each other out, leaving us with just 4. On the right side, we have (4/5) * n, which we can write as (4n)/5. Our equation now looks like this: 4 = (4n)/5. We've made progress! We've eliminated one fraction, but we still have a fraction on the right side. To get rid of this fraction, we can multiply both sides of the equation by the denominator, which is 5. This will cancel out the 5 in the denominator on the right side. So, let's multiply both sides of 4 = (4n)/5 by 5. On the left side, we have 4 * 5, which equals 20. On the right side, the 5 in the numerator and the 5 in the denominator cancel each other out, leaving us with just 4n. Our equation now looks like this: 20 = 4n. We're getting closer! We have n multiplied by 4 on the right side. To isolate n, we need to undo this multiplication. The opposite of multiplication is division, so we can divide both sides of the equation by 4. Let's do that. On the left side, we have 20 / 4, which equals 5. On the right side, we have (4n) / 4, which simplifies to just n. Our equation now looks like this: 5 = n. We've done it! We've isolated n and found its value. The solution to the equation 4/n = 4/5 is n = 5. This step-by-step approach not only helps us find the answer but also reinforces our understanding of the underlying mathematical principles. Remember, practice makes perfect, so keep working through these steps, and you'll become a pro at solving equations!

Analyzing the Options

Let's take a look at the options provided and see which one matches our solution. We found that n = 5, so we're looking for that answer among the choices. In this section, we'll carefully examine each of the provided options to determine which one aligns with the solution we derived in the previous section. Our step-by-step solution led us to the conclusion that n equals 5. Therefore, the correct answer should be the option that explicitly states n = 5. However, it's crucial to approach each option with a critical eye, ensuring that we're not overlooking any subtle nuances or potential distractions. We'll consider each choice individually, comparing it against our calculated value of n and explaining why it either matches or deviates from the correct solution. By thoroughly analyzing each option, we not only reinforce our understanding of the problem but also sharpen our attention to detail, a valuable skill in mathematics and beyond. Option (a) states that n = 5. This is exactly the value we found through our step-by-step solution. This option perfectly aligns with our derived answer and represents the correct value of n that satisfies the equation 4/n = 4/5. Therefore, option (a) appears to be the correct choice. However, we'll still examine the remaining options to ensure there are no other possibilities or any potential errors in our reasoning. Option (b) suggests that n = 5/16. This value is significantly different from our calculated value of 5. If we were to substitute 5/16 for n in the original equation, we would get 4 / (5/16) = 4 * (16/5) = 64/5, which is not equal to 4/5. Therefore, option (b) is incorrect. The fraction 5/16 is much smaller than 5, and substituting it for n would drastically alter the balance of the equation. Option (c) proposes that n = 3 1/5. This is a mixed number, which can be converted to an improper fraction. 3 1/5 is equivalent to (3 * 5 + 1) / 5 = 16/5. This value is also different from our calculated value of 5. Substituting 16/5 for n in the original equation would give us 4 / (16/5) = 4 * (5/16) = 20/16 = 5/4, which is not equal to 4/5. Thus, option (c) is incorrect. The value 3 1/5 is less than 5, but not by a large margin like option (b). However, it still does not satisfy the equation. Finally, option (d) states that n = 50. This value is much larger than our calculated value of 5. If we were to substitute 50 for n in the original equation, we would get 4/50 = 2/25, which is not equal to 4/5. Therefore, option (d) is incorrect. The large value of 50 would make the left side of the equation much smaller than the right side. By carefully analyzing each option and comparing it to our derived solution, we can confidently conclude that only option (a), n = 5, is the correct answer. This process highlights the importance of not only solving the equation but also verifying the solution against the given choices to ensure accuracy.

