Sum Of Ages: Pilar & Guadalupe
Hey guys, let's dive into a classic age puzzle that'll get your brains buzzing! We're talking about two friends, Pilar and Guadalupe, and we need to figure out their ages based on a couple of clues. It's a math problem, but don't let that scare you – we'll break it down step-by-step, making it super easy to follow. So, grab a drink, get comfy, and let's solve this riddle together!
The Problem at Hand
So, here's the deal: we know that Pilar's age is two-thirds of Guadalupe's age. That's our first big clue. Think about it – if Guadalupe is a certain age, Pilar is a fraction of that. The second crucial piece of information is that the sum of their ages is 30. Easy peasy, right? We just need to combine these two facts to unlock the mystery of how old each of them is. This isn't just about numbers; it's about logical deduction, and trust me, it's pretty satisfying when you crack it. We're going to use a bit of algebra, but don't sweat it if math isn't your strong suit. We'll keep it super simple and focus on understanding the 'why' behind each step. By the end of this, you'll not only know Pilar's and Guadalupe's ages but also feel a little more confident tackling similar problems. It’s all about setting up the right equations and solving them logically. We'll use variables to represent their unknown ages and then translate the word problem into mathematical language. This is the foundation for solving a lot of problems, not just in math class, but in real life too! So, let's get started on figuring out this age conundrum.
Setting Up the Equations
Alright, let's get down to business and set up the equations for our age problem. First off, we need to represent the unknown ages with variables. Let's say 'G' represents Guadalupe's age and 'P' represents Pilar's age. This is a common practice in algebra, and it helps us keep track of what we're trying to find. Now, let's translate the first clue into an equation: "Pilar has two-thirds the age of Guadalupe." This means that Pilar's age (P) is equal to 2/3 multiplied by Guadalupe's age (G). So, our first equation is: P = (2/3)G. Pretty straightforward, right? It directly reflects the relationship given in the problem. It tells us that whatever age Guadalupe is, Pilar's age will be a smaller portion of that. Think of it like slicing a pizza – Guadalupe has the whole pizza (her age), and Pilar has two out of three slices of that pizza. Our second clue is that "the sum of their ages is 30." This is even simpler to translate. It means that Guadalupe's age (G) plus Pilar's age (P) equals 30. So, our second equation is: G + P = 30. Now we have two equations, and two unknowns (G and P). This is what we call a system of equations, and it's exactly what we need to solve for both variables. The beauty of having two equations is that they give us enough information to find a unique solution. We can use substitution or elimination methods to solve these, but for this particular problem, substitution is super handy because we already have 'P' isolated in our first equation. We're going to take the expression for 'P' from the first equation and plug it into the second equation. This will give us an equation with only one variable (G), which we can then solve. This is the core of solving many algebraic problems – transforming a word problem into a solvable mathematical model. The setup is key, and once you've got that, the rest is just following the rules of algebra.
Solving for Guadalupe's Age
Now that we've got our equations, let's get solving! We're going to use the substitution method, which is super effective here. Remember our two equations?
- P = (2/3)G
- G + P = 30
We know from the first equation that 'P' is the same as '(2/3)G'. So, we can take this expression for 'P' and substitute it directly into the second equation wherever we see 'P'. This will eliminate 'P' from the second equation, leaving us with an equation that only involves 'G'.
So, let's substitute:
G + (2/3)G = 30
Now, we need to combine the 'G' terms. Think of 'G' as '1G'. To add 1 and 2/3, we need a common denominator. One whole is the same as 3/3. So, we have:
(3/3)G + (2/3)G = 30
Adding the fractions, we get:
(5/3)G = 30
Awesome! We're one step closer. To find 'G', we need to isolate it. Right now, 'G' is being multiplied by 5/3. To undo that, we can multiply both sides of the equation by the reciprocal of 5/3, which is 3/5.
** (3/5) * (5/3)G = 30 * (3/5)**
On the left side, (3/5) * (5/3) cancels out to 1, leaving us with just 'G'. On the right side, we calculate 30 * (3/5). You can think of this as (30 * 3) / 5, which is 90 / 5, or you can simplify first: 30 divided by 5 is 6, and then 6 * 3 is 18.
G = 18
So, there you have it! Guadalupe's age is 18. High five! We've successfully solved for one of the unknowns using our system of equations. This is a huge step, and it shows how breaking down a problem into smaller, manageable parts really works. The key was substituting the expression for Pilar's age into the sum equation, which allowed us to solve for Guadalupe's age directly. Now that we know Guadalupe's age, finding Pilar's age will be a piece of cake!
Finding Pilar's Age
Okay, guys, we've already done the heavy lifting. We figured out that Guadalupe is 18 years old. Now, finding Pilar's age is super simple, thanks to the information we already have. We can use either of our original equations, but the first one is the easiest: P = (2/3)G.
We know G = 18, so let's plug that number in:
P = (2/3) * 18
To calculate this, we multiply 2 by 18 and then divide by 3. So, (2 * 18) = 36. Then, 36 divided by 3 equals 12.
P = 12
And just like that, Pilar is 12 years old! See? It wasn't that difficult. We used the relationship given in the problem (Pilar is two-thirds Guadalupe's age) and the age we just calculated for Guadalupe to find Pilar's age. You could also use the second equation, G + P = 30, to double-check. If G = 18 and P = 12, then 18 + 12 = 30. It adds up perfectly, confirming our answers! This step-by-step approach, starting with setting up equations and then solving them systematically, is a powerful way to tackle word problems. It demystifies the math and shows that with a clear strategy, even seemingly complex problems can be solved. So, we've successfully determined that Guadalupe is 18 and Pilar is 12. Awesome work!
Conclusion
And there you have it, folks! We've successfully solved the age puzzle for Pilar and Guadalupe. Guadalupe is 18 years old, and Pilar is 12 years old. We figured this out by first setting up two simple algebraic equations based on the clues given: Pilar's age being two-thirds of Guadalupe's age, and the sum of their ages being 30. We then used the substitution method to solve for Guadalupe's age, finding it to be 18. With Guadalupe's age in hand, we easily calculated Pilar's age to be 12. We even did a quick check using the sum of their ages to confirm our results were correct. This problem is a fantastic example of how basic algebra can be used to solve real-world scenarios, or at least, scenarios that feel like they could be from a math textbook! It’s all about translating words into numbers and then using logical steps to find the unknown. Remember, the key steps were: identifying the unknowns, assigning variables, writing down the relationships as equations, and then solving the system of equations. Whether you're a math whiz or just getting started, breaking down problems like this makes them approachable and, dare I say, even fun! So, next time you encounter a word problem, don't be intimidated. Just follow these steps, and you'll be solving mysteries in no time. Keep practicing, keep exploring, and keep those math skills sharp! You guys totally crushed this!