Solving Math Problems: Finding The Number 'ab'

Hey everyone! Today, we're diving into a cool math problem. We're going to figure out a two-digit number, which we'll cleverly call 'ab'. The problem gives us a clue: if we add 1a5b to a3b7, we get 3590. Let's break it down and see how we can solve this together. It’s like a little puzzle, and trust me, it’s fun once you get the hang of it. Get ready to sharpen your math skills because we are going to dissect this math problem and get you a strong understanding of the techniques involved. This article is your guide to a solid understanding of how to approach and solve this type of mathematical puzzle. We will explore the methods step by step, making sure you grasp every concept along the way. So, grab your pencils and paper, and let's start solving this math mystery!

Understanding the Problem: What's 'ab' Anyway?

Okay, so we're looking for a number, and this number has two digits, 'a' and 'b'. That means we have a tens digit ('a') and a units digit ('b'). When we write it as 'ab', it's a shorthand way of saying (10 times 'a') + 'b'. For example, if 'a' was 2 and 'b' was 3, then 'ab' would actually be 23 (because 10*2 + 3 = 23). Now, the problem throws some other numbers at us: 1a5b and a3b7. What does this even mean? Well, the 'a' and 'b' are still digits, but they're sitting in different places in these larger numbers. 1a5b really means 1000 + (100 times 'a') + 50 + 'b', and a3b7 is (1000 times 'a') + 300 + (10 times 'b') + 7. Breaking down these numbers into their place values is key to solving this problem. This initial step might seem a bit tricky, but once you get comfortable with it, it becomes a piece of cake. Remember, in mathematics, as in many things, the better you understand the fundamentals, the easier it will be to tackle more complex issues. We are going to rewrite the addition operation to make it easier to calculate the result. The objective is to isolate 'a' and 'b' and figure out their values individually. It may seem overwhelming, but keep in mind that with patience and the right approach, even complex problems become solvable. Let's move forward with a positive attitude and clear goals!

Setting Up the Equation: The Foundation of Our Solution

Now we know what our numbers mean, let's put the information we have into an equation. The problem tells us that 1a5b + a3b7 = 3590. We're going to rewrite this by expanding those numbers based on their place values, as we talked about earlier. This will give us: (1000 + 100a + 50 + b) + (1000a + 300 + 10b + 7) = 3590. See? Now it looks a little more complicated, but don't worry. We'll simplify it. Our goal is to rearrange this equation so that we can isolate 'a' and 'b'. Let's combine the like terms (the numbers with 'a', the numbers with 'b', and the plain numbers). Doing this step by step helps in not making any errors! This step is crucial because it transforms a seemingly complex expression into a more manageable form. The power of simplifying expressions lies in making it easier to see the underlying structure of the problem. When you simplify, you make the process of solving the equation more efficient and less prone to errors. After we have simplified it, the equation should look much cleaner, which makes solving it much easier. This also boosts your ability to think logically and solve problems, which are invaluable skills in everyday life, not just math.

Simplifying and Solving: The Heart of the Matter

Alright, let's simplify that big equation. Combine all the 'a' terms (100a + 1000a = 1100a), combine all the 'b' terms (b + 10b = 11b), and add up all the regular numbers (1000 + 50 + 300 + 7 = 1357). This gives us: 1100a + 11b + 1357 = 3590. Now, to make things easier, let’s subtract 1357 from both sides of the equation. This will isolate the 'a' and 'b' terms, which is what we need to do to find their values. Subtracting 1357 from both sides gives us: 1100a + 11b = 2233. Now, we have to find what 'a' and 'b' are. Notice that the left side of the equation (1100a + 11b) has a common factor of 11. So, let's divide both sides by 11. Dividing both sides by 11 simplifies our equation. Dividing both sides by 11, we get 100a + b = 203. At this point, we have to think about the digits. Remember, 'a' and 'b' are single-digit numbers, meaning they can only be from 0 to 9. Now, we know that 'a' must be 2 because 100 multiplied by 2 is 200, which is the closest we can get to 203 without going over, because if a were 3, we would have 300, which is bigger than 203. So, if a is 2, then the equation becomes 200 + b = 203. Therefore, b must be 3 because 200 + 3 = 203. So, we've found our digits: a = 2 and b = 3! This process may seem tedious, but it is very important to carefully follow each step.

Finding the Solution and Checking Our Work

Now that we know a = 2 and b = 3, we can figure out the number 'ab'. It's simple: 'ab' is 23! We found it, guys! But, it's always a good idea to check your work to make sure you are correct. Let's go back to the original equation: 1a5b + a3b7 = 3590. Substitute the values of a and b (2 and 3, respectively) into this equation: 1253 + 2337 = 3590. Doing the math, 1253 + 2337 does indeed equal 3590! Voila, we are correct! Checking our work is an important part of the problem-solving process. It ensures that you understand the steps and haven't made any mistakes. Remember, understanding how to solve these types of problems is not just about getting the right answer; it's about understanding the process and being able to explain it clearly.

Conclusion: Math is Awesome!

So, there you have it! We successfully found the two-digit number 'ab', which is 23, given the conditions of the problem. We learned how to break down numbers based on their place values, set up and solve equations, and verify our solutions. I hope you guys enjoyed this tutorial. Remember, math can be really fun. Keep practicing, keep questioning, and keep exploring the amazing world of numbers! Keep in mind that practice makes perfect, so keep trying, and don't be afraid to ask for help. Congratulations, you have successfully solved this math problem! Always remember that the key is to approach each problem with a positive attitude and break it down into smaller, more manageable steps. Until next time, happy calculating!