Finding Two Numbers: Sum Is 138, One Is Twice The Other Plus 12

Hey guys! Today, we're diving into a fun math problem. We're going to figure out two numbers. The cool thing is, we know a few things about them. First off, if you add them together, you get 138. Secondly, one of the numbers is a bit special – it's actually 12 more than double the other number. Sounds like a riddle, right? But don't worry, we'll break it down step by step and crack the code to find those mystery numbers. This problem is a classic example of how algebra can make solving problems super easy. We’ll use a bit of algebra, a little bit of common sense, and, most importantly, some teamwork to get to the answer. Ready to roll up our sleeves and get started? Let’s jump in and make these numbers reveal themselves!

Understanding the Problem: The Sum and the Relationship

Okay, so the very first thing we need to do is really get our heads around the problem. We've got two numbers. Let's call them Number A and Number B. The problem tells us two key facts. The sum of these two numbers (A + B) is 138. That’s our first piece of the puzzle. Now comes the second clue, which tells us how the numbers relate to each other. It says that one number is 12 more than twice the other. Let’s say Number A is the one that's special. So, Number A is equal to 2 times Number B, plus 12 (A = 2B + 12).

See, the language is important! We have to read it carefully and transform the words into a math expression. Think of it like a secret code that we have to crack. We have all the necessary information, so by breaking down the sentence into smaller, manageable chunks, we can see exactly what's happening. The problem is like a story; we must understand the whole plot, including the characters and their relationship with each other. This is crucial as it helps us build our foundation to work with the formulas and calculations later on. We're going to create equations to help us find out the values of the two numbers. Don't worry if it sounds complicated right now. Once we write everything down, it will start to click in your head. It's like taking a recipe to create a delicious dish, right? With all the ingredients available, all we need to do is follow the instructions, and we’ll get the result.

Breaking Down the Clues

Let’s summarize the facts we've got in our hands.

• Fact 1: The sum of the two numbers (A + B) equals 138.
• Fact 2: One number (A) is equal to double the other number (B) plus 12. Written mathematically: A = 2B + 12.

These are our foundation. These are our starting points. We will use them to form equations, and from these equations, we will be able to find the numbers we need. Now, you’ll see the power of mathematics. It might seem tricky at first, but with patience and a systematic approach, we’re sure to reach the answer.

Setting Up the Equations: From Words to Math

Alright, it's time to translate those clues into equations. We already know the basics, so let’s get them written down properly. Our first equation is super straightforward: A + B = 138. This represents our first fact – the sum of the two numbers is 138. Nothing fancy here, just a direct translation. The second equation, which comes from the second clue, is A = 2B + 12. This tells us the relationship between the two numbers. Remember, A is equal to twice B, and then we add 12.

The Substitution Method

Now, here comes the fun part: solving the equations! There are several ways to do this, but we're going to use a method called substitution. This is like playing a puzzle where you find out what one piece looks like and then replace that piece with the new one. In our case, we already know that A = 2B + 12. So, we're going to take that information and put it into our first equation, A + B = 138. We replace the A in the first equation with (2B + 12). This changes our first equation into (2B + 12) + B = 138. See how we've used one equation to help us with another? We're taking one thing we know (A = 2B + 12) and putting it into an equation where A appears. The idea behind this is to get everything into terms of just one variable so we can solve for it. By doing this, we're now just working with one unknown (B) instead of two (A and B). This is a really powerful technique in algebra, and it can be used to solve many types of problems.

Solving for One Variable: Finding Number B

Okay, let's keep going. Now, we have one equation with just one variable, (2B + 12) + B = 138. We need to solve it so we can find what Number B is. First, let’s simplify the equation by combining like terms. On the left side, we have 2B and another B, so together they make 3B. Our equation now looks like this: 3B + 12 = 138. Next, we want to isolate the term with B. So, we'll subtract 12 from both sides of the equation. This gets rid of the +12 on the left side. What we do to one side of an equation, we must do to the other to keep it balanced. After subtracting 12 from both sides, we're left with 3B = 126. We are getting closer to the solution. Finally, to find the value of B, we need to get B all by itself. We do this by dividing both sides of the equation by 3. This gives us B = 42. Congratulations, we've found the value of one of the numbers! Number B is 42. We're over halfway there; the hardest part is done.

Steps to Find B

Let’s review the steps:

1. Simplify: (2B + 12) + B becomes 3B + 12 = 138.
2. Subtract 12 from both sides: 3B = 126.
3. Divide by 3: B = 42.

Finding the Second Number: Discovering Number A

Now that we know Number B is 42, we can easily find Number A. Remember our equation from earlier? A = 2B + 12. We'll simply plug in the value of B (which is 42) into this equation. So, A = 2 * 42 + 12. Let's do the math: 2 times 42 is 84. Then, we add 12 to 84, and we get 96. So, Number A is 96. We have successfully found both numbers! We used substitution, but we could also use the original equation, A + B = 138. This helps us to double-check our math. If we plug in the number of B (42), we get A + 42 = 138. To find A, we must subtract 42 from 138, and we get 96. This proves our solution. This is always a great way to check your work; once you get the answer, check to make sure that everything makes sense.

Putting it Together

• We know A = 96.
• We know B = 42.
• Let’s double-check: 96 + 42 = 138. And, 96 = (2 * 42) + 12. Both equations are right! So we can see that our answer is correct.

The Answer: The Two Numbers Revealed!

Drumroll, please! We have successfully solved the math problem! The two numbers are 96 and 42. When you add them together, you get 138. Also, 96 is indeed 12 more than double 42. We did it, guys! The process involved breaking down the problem, setting up equations, using the substitution method, and doing a little bit of algebra. It's really cool to see how math can help us solve real-world problems. Keep practicing and keep up the great work! You'll become a math whiz in no time. If you got stuck at any point, don't worry. Just review the steps, practice a few more examples, and you'll be well on your way to mastering these kinds of problems.

Conclusion: Mastering the Sum of Two Natural Numbers

So, we've successfully navigated the challenge of finding two numbers with a specific sum and a defined relationship! This was a great example of how mathematical thinking and a systematic approach can unlock seemingly complex problems. We've seen how to translate word problems into algebraic equations, a fundamental skill in math. We employed the substitution method to solve for our unknowns, showing that sometimes, a little trick can make a big difference.

Key Takeaways

• Translate words to equations: This is the first and most important step to solving any math problem.
• Understand the relationships: Make sure you know how the numbers relate to each other.
• Use the Substitution Method: When you have equations, try substituting one equation's values in another to solve for unknowns.

Now, you have a solid toolkit for tackling similar problems! Remember, practice makes perfect. Try creating more problems on your own. Change the total sum or the relationship between the numbers, and see if you can solve them. You’ll be surprised at how much your confidence grows. Keep exploring the world of math, and you'll find it's full of fascinating puzzles just waiting to be solved. And the more problems you solve, the more powerful you'll become! So, keep going, keep practicing, and enjoy the journey.