The Correct Answer

Drumroll, please! After solving the equation and analyzing the options, the correct answer is a. n = 5. We nailed it! The correct answer is option (a), which clearly states that n equals 5. This conclusion is a direct result of our meticulous step-by-step solution, where we carefully manipulated the equation 4/n = 4/5 to isolate n and determine its value. Our journey through the equation involved several key algebraic techniques, including multiplying both sides by n to eliminate the fraction, multiplying by 5 to remove the denominator, and finally, dividing by 4 to isolate n. Each step was performed with precision, ensuring that we maintained the balance of the equation and arrived at the accurate solution. The process of solving for n highlighted the fundamental principles of algebra, emphasizing the importance of inverse operations and the preservation of equality. By understanding these principles, we can confidently tackle a wide range of algebraic problems, not just this specific equation. Furthermore, our analysis of the provided options reinforced the importance of critical thinking and attention to detail. We didn't simply choose the first answer that looked plausible; instead, we systematically evaluated each option, comparing it against our derived solution and explaining why it either matched or deviated from the correct value. This rigorous approach not only confirmed the correctness of our solution but also enhanced our understanding of the equation and its properties. Choosing the correct answer is more than just selecting a letter; it's a validation of our problem-solving process and our understanding of the underlying mathematical concepts. In this case, option (a) not only matches our calculated value of n but also aligns with the logical flow of our solution. The equation 4/n = 4/5 essentially asks: what number, when placed in the denominator, will make the fraction equal to 4/5? Our solution, n = 5, provides the perfect answer to this question. When we substitute 5 for n, we get 4/5 = 4/5, a true statement that confirms the validity of our solution. The satisfaction of arriving at the correct answer is not just about the end result; it's about the journey we took to get there. It's about the application of mathematical principles, the careful execution of steps, and the confidence in our ability to solve problems. This experience not only strengthens our mathematical skills but also cultivates a problem-solving mindset that can be applied to various aspects of life.

Key Takeaways

Alright, guys, let's wrap things up with some key takeaways from this problem. Understanding these points will help you tackle similar equations in the future. This section is all about solidifying our understanding of the problem-solving process and extracting the most valuable lessons that we can apply to future mathematical challenges. Solving the equation 4/n = 4/5 wasn't just about finding the value of n; it was about honing our algebraic skills and developing a systematic approach to problem-solving. Let's break down the key takeaways that will empower us to confidently tackle similar equations in the future. One of the most crucial takeaways is the importance of isolating the variable. In this case, we needed to get n by itself on one side of the equation. This involved a series of algebraic manipulations, such as multiplying and dividing both sides by appropriate quantities. Isolating the variable is a fundamental technique in algebra, and mastering it is essential for solving a wide range of equations. It's like finding the key to unlock a puzzle; once the variable is isolated, its value is revealed. Another key takeaway is the use of inverse operations. To isolate n, we had to undo the operations that were being performed on it. For example, since n was in the denominator of a fraction, we multiplied both sides of the equation by n to eliminate the fraction. Similarly, we divided both sides by 4 to undo the multiplication of n by 4. Understanding inverse operations is like having a set of tools that allow us to reverse the effects of mathematical operations. The concept of maintaining balance in an equation is also paramount. Whatever operation we perform on one side of the equation, we must perform on the other side to keep the equation true. This principle ensures that we're not changing the fundamental relationship expressed by the equation. It's like a seesaw; if we add weight to one side, we must add the same weight to the other side to keep it balanced. Furthermore, this problem highlights the power of proportions. The equation 4/n = 4/5 is a proportion, which is a statement that two ratios are equal. Proportions are prevalent in mathematics and have numerous real-world applications, from scaling recipes to calculating distances on maps. Recognizing proportions and understanding how to solve them is a valuable skill. In addition to the mathematical concepts, this problem also emphasizes the importance of careful analysis and attention to detail. We didn't simply stop at finding a solution; we also analyzed the provided options, comparing each one against our derived value of n and explaining why it either matched or deviated from the correct answer. This rigorous approach not only confirmed the correctness of our solution but also enhanced our understanding of the equation and its properties. Remember, mathematics is not just about memorizing formulas; it's about developing a logical and systematic approach to problem-solving. By understanding these key takeaways and applying them to future challenges, we can build confidence in our mathematical abilities and tackle even the most complex equations with ease.

Hope this helps you guys understand how to solve this type of problem! Keep practicing, and you'll become math whizzes in no time!