Find The Two-Digit Number Where The Sum Of Its Digits Is One-Sixth Of The Number
Introduction
Hey guys! Ever stumbled upon a math problem that seems like a riddle wrapped in an enigma? Well, today, we're diving headfirst into one such brain-teaser. Our mission, should we choose to accept it (and we totally do!), is to find a two-digit number that has a unique property. This property? The sum of its digits must be equal to one-sixth of the number itself. Sounds intriguing, right? Let's put on our detective hats and embark on this mathematical adventure together. We'll break down the problem step by step, explore different strategies, and finally, crack the code to reveal the hidden number. So, buckle up, math enthusiasts, because we're about to unravel a fascinating numerical mystery!
This isn't just about finding the right answer; it's about the journey of problem-solving. We'll delve into the logic behind the solution, understand the concepts involved, and sharpen our mathematical intuition along the way. Think of it as a mental workout, a chance to flex our brain muscles and expand our mathematical horizons. So, let's dive in and discover the magic of numbers together!
Understanding the Problem
Okay, before we jump into calculations, let's make sure we truly understand what the problem is asking. The core of the problem revolves around a two-digit number. Remember, a two-digit number is any number from 10 to 99. Each of these numbers has two digits: a tens digit and a units digit. For example, in the number 42, the tens digit is 4, and the units digit is 2. The problem states a crucial relationship between these digits and the number itself. The sum of these digits must be equal to one-sixth of the original two-digit number. This is the key piece of information that will guide our search. Let's break this down further with an example. Suppose we have a number, say 36. The sum of its digits is 3 + 6 = 9. Now, one-sixth of 36 is 36 / 6 = 6. Since 9 is not equal to 6, 36 is not our answer. This simple check helps us understand the condition we need to satisfy.
To solve this, we need to find a number where the sum of its digits perfectly aligns with one-sixth of its value. This requires a bit of algebraic thinking and systematic checking. The beauty of this problem lies in its simplicity and the elegance of the solution. It's a perfect example of how basic mathematical principles can be used to solve intriguing puzzles. So, let's keep this core understanding in mind as we explore different approaches to finding our mystery number.
Exploring Possible Strategies
Now that we've got a firm grasp on the problem, let's brainstorm some strategies to tackle it. There are a couple of main approaches we can take. One method is a bit more hands-on: we can systematically test the given options. We have a list of potential answers (42, 54, 72, 64, and 96), so we can go through each one, calculate the sum of its digits, find one-sixth of the number, and see if they match. This is a straightforward, brute-force approach, but it's reliable and ensures we find the correct answer if it's among the options. Another strategy involves a more algebraic approach. We can represent the two-digit number using variables and form an equation based on the problem's condition. This might sound a bit intimidating, but it's a powerful technique that can help us solve a wide range of mathematical problems. Let's say the tens digit is 'x' and the units digit is 'y'. Then, the two-digit number can be represented as 10x + y (since the tens digit contributes 10 times its value). The sum of the digits is simply x + y. The problem states that x + y = (1/6)(10x + y). This equation captures the essence of the problem in a concise mathematical form.
By solving this equation, we can find the values of x and y, which will reveal our two-digit number. Both strategies have their merits, and the best approach often depends on personal preference and the specific problem. In this case, since we have multiple-choice options, the systematic testing method might be the quickest way to find the answer. However, understanding the algebraic approach is valuable for developing our problem-solving skills and tackling more complex problems in the future. So, let's keep both strategies in mind as we move forward and start crunching some numbers!
