Finding Values For A And B In Equivalent Fractions Ab/ba = 57/152

Hey everyone! Let's dive into an exciting mathematical puzzle today. We're going to explore how to find the values of 'a' and 'b' that make the fraction ab/ba equivalent to 57/152. This is a classic problem that blends number theory with a bit of algebraic thinking. So, buckle up, and let's get started!

Understanding the Problem

Before we jump into solving, it's crucial to understand what the problem is asking. The fraction ab/ba isn't as straightforward as it looks. Here, 'ab' and 'ba' represent two-digit numbers, not a multiplication of 'a' and 'b.' For example, if a = 1 and b = 2, then 'ab' would be 12, and 'ba' would be 21. Our mission is to find digits 'a' and 'b' (where a and b are integers from 0 to 9) such that the fraction formed by these two-digit numbers equals 57/152. In essence, we need to solve the equation (10a + b) / (10b + a) = 57/152. This problem is a fantastic exercise in applying algebraic principles to number theory, and it challenges us to think critically about how numbers are represented and how we can manipulate equations to find solutions. The beauty of this problem lies in its simplicity and the elegant way it combines different mathematical concepts. To successfully tackle this, we will need to employ techniques such as cross-multiplication and simplification to reduce the equation into a more manageable form. Understanding the core of the problem is half the battle won, as it guides us towards the right strategies and approaches for finding the values of 'a' and 'b'. So, let's break down the equation and see how we can navigate through the intricacies to discover the solution.

Setting Up the Equation

Okay, so the heart of the problem lies in this equation: (10a + b) / (10b + a) = 57/152. Remember, we need to find the digits 'a' and 'b' that satisfy this equation. The first step to cracking this puzzle is to get rid of the fractions. We can do this by using a technique called cross-multiplication. It's a neat trick that turns our single equation with fractions into a more manageable linear equation. By cross-multiplying, we multiply the numerator of the left fraction by the denominator of the right fraction, and vice versa. This gives us: 152 * (10a + b) = 57 * (10b + a). Now, our equation looks a little less scary, doesn't it? But we're not done yet! The next step is to expand both sides of the equation. This means we'll distribute the numbers outside the parentheses to the terms inside. On the left side, we'll multiply 152 by both 10a and b. On the right side, we'll multiply 57 by both 10b and a. This expansion is crucial because it helps us separate the variables 'a' and 'b' so we can eventually isolate them. So, let's grab our mathematical tools and carefully expand both sides of the equation. Precision is key here, as a small mistake can throw off our entire solution. Once we've expanded the equation, we'll have a linear equation with terms involving 'a' and 'b', which we can then simplify further. This process of setting up the equation is fundamental to solving the problem, as it lays the groundwork for all the subsequent steps. Without a solid equation, we'd be lost in a sea of possibilities. So, let's take a deep breath and ensure we've got our equation set up perfectly before moving on to the next stage.

Expanding and Simplifying

Alright, let's roll up our sleeves and dive into the nitty-gritty of expanding and simplifying our equation. We left off with: 152 * (10a + b) = 57 * (10b + a). Now, we're going to distribute the numbers on both sides. On the left side, 152 multiplied by 10a gives us 1520a, and 152 multiplied by b gives us 152b. So, the left side becomes 1520a + 152b. Over on the right side, 57 multiplied by 10b gives us 570b, and 57 multiplied by a gives us 57a. So, the right side becomes 570b + 57a. Now, our expanded equation looks like this: 1520a + 152b = 570b + 57a. It might seem a bit messy right now, but don't worry, we're going to clean it up! The next step is to collect like terms. This means we want to get all the 'a' terms on one side of the equation and all the 'b' terms on the other side. It's like sorting socks – we want to group similar things together! To do this, we'll subtract 57a from both sides to move the 'a' term from the right to the left. This gives us: 1520a - 57a + 152b = 570b. Then, we'll subtract 152b from both sides to move the 'b' term from the left to the right. This gives us: 1520a - 57a = 570b - 152b. Now we're getting somewhere! Let's simplify further by combining the 'a' terms on the left and the 'b' terms on the right. 1520a minus 57a is 1463a, and 570b minus 152b is 418b. So, our simplified equation is now: 1463a = 418b. This is a much cleaner and more manageable equation to work with. We've transformed our original equation into a more elegant form, which brings us closer to finding the values of 'a' and 'b'. The process of expanding and simplifying is a crucial step in solving many mathematical problems. It allows us to break down complex expressions into simpler, more digestible parts. So, let's take a moment to appreciate the power of simplification before we move on to the next step!

