Calculating The Sum Of Digits Of A Number A Mathematical Exploration
Let's dive into a fun mathematical problem! We're tasked with finding a number, but not just any number. This number has a special condition: when its quintuple (five times the number) is increased by 624, the result is 2714. Once we find this mysterious number, we need to calculate the sum of its digits. Sounds like a plan? Awesome, let's get started!
Setting up the Equation
To crack this mathematical puzzle, our first step is to translate the word problem into a mathematical equation. This is a crucial skill in algebra, and it makes solving problems like these much easier. Let's break down the information we have: we're looking for a number, let's call it x. The problem states that five times this number (its quintuple) increased by 624 equals 2714. We can write this as:
5x + 624 = 2714
Now we have a neat little equation that we can solve for x. This equation represents the core relationship described in the problem, and solving it will reveal the number we're after. The beauty of algebra lies in its ability to transform complex word problems into concise and solvable equations. It's like having a secret code that unlocks the answer! By setting up the equation correctly, we've laid a solid foundation for the rest of the solution. This initial step is often the most critical, as an incorrect equation will lead to a wrong answer. So, always double-check your setup to ensure it accurately reflects the problem's conditions. Now that we have our equation, let's move on to the next step: isolating the variable and finding the value of x. Remember, the key is to maintain balance on both sides of the equation as we perform our operations. This is the golden rule of algebra, and it ensures that our solution remains accurate. With our equation in place, we're well on our way to uncovering the mystery number and calculating the sum of its digits.
Solving for the Unknown Number
Now comes the fun part: solving the equation for x. Our equation, as we established, is 5x + 624 = 2714. To find x, we need to isolate it on one side of the equation. Think of it like peeling away the layers of an onion – we'll undo the operations one by one until we're left with just x. The first step is to get rid of the 624. Since it's being added to 5x, we'll subtract 624 from both sides of the equation. Remember, what we do to one side, we must do to the other to keep the equation balanced. This gives us:
5x + 624 - 624 = 2714 - 624
Simplifying this, we get:
5x = 2090
Great! We're one step closer. Now, x is being multiplied by 5. To undo this, we'll divide both sides of the equation by 5:
(5x) / 5 = 2090 / 5
This simplifies to:
x* = 418
Eureka! We've found our number. x is equal to 418. This means that the mystery number we've been searching for is 418. But our journey isn't over yet. The original problem asked us to calculate the sum of the digits of this number. So, we have one more step to complete. But before we move on, let's just appreciate the power of algebra for a moment. We took a word problem, translated it into an equation, and then systematically solved for the unknown. It's like being a detective, using clues to uncover the truth. And in this case, the truth is that the number is 418. Now, let's move on to the final step and calculate the sum of its digits.
Calculating the Sum of Digits
Alright, we've discovered that our number is 418. The final piece of the puzzle is to calculate the sum of its digits. This is a straightforward process, but it's important to be careful and accurate. We simply need to add up each digit in the number. Our number, 418, has three digits: 4, 1, and 8. So, to find the sum, we add them together:
4 + 1 + 8 = 13
And there we have it! The sum of the digits of the number 418 is 13. We've successfully completed our mission. We started with a word problem, transformed it into an algebraic equation, solved for the unknown number, and then calculated the sum of its digits. It's a testament to the power of mathematical thinking and problem-solving skills. Each step built upon the previous one, leading us to the final answer. And that's the beauty of mathematics – it's a logical and systematic approach to unraveling mysteries and finding solutions. So, next time you encounter a mathematical challenge, remember the steps we've taken here. Break the problem down, set up an equation, solve for the unknown, and then perform any necessary calculations. With practice and perseverance, you'll become a mathematical whiz in no time! Now, let's take a moment to reflect on our journey and appreciate the elegance of the solution we've found. We've not only solved a problem, but we've also reinforced our understanding of algebraic principles and problem-solving strategies.
Conclusion
So, to recap, we started with a mathematical word problem, translated it into an equation (5x + 624 = 2714), solved for x (finding x = 418), and finally, calculated the sum of the digits of 418, which is 13. This entire process showcases the power of mathematical thinking and problem-solving. We've demonstrated how to break down a complex problem into smaller, manageable steps, and how to apply algebraic principles to find a solution. The key takeaways here are the importance of translating word problems into equations, the systematic approach to solving equations, and the careful execution of calculations. Remember, mathematics is not just about numbers and formulas; it's about logical reasoning and problem-solving skills that can be applied in various aspects of life. By mastering these skills, we become better equipped to tackle challenges and make informed decisions. So, keep practicing, keep exploring, and keep enjoying the beauty and power of mathematics! We've successfully navigated this mathematical puzzle, and hopefully, you've gained a deeper appreciation for the elegance and utility of mathematics. Remember, every problem is an opportunity to learn and grow, and with the right approach, even the most challenging problems can be solved. So, embrace the challenge, sharpen your skills, and continue your mathematical journey with confidence and enthusiasm. And who knows, maybe you'll even inspire others to join in the fun of mathematical exploration!
