Unlocking The Age Mystery: Marcela And Tania's Years

Hey there, math enthusiasts! Today, we're diving into a classic age problem involving two sisters, Marcela and Tania. This kind of problem is a fun way to flex those algebra muscles and figure out how to solve real-world scenarios. We'll break down the clues step-by-step and uncover the ages of these two siblings. Let's get started and unravel the mystery of their ages! If you have any questions along the way, don't hesitate to shout them out! So, grab your pencils and let's decode this age puzzle together. This is going to be fun, and you'll be surprised at how easy it can be once you know the right approach. Let’s get started and make this age-old problem a piece of cake. This age problem provides a great opportunity to practice your problem-solving skills and enhance your understanding of mathematical concepts. Ready to solve some equations? Let’s do this!

Decoding the Initial Clues and Setting up the Equations

Alright, guys, let's start by laying out the information we have. The core of this problem revolves around the ages of Marcela and Tania, and we have a couple of vital clues. First off, we know that the sum of their current ages is 48 years. This is our starting point and a key piece of the puzzle. Mathematically, we can represent this as: M + T = 48, where 'M' represents Marcela's current age and 'T' represents Tania's current age. This is our first equation, and it’s super important. Keep it in mind; it will be essential for our future calculations and problem-solving. But wait, there's more! The second clue is a bit more involved, so pay attention. We're told that in 15 years, Marcela's age will be double the age Tania was 3 years ago. This gives us another equation that helps connect their ages across time. To represent this mathematically: Marcela's age in 15 years will be M + 15. Tania's age 3 years ago was T - 3. So, our second equation becomes M + 15 = 2(T - 3). Now, we have two equations: M + T = 48 and M + 15 = 2(T - 3). These equations are the keys to unlocking the solution. They are the tools we'll use to find the ages of Marcela and Tania, and by carefully using them, we will find our solution. Let's get to it! Don't worry, we're going to break this down into smaller, manageable steps. Remember, mathematics is all about clarity and following the steps. We'll make sure that all the steps are clear, so you don't miss anything. Alright, let’s keep going. We're well on our way to solving this age-old problem (pun intended!).

Simplifying the Equations and Getting Ready to Solve

Okay, now that we have our equations set up, it’s time to simplify the second one. This is all about making the equations easier to work with, guys. It helps us get closer to solving for M and T. So, let’s revisit the second equation, which is M + 15 = 2(T - 3). First, distribute the 2 on the right side of the equation: M + 15 = 2T - 6. Now, let's rearrange it. We want to isolate variables to one side and constants to the other, so we have something easier to work with. Let's subtract 15 from both sides: M = 2T - 6 - 15. This simplifies to M = 2T - 21. And there you have it, a simplified version of our second equation! This makes things a whole lot easier. Now, we have:

• Equation 1: M + T = 48
• Equation 2 (simplified): M = 2T - 21

We are now ready to solve this! Now, we have two different forms of our equations, which makes solving a lot easier. It also makes things easier for substitution, allowing us to find the answers. Keep in mind that solving the equations requires careful attention to detail. This is where we use the power of algebra to find the individual ages of Marcela and Tania. This process can be so much fun and fulfilling. Let's get to the fun part!

Solving for the Ages: Substitution and Unveiling the Truth

Alright, folks, it’s time to solve for the ages! Here's where we get to put those algebra skills to good use and find out exactly how old Marcela and Tania are. Since we have one equation already solved for 'M' (M = 2T - 21), we can use the method of substitution. This is where we take the expression for 'M' and substitute it into the first equation, which is M + T = 48. So, we replace 'M' with (2T - 21), resulting in: (2T - 21) + T = 48. Now, let's solve for 'T'. First, combine like terms: 3T - 21 = 48. Next, add 21 to both sides: 3T = 48 + 21, which simplifies to 3T = 69. Finally, divide both sides by 3: T = 69 / 3, thus T = 23. Congratulations, we've found Tania's age! Tania is currently 23 years old. But hey, we’re not done yet, we still need to find Marcela’s age. Now that we know Tania's age, we can easily find Marcela's age. Remember our first equation? M + T = 48. Since we know T = 23, we can substitute that into the equation: M + 23 = 48. To find M, we subtract 23 from both sides: M = 48 - 23, thus M = 25. Tada! Marcela is 25 years old. This is the moment we've all been waiting for. We have found the ages of both Marcela and Tania. Now that we've found the solution, let's summarize it. Both our answers make sense because they're based on the information we were given. Let's do it!

Verifying the Solution: Checking Our Answers

Before we pop the champagne (or, you know, just high-five each other), it’s always a good idea to verify our answers. This helps us ensure that we haven’t made any mistakes along the way. Let's start by checking if our ages meet the conditions of the original problem. We know that the sum of their ages is 48. So, let’s add Marcela's age and Tania's age: 25 + 23 = 48. Boom! Our answer checks out with the first part of the problem. That's a great start. Next, let’s check if the second condition holds true. Remember, in 15 years, Marcela's age will be double Tania's age three years ago. Marcela's age in 15 years will be 25 + 15 = 40. Tania's age three years ago was 23 - 3 = 20. Now, let’s see if Marcela’s age in 15 years is double Tania’s age 3 years ago: 40 = 2 * 20. Yup, it checks out! So, based on both the initial conditions, it seems our calculations are accurate. This step is super important, guys, it is vital to double-check our work. It's like proofreading your essay to make sure you didn’t miss anything. Always verify and validate your results. This step can save you a lot of trouble. That’s how you ace a math problem. By verifying, we have high confidence in our answer. Now that we have verified, we can say with confidence that our solution is correct. Great job!

Conclusion: Wrapping Up the Age Problem

So, there you have it, folks! We've successfully solved the age problem and uncovered the ages of Marcela and Tania. Marcela is 25 years old, and Tania is 23 years old. It's satisfying, isn’t it? Age problems like this one are a great way to practice your algebra skills and enhance your logical thinking. Remember, the key is to break down the problem into smaller parts, set up your equations, and solve systematically. With practice, you'll be able to tackle even more complex problems. Isn’t it amazing what we can achieve with the right approach and a bit of determination? Feel free to try out some more age problems and see if you can solve them! This problem demonstrates how mathematical concepts can be applied to real-world scenarios. We hope you enjoyed this journey into the world of age problems. Always remember to break it down, write it down, and solve! Keep practicing, and you'll get better and better at these types of problems. Thanks for joining me on this math adventure, and remember, keep practicing and stay curious. If you found this helpful, share it with your friends, and keep exploring the amazing world of mathematics! Keep up the great work, and don't be afraid to try new things. Math is fun when you approach it with the right mindset! Now that we have the answer, we can be proud of our work. Keep in mind that math is all about persistence, and you can achieve whatever you set your mind to.