Solving For 'n': A Math Adventure With Stanley

Hey math enthusiasts! Let's dive into a fun problem where we'll figure out a positive number, represented by 'n', that's less than 1. This number has a special relationship with its reciprocal – when you add them together, you get 2.05! We're going to act like detectives, unraveling this numerical mystery and expressing our answer as a simplified fraction. Get ready to put on your thinking caps, because it's going to be an exciting ride. We are going to break down this problem, step-by-step, making sure that even if math isn't your favorite thing, you'll still be able to follow along. Trust me, it's easier than it sounds, and we'll learn some cool math tricks along the way. So, let's jump right in and see how we can solve this problem. We'll start by understanding what the problem is asking, then we'll break it down into smaller, more manageable steps. Don't worry, I'll be here to guide you through every twist and turn. Consider it a friendly math lesson, where we explore the fascinating world of numbers and equations. Now, grab a pen and paper, and let's get started. By the end of this, you'll feel like a math whiz, ready to tackle any problem that comes your way. This is not just about finding an answer; it's about understanding the process and building your problem-solving skills. Remember, the journey is just as important as the destination. We'll be using some basic algebra, but don't worry if you're not an expert. I'll explain everything in a way that's easy to understand. So, are you ready to embark on this mathematical adventure? Let's go!

Unpacking the Problem: What Does It All Mean?

Alright, guys, before we start crunching numbers, let's make sure we fully understand what we're dealing with. Stanley, in our problem, is thinking of a positive number. This number, we know, is less than 1. Let's call this mysterious number 'n'. Now, there's another concept at play here called the 'reciprocal'. The reciprocal of a number is simply 1 divided by that number. So, the reciprocal of 'n' is 1/n. The problem tells us that when we add 'n' to its reciprocal (1/n), the result is 2.05. We're asked to find the value of 'n' and give our answer as a simplified fraction. This means we need to find the specific fraction that, when added to its reciprocal, equals 2.05. To get started, we need to convert the decimal, 2.05, into a fraction. Remember, working with fractions often makes these kinds of problems easier to solve. Also, keep in mind that the value of 'n' has to be positive and less than 1, so the solution needs to fit that description. Always keep an eye on these conditions as you progress through the math. We'll need to use this information to create an equation that we can solve. It’s like setting up a puzzle where we know a little bit about each piece, and our task is to fit them together to complete the picture. This involves understanding how to manipulate equations, knowing the rules of fractions, and being patient. Let’s break it down further so that we understand what we are dealing with. We need to clearly understand what we are dealing with before we start to solve this problem.

Transforming the Problem into an Equation

Okay, team, let's translate the words of the problem into a mathematical equation. We know that 'n' plus its reciprocal (1/n) equals 2.05. We can write this as: n + 1/n = 2.05. But before we get cracking, let's convert 2.05 into a fraction. The easiest way to do this is to recognize that 2.05 is the same as 2 and 5/100, which simplifies to 2 and 1/20, which is equal to 41/20. Now our equation looks like this: n + 1/n = 41/20. Our next step is to eliminate the fraction by multiplying every term in the equation by 'n'. This gives us: n² + 1 = (41/20)n. Now, we're not going to be intimidated by this new equation; we are just going to rearrange the terms so that everything is on one side, and we have a zero on the other side. So, let’s subtract (41/20)n from both sides, so we get: n² - (41/20)n + 1 = 0. This is a quadratic equation, and we can solve it by getting rid of the fraction in the middle term by multiplying every term by 20. Then we get: 20n² - 41n + 20 = 0. We've got our equation, and now we need to figure out how to find the value of 'n'. This can be done by using the quadratic formula, by completing the square, or by factoring the quadratic expression, if possible. Don't worry if it sounds complicated – we're going to take it one step at a time. The goal is to isolate 'n' and reveal its value. This is where your algebra skills come in handy! This whole process is similar to creating a map to find a treasure. Each step we take brings us closer to the final solution. The key is to be precise and patient, and to double-check your work along the way. Trust me, it's not as hard as it looks. You've got this!

Solving the Quadratic Equation

Alright, folks, we've got a quadratic equation: 20n² - 41n + 20 = 0. To solve this, we can use the quadratic formula, which is a powerful tool for finding the solutions to any quadratic equation. The quadratic formula is: n = (-b ± √(b² - 4ac)) / 2a, where a, b, and c are the coefficients from our equation. In our case, a = 20, b = -41, and c = 20. Let's plug these values into the formula. This gives us: n = (41 ± √((-41)² - 4 * 20 * 20)) / (2 * 20). Simplifying this, we get: n = (41 ± √(1681 - 1600)) / 40. Which further simplifies to: n = (41 ± √81) / 40. The square root of 81 is 9, so we have: n = (41 ± 9) / 40. Now, we have two possible solutions, because of the ± sign. Let's calculate them separately. First, using the plus sign: n = (41 + 9) / 40 = 50 / 40 = 5/4. Second, using the minus sign: n = (41 - 9) / 40 = 32 / 40 = 4/5. Now, we have two possible values for 'n', 5/4 and 4/5. But remember, the problem stated that 'n' must be less than 1. So, let's analyze these answers to figure out which one fits the bill. The first solution is 5/4, which is 1.25. Since it's greater than 1, we can toss that one out. The second solution is 4/5, which is 0.8. Bingo! This number is less than 1 and fits all the criteria of our problem. This step of solving the quadratic formula is really about precision, following the rules, and keeping track of all the numbers. Don't worry if you need to double-check a few times. It's totally normal. After calculating this, we're almost at the finish line!

Finding the Value of 'n' as a Simplified Fraction

Alright, we've done it, guys! We've crunched the numbers and found the value of 'n'. From the previous step, we found that n = 4/5. This is already a fraction. But, can it be simplified? Let's check. A fraction is in its simplest form when the numerator (the top number) and the denominator (the bottom number) have no common factors other than 1. In our case, the numerator is 4 and the denominator is 5. The only factors of 4 are 1, 2, and 4, and the only factors of 5 are 1 and 5. Since the only common factor is 1, the fraction 4/5 is already in its simplest form. So, the value of n that Stanley was thinking of is 4/5. We did it! We have solved the problem, and we've expressed 'n' as a simplified fraction. We started with a word problem, transformed it into a mathematical equation, applied the necessary formulas, and finally arrived at the correct answer. Now, we've successfully navigated the mathematical maze, and we've done so by using a systematic approach, understanding the basic concepts, and performing each calculation with care. That feeling of accomplishment is worth celebrating, and it is also a reminder that with practice and persistence, any problem can be solved. Great job!

Conclusion: We Solved the Mystery!

And there you have it, folks! We've successfully solved the puzzle and found the value of 'n'. Remember, 'n' is a positive number less than 1, and when added to its reciprocal, it equals 2.05. The answer, in its simplest form, is 4/5. This journey was an excellent exercise in understanding equations, fractions, and how to use the quadratic formula. You've shown that you can translate a word problem into a mathematical expression and solve it step-by-step. Pat yourselves on the back, because you've conquered a math challenge! Feel proud of the work you've put in, and remember that with each problem you solve, you are building a stronger understanding of mathematics. Keep practicing, keep exploring, and keep the curiosity alive. Math can be fun, and with a bit of effort, you can solve any problem. If you ever come across a similar problem, you'll know exactly how to tackle it. You have the tools, the knowledge, and now the experience. Congratulations on completing this math adventure! We hope you enjoyed the ride, and we can’t wait to see you back for our next problem. Until then, keep those math muscles flexed, and keep exploring the amazing world of numbers. You are all math wizards now!