Solving Number Puzzles Finding Three Unknown Numbers
Hey guys, ever stumbled upon a math problem that seems like a puzzle? Well, let’s break down one of those puzzles together! This is a classic algebra problem that involves setting up equations and solving for unknowns. It might sound intimidating, but trust me, we’ll make it super clear and easy to understand.
Understanding the Problem
Okay, so the problem states: “One number is 5 times a first number. A third number is 100 more than the first number. If the sum of the three numbers is 611, find the numbers.” Sounds like a riddle, right? But let’s translate this into math we can work with. The key here is to identify the unknowns and how they relate to each other. The first step in tackling this problem is to break it down into manageable parts. We need to identify the key pieces of information and what we're trying to find. In this case, we're dealing with three numbers, and we have some clues about how they relate to each other. We know that one number is five times another, and a third number is 100 more than the first. Our goal is to find the values of these three numbers, given that their sum is 611. This is where algebra comes in handy. By assigning variables to the unknowns and setting up equations, we can systematically solve for the values we need. Remember, the beauty of algebra lies in its ability to transform word problems into mathematical expressions, making them easier to solve. So, let's dive in and see how we can use algebra to crack this number puzzle.
Defining Our Variables
The most important thing to do when starting an algebra problem like this is to define our variables. This means assigning a letter to represent each unknown number. Let’s keep it simple:
- Let's call the first number “x”.
- Now, the problem says “one number is 5 times a first number.” So, the second number will be “5x”.
- The problem also says “a third number is 100 more than the first number.” So, the third number is “x + 100”.
See? We've turned words into algebraic expressions! By carefully defining our variables, we've laid the groundwork for solving the problem. This is a crucial step in algebra, as it allows us to translate the word problem into mathematical equations. Think of it as creating a roadmap for our solution. Each variable represents a piece of the puzzle, and by understanding how these variables relate to each other, we can start to see the bigger picture. Now that we have our variables defined, we can move on to the next step: setting up an equation that represents the information given in the problem. This will involve using the fact that the sum of the three numbers is 611. So, let's get ready to transform our variables into an equation and continue our journey toward finding the solution.
Setting Up the Equation
Now comes the crucial part where we build our equation. We know that the sum of the three numbers is 611. So, we can write this as:
x + 5x + (x + 100) = 611
This equation is the heart of our problem. It represents the relationship between our three numbers and the total sum. It's like a blueprint that guides us toward the solution. Each term in the equation corresponds to one of our numbers, and the equal sign signifies that the total value on the left side is the same as the value on the right side. Setting up the equation correctly is paramount, as it ensures that we're working with an accurate representation of the problem. A small mistake here can lead to a completely wrong answer. But don't worry, we've got this! We've carefully defined our variables and translated the word problem into a mathematical statement. Now, the next step is to simplify and solve this equation. This will involve combining like terms and isolating the variable x. So, let's roll up our sleeves and get ready to do some algebraic maneuvering!
Solving the Equation
Time to put on our algebra hats and solve this equation! The first thing we want to do is simplify the equation by combining like terms. This means adding together all the 'x' terms:
x + 5x + x + 100 = 611 becomes 7x + 100 = 611
See how we combined the x's? Now it looks a bit cleaner. Combining like terms is a fundamental step in solving equations, and it's essential for simplifying complex expressions. Think of it as organizing your tools before starting a project – it makes the whole process smoother and more efficient. By grouping similar terms together, we reduce the complexity of the equation and make it easier to work with. In our case, we combined the x terms to get 7x, which simplifies the equation significantly. Now that we've combined like terms, we can move on to the next step: isolating the variable x. This will involve performing inverse operations to get x by itself on one side of the equation. So, let's continue our journey toward finding the value of x.
Isolating 'x'
Our goal now is to get 'x' all by itself on one side of the equation. To do this, we need to undo the operations that are being done to 'x'. Right now, we have “7x + 100 = 611”. The first thing to undo is the “+ 100”. We do this by subtracting 100 from both sides of the equation:
7x + 100 - 100 = 611 - 100, which simplifies to 7x = 511
Remember, whatever we do to one side of the equation, we have to do to the other to keep it balanced. This is a crucial concept in algebra, as it ensures that the equation remains true throughout the solving process. It's like maintaining equilibrium on a seesaw – if you add or remove weight from one side, you need to adjust the other side accordingly. In our case, we subtracted 100 from both sides to isolate the term with x. Now, we're one step closer to finding the value of x. The next step is to undo the multiplication by 7. So, let's continue our algebraic journey and see how we can isolate x completely.
Final Step: Solving for 'x'
Now we have “7x = 511”. To get 'x' by itself, we need to undo the multiplication by 7. We do this by dividing both sides of the equation by 7:
7x / 7 = 511 / 7, which gives us x = 73
Woohoo! We found 'x'! This is a major milestone in solving the problem. Finding the value of x is like discovering the key that unlocks the rest of the puzzle. It's the foundation upon which we can build to find the other numbers. By isolating x, we've successfully determined its value, which is 73 in this case. But our journey doesn't end here. We still need to find the other two numbers using the relationships we defined earlier. So, let's take a moment to celebrate our progress and then move on to the final phase of solving the problem: finding the values of the remaining numbers.
Finding the Other Numbers
Okay, so we know the first number (x) is 73. Now we can use this to find the other two numbers.
- The second number is 5 times the first number, so it’s 5 * 73 = 365.
- The third number is 100 more than the first number, so it’s 73 + 100 = 173.
See how easy that was? Once we found 'x', the rest fell into place! This is a common pattern in algebra problems – solving for one variable often opens the door to finding the others. It's like a domino effect, where one solution leads to the next. By using the relationships we established earlier, we were able to quickly calculate the values of the second and third numbers. Now, we have all the pieces of the puzzle. We know the values of all three numbers, but it's always a good idea to double-check our work to ensure accuracy.
Double-Checking Our Work
It’s always a good idea to check our answers to make sure they're correct. We can do this by adding the three numbers together and seeing if they sum up to 611:
73 + 365 + 173 = 611
It checks out! We got the right answer. Double-checking our work is like putting a safety net in place. It helps us catch any errors we might have made along the way and ensures that our solution is accurate. In this case, we verified that the sum of the three numbers is indeed 611, confirming that our calculations are correct. This step is crucial for building confidence in our solution and avoiding mistakes. So, always remember to take the time to double-check your work, especially in math problems. Now that we've confirmed our solution, we can confidently present our final answer.
The Answer
The three numbers are 73, 365, and 173.
We did it! We solved the problem step by step, and now we have our answer. Solving a math problem like this is like embarking on a journey. It requires patience, persistence, and a willingness to break down complex tasks into smaller, manageable steps. Along the way, we learned how to define variables, set up equations, solve for unknowns, and double-check our work. These are valuable skills that can be applied to many other areas of mathematics and beyond.
Final Thoughts
So, there you have it! We cracked the code and found the three numbers. Remember, guys, math problems might seem tough at first, but with a little bit of algebra and a step-by-step approach, you can conquer them. Keep practicing, and you'll become a math whiz in no time!