Solving For The Unknown Number A Step-by-Step Math Guide
Hey guys! Ever stumbled upon a math problem that looks like a puzzle? Today, we're going to tackle one of those together. We'll break it down step-by-step, so you'll not only get the answer but also understand how we got there. This isn't just about finding the right number; it's about building your problem-solving skills. So, let's jump right into it!
Understanding the Problem
Let's dive into the heart of the problem: "If you subtract 1/2 from a number and multiply the result by 1/2, you get 1/8. What is the number?" The initial step in unraveling any mathematical problem lies in thoroughly grasping its essence. We need to pinpoint precisely what the question is asking us to unearth. In this scenario, we are presented with a conundrum that involves a series of operations performed on an unknown number, eventually resulting in a known value. Our mission, should we choose to accept it, is to reverse-engineer the process and pinpoint the elusive original number.
The problem is elegantly crafted, presenting a sequence of actions performed on an unknown entity, which we'll soon christen with a variable. The narrative commences with the act of subtracting 1/2 from this mysterious number. This subtraction forms the bedrock of our journey, setting the stage for the subsequent operation. Following this subtraction, the plot thickens as we're instructed to multiply the outcome by 1/2. This multiplication serves as the pivotal link connecting our initial action to the ultimate revelation of 1/8. It's akin to a mathematical breadcrumb trail, guiding us from the unknown back to the known.
The grand finale of our problem statement unveils the crucial piece of information: the ultimate result of all these mathematical machinations is 1/8. This fraction serves as our anchor, the fixed point in our quest to solve for the unknown. It's the destination on our mathematical map, the place we're trying to reach by retracing our steps.
Therefore, the core of the problem boils down to this: we must discover the original number that, when subjected to the specified operations (subtracting 1/2 and multiplying by 1/2), yields the final result of 1/8. It's a bit like being handed a recipe and the finished dish, and then being asked to figure out the original ingredients and their quantities. To conquer this challenge, we'll need to employ our algebraic prowess, translating the words into a mathematical equation that we can then solve. This equation will be our vehicle, carrying us through the twists and turns of the problem until we arrive at our destination: the value of the unknown number. Are you ready to embark on this mathematical adventure with me? Let's roll up our sleeves and get started!
Translating Words into Math
Now, guys, let's translate those words into a mathematical equation. This is where the magic happens! We're going to take the problem and turn it into something we can actually solve. Think of it like learning a new language – in this case, the language of mathematics. It's all about understanding the symbols and how they relate to each other.
First things first, let's give our unknown number a name. In math, we often use the letter 'x' for this, but you can use any letter you like. It's just a placeholder until we figure out the actual value. So, let's say 'x' is our mystery number. Now, let’s go back to the problem statement: "If you subtract 1/2 from a number...". In mathematical terms, this translates directly to "x - 1/2". See how we're turning the words into symbols? It's like decoding a secret message!
Next up, we have "...and multiply the result by 1/2". This is super important – we're multiplying the entire result of our subtraction by 1/2. To show this in math, we need to use parentheses. So, it becomes "(x - 1/2) * (1/2)". The parentheses tell us to do the subtraction first, and then multiply the whole thing by 1/2. Think of it as the order of operations – we need to follow the rules to get the right answer.
Finally, we have "...you get 1/8". This is the easiest part to translate! "You get" simply means "equals", so we write "= 1/8". Now, let's put it all together. Our complete equation looks like this:
(x - 1/2) * (1/2) = 1/8
Boom! We've done it! We've taken a word problem and turned it into a mathematical equation. This is a huge step because now we can use our algebra skills to solve for 'x'. It's like we've built a bridge from the problem to the solution. This equation is our roadmap, guiding us through the steps we need to take to find our unknown number. So, give yourself a pat on the back – you've just mastered a key skill in math. Now, let's move on to the next exciting part: solving the equation!
Solving the Equation Step-by-Step
Alright, let's tackle the equation we just built: (x - 1/2) * (1/2) = 1/8. This is where the fun really begins because we get to put our problem-solving skills to the test. Don't worry if it looks a bit intimidating at first; we'll break it down into manageable steps, just like solving a puzzle.
