Solve Math Problem: Sum Of 1/10 And 1/100 Is 770

Have you ever stumbled upon a math problem that seems like a riddle wrapped in an enigma? This is one of those! We're tasked with finding a number, but not just any number – a number whose one-tenth and one-hundredth, when added together, magically equal 770. Sounds intriguing, right? Let's put on our detective hats and dive into the world of fractions, equations, and problem-solving strategies to crack this numerical puzzle.

Understanding the Core Concept

Before we jump into calculations, let's make sure we understand what the question is asking. The key here is fractions: one-tenth (1/10) and one-hundredth (1/100). We need to visualize what these fractions represent in relation to the whole number we're trying to find. Think of it like slicing a pie: one-tenth is a larger slice than one-hundredth. Our mission is to figure out the size of the whole pie, knowing that the combined size of these two slices is 770. To effectively tackle this problem, we need to translate the words into a mathematical equation. This involves representing the unknown number with a variable (let's use 'x'), and then expressing the given information as a relationship between these fractions and the number 770. This is where our algebraic skills come into play, allowing us to manipulate the equation and isolate the variable, ultimately revealing the solution. Remember, the beauty of mathematics lies in its ability to transform real-world scenarios into abstract representations, enabling us to solve complex problems with logical precision.

The Art of Equation Formulation

Now, let’s translate the problem into a mathematical equation. This is a crucial step because a well-formed equation is half the battle won. We'll represent the unknown number as 'x'. One-tenth of x can be written as (1/10)x or x/10, and one-hundredth of x is (1/100)x or x/100. The problem states that the sum of these two fractions equals 770. Therefore, our equation looks like this:

x/10 + x/100 = 770

This equation is the heart of our problem. It encapsulates all the information we have in a concise mathematical statement. Now, our focus shifts to solving this equation for 'x'. This involves algebraic manipulation, a set of techniques used to isolate the variable on one side of the equation. We'll need to find a common denominator for the fractions, combine like terms, and then perform operations on both sides of the equation to get 'x' by itself. The goal is to maintain the equality while simplifying the equation step by step, until we arrive at a clear value for our unknown number. This process highlights the elegance and power of algebra, allowing us to solve for unknowns in a systematic and logical way.

Solving the Equation: A Step-by-Step Journey

Here’s where the fun begins – solving the equation! Our equation is:

x/10 + x/100 = 770

To add the fractions, we need a common denominator. The least common multiple of 10 and 100 is 100. So, we'll convert x/10 to have a denominator of 100 by multiplying both the numerator and denominator by 10:

(10x/100) + x/100 = 770

Now, we can combine the fractions:

(10x + x) / 100 = 770

11x / 100 = 770

To isolate 'x', we'll multiply both sides of the equation by 100:

11x = 770 * 100

11x = 77000

Finally, we divide both sides by 11:

x = 77000 / 11

x = 7000

And there you have it! The number we've been searching for is 7000. This step-by-step solution demonstrates the power of algebraic manipulation in unraveling mathematical mysteries. Each step builds upon the previous one, guided by the principles of equality and the goal of isolating the unknown variable. The journey from the initial equation to the final answer showcases the methodical and logical nature of mathematics.

The Grand Verification: Checking Our Answer

But wait, we're not quite done yet! It's crucial to verify our answer to make sure we haven't made any mistakes along the way. This is a critical step in problem-solving, ensuring that our solution is not only mathematically sound but also logically consistent with the original problem statement. To verify, we'll plug our solution, x = 7000, back into the original equation:

(7000/10) + (7000/100) = 770

Let's simplify:

700 + 70 = 770

770 = 770

Success! The equation holds true. This verification step not only confirms the correctness of our solution but also reinforces our understanding of the problem and the steps we took to solve it. It's like the final piece of a puzzle clicking into place, providing a sense of accomplishment and solidifying our confidence in our mathematical abilities.

Real-World Relevance: Why This Matters

Now, you might be wondering, "Why is this type of problem important in the real world?" Well, the principles we've used here – understanding fractions, setting up equations, and solving for unknowns – are fundamental to many real-life situations. From calculating proportions in recipes to determining financial investments, these skills are invaluable. Imagine you're trying to figure out a discount on a sale item or calculating the percentage of ingredients in a mixture. These scenarios all involve mathematical concepts similar to the one we've tackled today. The ability to translate real-world problems into mathematical models and solve them empowers us to make informed decisions and navigate the complexities of everyday life. By mastering these skills, we not only become better problem-solvers but also more confident and capable individuals.

Conclusion: The Joy of Mathematical Discovery

So, we've successfully navigated the world of fractions and equations to uncover the mystery number: 7000. This problem, while seemingly abstract, highlights the power and elegance of mathematics in solving puzzles and real-world challenges. Remember, the key to tackling these types of problems is to break them down into smaller, manageable steps, translate them into mathematical language, and then apply the appropriate techniques. And most importantly, don't forget to verify your answer! Math isn't just about finding the right answer; it's about the journey of discovery and the satisfaction of knowing you've cracked the code. Keep practicing, keep exploring, and keep enjoying the beautiful world of mathematics!

For more practice and deeper understanding of algebraic equations, you might find helpful resources on websites like Khan Academy Algebra.