Method 1: Testing the Options
Alright, let's get our hands dirty and start testing those options! We've got five potential answers staring back at us: 42, 54, 72, 64, and 96. Our strategy here is simple but effective. For each number, we'll follow these steps: First, we'll calculate the sum of its digits. Then, we'll find one-sixth of the number. Finally, we'll compare the two results. If they match, we've found our mystery number! If not, we move on to the next option. Let's start with 42. The sum of the digits is 4 + 2 = 6. One-sixth of 42 is 42 / 6 = 7. Since 6 is not equal to 7, 42 is not our answer. On to the next one! Let's try 54. The sum of the digits is 5 + 4 = 9. One-sixth of 54 is 54 / 6 = 9. Bingo! The sum of the digits and one-sixth of the number are the same. It looks like we've found our answer. But just to be sure, let's quickly check the remaining options. For 72, the sum of the digits is 7 + 2 = 9. One-sixth of 72 is 72 / 6 = 12. Not a match. For 64, the sum of the digits is 6 + 4 = 10. One-sixth of 64 is 64 / 6 = 10.67 (approximately). Again, no match. Finally, for 96, the sum of the digits is 9 + 6 = 15. One-sixth of 96 is 96 / 6 = 16. Nope, not this one either. So, after systematically testing each option, we've confirmed that 54 is indeed the two-digit number we were looking for. This method demonstrates the power of direct application and methodical checking in problem-solving.
Method 2: The Algebraic Approach
Now, let's switch gears and explore the algebraic approach. This method might seem a bit more abstract, but it offers a deeper understanding of the problem's structure and can be incredibly useful for solving similar challenges in the future. Remember, the key is to represent the problem's information using variables and equations. Let's break down the steps. First, we need to represent our two-digit number algebraically. Let's say the tens digit is represented by the variable 'x' and the units digit by the variable 'y'. The value of the two-digit number is then 10x + y. Think about it: if x is 4 and y is 2, the number is 10(4) + 2 = 42. Next, we need to express the sum of the digits, which is simply x + y. The problem states that the sum of the digits is equal to one-sixth of the number itself. This translates to the equation x + y = (1/6)(10x + y). Now, we have a beautiful algebraic equation that captures the essence of our problem. Our goal is to solve this equation for x and y. To do this, let's first get rid of the fraction by multiplying both sides of the equation by 6: 6(x + y) = 10x + y. Expanding the left side gives us 6x + 6y = 10x + y. Now, let's rearrange the terms to group the x's and y's together. Subtracting 6x from both sides gives us 6y = 4x + y. Then, subtracting y from both sides gives us 5y = 4x. This is a simpler equation that relates x and y. Now, we need to find integer values for x and y that satisfy this equation. Remember, x and y are digits, so they must be whole numbers between 0 and 9. We can rewrite the equation as y = (4/5)x. This tells us that y is four-fifths of x. Since y must be a whole number, x must be a multiple of 5. The possible values for x are therefore 0 and 5. If x is 0, then y is also 0, which doesn't give us a two-digit number. If x is 5, then y = (4/5) * 5 = 4. So, we have x = 5 and y = 4. This means our two-digit number is 10x + y = 10(5) + 4 = 54. Voila! We've arrived at the same answer using the algebraic approach. This method not only solves the problem but also provides a deeper understanding of the relationships between the digits and the number itself. It's a testament to the power of algebra in problem-solving.
Conclusion
And there you have it, folks! We've successfully navigated the mathematical puzzle and found our two-digit number: 54. We explored two different strategies to crack this problem: the direct approach of testing the options and the more abstract but powerful algebraic method. The direct method allowed us to quickly verify each option and arrive at the answer through systematic checking. This approach is particularly useful when dealing with multiple-choice questions, as it provides a straightforward way to eliminate incorrect answers and pinpoint the correct one. On the other hand, the algebraic approach provided us with a deeper understanding of the problem's underlying structure. By representing the problem using variables and equations, we were able to establish a relationship between the digits and the number itself. This not only helped us find the solution but also enhanced our problem-solving skills and our ability to tackle similar challenges in the future. The beauty of mathematics lies in its versatility and the multiple ways we can approach a single problem. Both methods we used are valuable tools in our mathematical arsenal, and the choice of which one to use often depends on the specific problem and our personal preferences.
This exercise wasn't just about finding the right answer; it was about the journey of problem-solving. We honed our logical thinking, sharpened our algebraic skills, and reinforced the importance of understanding the problem before attempting to solve it. So, the next time you encounter a mathematical puzzle, remember the strategies we've discussed today. Embrace the challenge, explore different approaches, and most importantly, enjoy the process of discovery! Keep those brain cells firing, guys!