Our main goal here is to isolate 'x' – to get it all by itself on one side of the equation. Think of it like untangling a knot; we need to carefully undo each step until we reveal what 'x' truly is. To do this, we'll use the magic of inverse operations. Inverse operations are like the opposite moves in a game; they undo what's already been done.
Looking at our equation, the first thing we see is that the entire expression (x - 1/2) is being multiplied by (1/2). So, to undo this, we need to do the opposite of multiplication, which is division. We're going to divide both sides of the equation by (1/2). Remember, whatever we do to one side of the equation, we have to do to the other to keep things balanced. It's like a seesaw – we need to keep the balance to maintain the equality.
Dividing by a fraction can be a bit tricky, but here's a cool trick: dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of (1/2) is (2/1), which is just 2. So, we're going to multiply both sides of the equation by 2. This gives us:
(x - 1/2) * (1/2) * 2 = (1/8) * 2
On the left side, the (1/2) and the 2 cancel each other out (they become 1), leaving us with:
x - 1/2 = 1/4
See how much simpler things are looking? We've gotten rid of the multiplication, and now we're one step closer to isolating 'x'. Now, we have "x - 1/2 = 1/4". We need to get rid of that - 1/2. The opposite of subtraction is addition, so we're going to add (1/2) to both sides of the equation:
x - 1/2 + 1/2 = 1/4 + 1/2
On the left side, the - 1/2 and the + 1/2 cancel each other out, leaving us with just 'x'. On the right side, we need to add (1/4) and (1/2). To add fractions, they need to have a common denominator. The least common denominator of 4 and 2 is 4, so we'll rewrite (1/2) as (2/4). This gives us:
x = 1/4 + 2/4
Now we can easily add the fractions:
x = 3/4
Woohoo! We've done it! We've solved for 'x'! This means that the original number we were looking for is 3/4. Give yourself a huge round of applause – you've conquered the equation! We've taken a complex-looking problem and broken it down into simple, manageable steps. Now, let's move on to the final step: checking our answer to make sure we've nailed it.
Checking the Solution
Okay, guys, we've arrived at a potential solution, x = 3/4, but our journey isn't quite over yet. In mathematics, just like in life, it's always a brilliant idea to double-check your work. Think of it as the final polish on a masterpiece, ensuring that every detail is perfect. In this case, checking our solution is going to confirm whether 3/4 truly fits the criteria laid out in the original problem. It's like fitting a key into a lock to see if it turns – we want to make sure our answer unlocks the mystery!
To verify our solution, we're going to take our value for 'x', which is 3/4, and plug it back into the original equation we crafted: (x - 1/2) * (1/2) = 1/8. This process is often called substitution, and it's a powerful tool for validating our answers.
So, let's replace 'x' with 3/4 in the equation. This gives us:
(3/4 - 1/2) * (1/2) = 1/8
Now, we need to simplify the left side of the equation, following the order of operations (PEMDAS/BODMAS). First up are the parentheses. We need to subtract (1/2) from (3/4). Just like before, to subtract fractions, we need a common denominator. The least common denominator of 4 and 2 is 4, so we'll rewrite (1/2) as (2/4). This gives us:
(3/4 - 2/4) * (1/2) = 1/8
Now we can subtract the fractions inside the parentheses:
(1/4) * (1/2) = 1/8
Next, we multiply (1/4) by (1/2). To multiply fractions, we simply multiply the numerators (the top numbers) and the denominators (the bottom numbers):
(1 * 1) / (4 * 2) = 1/8
1/8 = 1/8
Ta-da! Look at that! The left side of the equation simplifies to 1/8, which is exactly what we have on the right side. This means our equation balances perfectly, like a perfectly balanced scale. This is the moment of truth, guys – it confirms that our solution, x = 3/4, is indeed the correct answer. We've cracked the code, solved the puzzle, and emerged victorious!
So, give yourself another pat on the back, a high-five, or maybe even a little victory dance. You've not only solved a tricky math problem, but you've also reinforced the importance of checking your work. This is a skill that will serve you well in all areas of life, not just math. You've shown that with careful steps, perseverance, and a little bit of checking, you can conquer any mathematical challenge that comes your way. What's the number? We now know it's 3/4! Congratulations, math whizzes! Now, let’s celebrate our success and gear up for the next mathematical adventure!