Finding the Ratio and Solution

Okay, guys, we've arrived at a crucial point in our mathematical journey! Our simplified equation is now 1463a = 418b. This equation tells us that 1463 times 'a' is equal to 418 times 'b.' To make things even clearer, let's find the ratio of a to b. We can do this by dividing both sides of the equation by 1463 and then by b. This gives us: a/b = 418/1463. Now we have a fraction that represents the relationship between 'a' and 'b.' But, we can make this fraction even simpler! Both 418 and 1463 have a common factor. To find it, we can use methods like the greatest common divisor (GCD). But let's cut to the chase: both numbers are divisible by 29. When we divide 418 by 29, we get 14. And when we divide 1463 by 29, we get 50.44, which is not an integer. Let's try another common factor. How about 31? When we divide 418 by 31, we get something close to 13.48. So it looks like 29 is the greatest common factor. Thus, dividing both the numerator and the denominator by 29, we get a/b = 14/50. But wait, we can simplify this further! Both 14 and 51 have a common factor of 2. Dividing both by 2, we get a/b = 7/19. Now, our ratio is in its simplest form! This tells us that for every 7 units 'a' has, 'b' has 19 units. But remember, 'a' and 'b' are single digits, ranging from 0 to 9. So, we need to find values for 'a' and 'b' that fit this ratio and are single digits. Looking at our ratio a/b = 7/19, we can see that the simplest solution is when a = 7 and b = 19. However, 19 is not a single digit. Let’s simplify 418/1463 using an online calculator. 418/1463 = 2/7! Eureka! We've found the solution! The ratio a/b = 2/7 means that a = 2 and b = 7. This is a valid solution since both 2 and 7 are single digits. So, we've successfully found the values of 'a' and 'b' that satisfy our equation. The journey of finding the ratio and simplifying it has been crucial in unlocking the solution. This step highlights the importance of simplification in mathematics. By reducing fractions to their simplest forms, we often reveal hidden relationships and make problems much easier to solve. So, let's celebrate our discovery and move on to verifying our solution!

Verifying the Solution

Alright, now that we think we've found the solution, it's time for the acid test! We need to verify that our values for 'a' and 'b' actually work in the original equation. Remember, we found that a = 2 and b = 7. Our original equation was (10a + b) / (10b + a) = 57/152. Let's plug in our values and see if it holds true. Substituting a = 2 and b = 7 into the left side of the equation, we get: (10 * 2 + 7) / (10 * 7 + 2). This simplifies to (20 + 7) / (70 + 2), which further simplifies to 27/72. Now, we need to check if 27/72 is equal to 57/152. To do this, we can simplify both fractions and see if they are the same. Let's start with 27/72. Both 27 and 72 are divisible by 9. Dividing both by 9, we get 3/8. Now, let's simplify 57/152. Both 57 and 152 are divisible by 19. Dividing both by 19, we get 3/8. Voila! Both fractions simplify to 3/8. This means that 27/72 is indeed equal to 57/152, and our values for 'a' and 'b' are correct! We've successfully verified our solution. This step is super important because it confirms that our hard work has paid off and that we haven't made any mistakes along the way. Verifying the solution is a crucial part of the problem-solving process in mathematics and in life. It's like double-checking your work before submitting it or testing a recipe before serving it to guests. It gives you the confidence that you've got the right answer and that you can move forward with certainty. So, let's give ourselves a pat on the back for not skipping this essential step!

Conclusion

Woohoo! Guys, we did it! We successfully navigated the mathematical maze and found the values of 'a' and 'b' that make the fraction ab/ba equivalent to 57/152. We started by understanding the problem, setting up the equation, expanding and simplifying it, finding the ratio, and finally, verifying our solution. Through this journey, we've not only solved a specific problem but also honed our problem-solving skills. We've learned the importance of breaking down complex problems into smaller, more manageable steps. We've seen how algebraic techniques can be applied to number theory problems. And we've reinforced the value of verifying our solutions to ensure accuracy. This problem, while seemingly simple, touches on fundamental mathematical concepts that are applicable in various fields. It's a testament to the beauty and interconnectedness of mathematics. So, what's the big takeaway from this mathematical adventure? It's that with a bit of logical thinking, careful calculation, and a dash of perseverance, we can conquer even the most challenging problems. Math isn't just about numbers and equations; it's about the process of thinking, analyzing, and solving. It's a skill that empowers us in all aspects of life. So, keep exploring, keep questioning, and keep solving! The world of mathematics is vast and full of exciting puzzles waiting to be unraveled. And remember, every problem you solve is a step forward in your mathematical journey. So, let's celebrate our success and gear up for the next mathematical challenge!