Solve: 5/8 Of A Number's Successor Equals Half Of 25

Hey guys! Today, let's dive into an interesting math problem that might seem a little tricky at first, but I promise, it's totally solvable. We're going to figure out a number where five-eighths of its successor is equal to half of 25. Sounds like a puzzle, right? Well, let's break it down step by step and make it super clear. So, grab your thinking caps, and let's get started!

Understanding the Problem

Okay, first things first, let's make sure we all understand what the problem is asking. The core of the problem, finding the number, lies in decoding the sentence: "The five-eighths of the successor of a number is equal to half of 25." The key components we need to focus on are: the successor of a number, which means the number plus one; five-eighths, which is a fraction (5/8); and half of 25, which is simply 25 divided by 2. To tackle this, we'll use a bit of algebra, which is just a fancy way of using letters to represent numbers we don't know yet.

The most effective method for solving this kind of problem is to convert the word problem into a mathematical equation. This involves assigning a variable (usually 'x') to the unknown number. The "successor of the number" will then be 'x + 1'. Next, we translate "five-eighths of the successor" into (5/8) * (x + 1). The phrase "equal to half of 25" translates to 25/2. Putting it all together, we get the equation (5/8) * (x + 1) = 25/2. This equation is our roadmap to solving the problem. By solving for 'x', we'll find the number that fits the conditions described in the problem. It’s like we’re translating from English to Math, turning a sentence into a solvable code!

Now that we've got our equation, it's all about solving it. Stick with me, and you'll see how straightforward it can be. Remember, the goal is to isolate 'x' on one side of the equation, so we know exactly what number it represents. To do that, we're going to use some algebraic techniques, which are just tools to help us keep the equation balanced while we simplify it. We will first multiply both sides of the equation by 8/5 to eliminate the fraction on the left side, then we will subtract 1 from both sides. After that, we will have the value of x. This is like peeling an onion, layer by layer, until we get to the core – which, in this case, is the value of 'x'.

Setting Up the Equation

So, let's dive into the nitty-gritty of turning this word problem into a neat, solvable equation. The first step in setting up the equation is assigning a variable. Let's call our mystery number "x". This is classic algebra – we use a letter to stand in for a value we don't know yet. Now, when the problem mentions the "successor of a number," that simply means the number plus one. So, the successor of x is x + 1. This part is crucial because it sets the stage for the rest of our equation. We're building a mathematical sentence that captures the essence of the problem.

Next, we need to translate "five-eighths of the successor" into math terms. "Five-eighths" is straightforward – it's the fraction 5/8. And in math, "of" often means multiplication. So, "five-eighths of the successor" translates to (5/8) multiplied by (x + 1), which we write as (5/8)(x + 1). This is where the problem starts to take shape as an equation. We're taking pieces of the word problem and turning them into mathematical expressions. It's like translating a sentence from one language to another, but in this case, we're going from English to Math!

Now, let's tackle the other side of the equation. The problem says this whole expression is "equal to half of 25." "Half of 25" is simply 25 divided by 2, which is 25/2. So, we've got the two halves of our equation. We know that (5/8)(x + 1) is equal to 25/2. This is the final piece of the puzzle. By carefully translating each part of the word problem, we've constructed a complete equation that we can solve for x. It's like building a bridge – we've laid the foundation, and now we're ready to cross over to the solution.

Putting it all together, we get our equation: (5/8)(x + 1) = 25/2. This is the mathematical representation of our word problem. It’s like having a treasure map – the equation shows us the path to finding the value of x. This equation is the key to unlocking the mystery number, and now that we have it, we’re ready for the next step: solving it. Think of this as the blueprint for solving the problem. We've laid out all the information in a clear, mathematical form, and now we can use our algebra skills to find the answer.

Solving the Equation

Alright, guys, we've got our equation all set up: (5/8)(x + 1) = 25/2. Now comes the fun part – actually solving it! Remember, our goal here is to get 'x' all by itself on one side of the equation. To do this, we're going to use some basic algebraic techniques. Think of it like a puzzle where we need to carefully move pieces around without upsetting the balance. The first step in solving the equation is to eliminate the fraction on the left side. We can do this by multiplying both sides of the equation by the reciprocal of 5/8, which is 8/5.

Why do we do this? Well, multiplying by the reciprocal is like undoing the fraction. When we multiply (5/8)(x + 1) by 8/5, the 5/8 and 8/5 cancel each other out, leaving us with just (x + 1) on the left side. But remember, in algebra, whatever we do to one side of the equation, we have to do to the other side to keep things balanced. So, we also multiply 25/2 by 8/5. This step is crucial because it simplifies the equation significantly. It's like clearing away the clutter so we can see the solution more clearly.

