Finding The Original Number After Subtraction And Multiplication A Step-by-Step Solution

Have you ever encountered a mathematical puzzle that seems to twist your brain into knots? These types of mathematical problems can be both challenging and rewarding, and this article will show you how to solve one of these puzzles by applying logic and basic algebraic principles.

Decoding the Puzzle: Subtracting, Multiplying, and Finding the Original Number

Let's dive into the problem. We are given a series of operations performed on an unknown number, and our mission is to backtrack these steps to reveal the original value. The puzzle unfolds as follows: First, we subtract 10 from a certain number. Then, we multiply the result by 6. And finally, we arrive at the answer 84. The challenge now is to discover the original number that set this mathematical journey in motion.

This type of problem is a staple in basic algebra, serving as an excellent introduction to the world of equations and variables. It allows us to practice the crucial skill of working backward through a series of operations, a technique that is widely applicable in various mathematical and scientific fields. By carefully dissecting the puzzle and employing reverse operations, we can successfully isolate the unknown and unveil its true value. So, let's embark on this mathematical quest together and unlock the secret of the original number.

Transforming the Puzzle into an Equation: The Language of Algebra

To effectively tackle this puzzle, we need to translate the words into the language of mathematics – an equation. Equations are the cornerstone of algebra, providing a concise and powerful way to represent relationships between numbers and variables. In this case, our puzzle involves an unknown number, which we can represent using a variable, say 'x'. This simple step of assigning a variable to the unknown is the key to unlocking the solution.

Now, let's carefully dissect the puzzle's instructions and translate them into algebraic expressions. The first operation is subtracting 10 from the number, which can be written as 'x - 10'. The next operation involves multiplying the result by 6, represented as '6 * (x - 10)'. Finally, we are told that the final answer is 84, which completes our equation: '6 * (x - 10) = 84'.

This equation now encapsulates the entire puzzle in a compact and symbolic form. It's a powerful tool that allows us to manipulate the problem algebraically and systematically isolate the variable 'x'. This is the essence of algebra – transforming word problems into manageable equations that can be solved using established rules and techniques. Now that we have our equation, we are ready to embark on the next step: solving for 'x'.

Undoing the Operations: Solving the Equation Step-by-Step

With our equation '6 * (x - 10) = 84' in hand, we are now ready to solve for 'x', the original number. The key to solving equations lies in the principle of inverse operations – performing the opposite operation to isolate the variable. In this case, we have a series of operations acting on 'x', and we need to systematically undo them.

The first operation to undo is the multiplication by 6. To do this, we divide both sides of the equation by 6. This maintains the balance of the equation while simplifying it. Dividing both sides by 6 gives us: '(6 * (x - 10)) / 6 = 84 / 6', which simplifies to 'x - 10 = 14'. Notice how we have successfully peeled away the multiplication, bringing us closer to isolating 'x'.

Now, we are left with a simpler equation: 'x - 10 = 14'. The only remaining operation is the subtraction of 10. To undo this, we perform the inverse operation: addition. We add 10 to both sides of the equation, ensuring that the balance is preserved. This gives us: 'x - 10 + 10 = 14 + 10', which simplifies to 'x = 24'.

And there we have it! Through the systematic application of inverse operations, we have successfully isolated 'x' and found its value. The original number, the solution to our puzzle, is 24. This step-by-step approach is the essence of solving algebraic equations, and it's a skill that becomes more refined with practice.

Verifying the Solution: Ensuring Accuracy in Our Answer

In mathematics, it's always a good practice to verify our solutions. This ensures that we haven't made any errors in our calculations and that our answer truly satisfies the conditions of the problem. Verification is like a final checkmark, giving us confidence in our result. In this case, we found that the original number is 24. To verify, we'll plug this value back into the original puzzle's instructions and see if we arrive at the final answer of 84.

Let's retrace the steps with x = 24. First, we subtract 10 from 24, which gives us 24 - 10 = 14. Then, we multiply the result by 6: 14 * 6 = 84. And indeed, we arrive at the final answer given in the puzzle! This confirms that our solution, x = 24, is correct.

Verification is not just about getting the right answer; it's about developing a habit of critical thinking and ensuring the accuracy of our work. It reinforces our understanding of the problem and the steps we took to solve it. So, always remember to verify your solutions whenever possible – it's a valuable skill that will serve you well in mathematics and beyond.

Embracing the Power of Algebra: Real-World Applications and Beyond

This puzzle, while seemingly simple, exemplifies the power and versatility of algebra. Algebra is not just an abstract set of rules and symbols; it's a language for describing and solving real-world problems. The ability to translate situations into equations and manipulate them to find solutions is a fundamental skill that has countless applications in various fields.

From engineering and physics to economics and computer science, algebra is the foundation upon which many disciplines are built. Engineers use algebraic equations to design structures and circuits, physicists use them to model the behavior of matter and energy, economists use them to analyze markets and predict trends, and computer scientists use them to develop algorithms and software. The possibilities are endless.

Moreover, the problem-solving skills honed through algebra extend far beyond the realm of mathematics. The logical thinking, analytical reasoning, and systematic approach that we develop in solving algebraic problems are transferable to many aspects of life. Whether we're planning a project, making a decision, or troubleshooting a problem, the ability to break down complex situations, identify key variables, and devise a solution is invaluable.

So, the next time you encounter an algebraic puzzle, remember that you're not just solving for 'x'; you're honing skills that will empower you in countless ways. Embrace the power of algebra, and unlock a world of possibilities.

Conclusion: The Original Number Revealed and the Beauty of Problem-Solving

In this article, we embarked on a mathematical journey to unravel a puzzle involving subtracting, multiplying, and ultimately finding the original number. By translating the puzzle into an algebraic equation, systematically solving it using inverse operations, and verifying our solution, we successfully discovered that the original number is 24.

This exercise highlights the beauty and power of problem-solving in mathematics. It demonstrates how we can take a seemingly complex situation, break it down into manageable steps, and arrive at a solution through logical reasoning and algebraic techniques. The satisfaction of cracking a mathematical puzzle is akin to solving a real-world challenge, and the skills we develop along the way are invaluable.

So, keep practicing, keep exploring, and keep embracing the joy of problem-solving. Whether it's a mathematical puzzle or a real-life challenge, the ability to think critically, analyze situations, and devise solutions will always be a valuable asset. And remember, every solved problem is a step forward in your journey of learning and growth. Let's continue to explore the fascinating world of mathematics and unlock its endless possibilities!