So, after multiplying both sides by 8/5, our equation looks like this: x + 1 = (25/2) * (8/5). Now, let's simplify the right side of the equation. When we multiply fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, (25/2) * (8/5) becomes (25 * 8) / (2 * 5), which equals 200/10. And 200/10 simplifies to 20. This is where the arithmetic comes in. We're doing the calculations to make the equation easier to manage. It's like crunching the numbers to reveal the hidden value.

Now our equation is even simpler: x + 1 = 20. We're so close to finding x! To get x by itself, we need to get rid of the +1 on the left side. We can do this by subtracting 1 from both sides of the equation. This is another fundamental algebraic move – using inverse operations to isolate the variable. So, x + 1 - 1 = 20 - 1, which simplifies to x = 19. And there we have it! We've solved for x. It's like we've followed the clues and finally arrived at the treasure.

So, x = 19 is our solution. This means that the number we were looking for is 19. We've gone from a word problem to an equation and then solved it step by step. Now, let's make sure our answer makes sense in the context of the original problem. This final step is important to make sure we haven't made any mistakes along the way. It’s like checking our map to make sure we’ve reached the right destination. We started with a puzzle, and now we've successfully pieced it together!

Checking the Solution

We've arrived at our solution, x = 19, but before we declare victory, it's super important to check our work. Think of this as the final exam for our solution – we want to make sure it holds up under scrutiny. Plugging our solution back into the original problem helps us ensure that we haven't made any sneaky errors along the way. It's like double-checking our calculations on a calculator, just to be sure.

So, let's go back to the original problem: "The five-eighths of the successor of a number is equal to half of 25." We found that the number is 19. The successor of 19 is 19 + 1, which equals 20. Now, we need to find five-eighths of 20. To do this, we multiply 20 by 5/8. So, (5/8) * 20 equals (5 * 20) / 8, which is 100/8. And 100/8 simplifies to 12.5. We're retracing our steps, but this time with a specific number in mind.

Next, we need to check if this result is equal to half of 25. Half of 25 is 25/2, which is also 12.5. So, we've found that five-eighths of the successor of 19 (which is 12.5) is indeed equal to half of 25 (which is also 12.5). This is the moment of truth! We're comparing our calculated value with the problem's condition.

Since both sides of the equation match up, we can confidently say that our solution, x = 19, is correct. We've successfully navigated the problem, solved the equation, and verified our answer. This step is not just about confirming our answer; it's about solidifying our understanding of the problem and the process we used to solve it. It’s like putting the final piece in a jigsaw puzzle – everything clicks into place, and we can see the whole picture.

Checking the solution is a crucial part of problem-solving in mathematics. It ensures accuracy and reinforces our understanding of the concepts involved. By verifying our solution, we not only confirm that we've found the right answer but also gain confidence in our problem-solving abilities. So, always remember to check your work – it's the best way to ensure success in math!

Conclusion

So, there you have it, guys! We've successfully navigated through this math problem, turning a word puzzle into a clear equation and finding our answer. We started with the challenge of figuring out a number where five-eighths of its successor equals half of 25. By breaking the problem down step by step, we transformed it into an algebraic equation: (5/8)(x + 1) = 25/2. This is the power of math – taking complex ideas and simplifying them into manageable steps.

We then solved this equation by carefully applying algebraic techniques. We multiplied both sides by the reciprocal of 5/8 to eliminate the fraction, simplified the equation, and isolated 'x' to find its value. Through these steps, we discovered that x = 19 is the solution to our equation. This process showcases the beauty of algebra – using tools and techniques to unravel the unknown.

But we didn't stop there! We checked our solution by plugging it back into the original problem. This crucial step confirmed that our answer was correct and gave us confidence in our problem-solving process. This is a vital lesson in mathematics – always verify your solution to ensure accuracy and understanding. It’s like proofreading an essay to catch any errors.

Through this exercise, we've not only solved a specific math problem but also reinforced important concepts in algebra and problem-solving strategies. We've seen how to translate word problems into equations, solve those equations, and verify our solutions. These are skills that will serve us well in many areas of mathematics and beyond. Math isn't just about numbers; it's about logical thinking and problem-solving.

So, the next time you encounter a tricky math problem, remember the steps we've covered today. Break the problem down, translate it into an equation, solve the equation carefully, and always check your answer. With practice and persistence, you'll be able to tackle any mathematical challenge that comes your way. Keep practicing, keep exploring, and most importantly, keep enjoying the world of math! You guys nailed it!