Solving Age Problems In Algebra Cassie And Kara's Ages
Hey everyone! Today, we're going to dive into a classic algebra problem that involves figuring out ages. These types of problems might seem tricky at first, but with a systematic approach, they become super manageable. We'll break down a specific example step by step, and by the end, you'll be able to tackle similar age-related algebraic challenges with confidence. So, let's get started and unravel the mysteries of Cassie and Kara's ages!
Understanding the Problem
Before we jump into the equation, let's really break down the word problem we're dealing with. This is a crucial step because understanding the relationships between the ages is key to setting up the right equation. In this case, we have two people, Cassie and Kara, and we're given two key pieces of information that connect their ages.
First, we know that Cassie's age is related to Kara's age in a specific way: it's "3 less than twice Kara's age." This means we're dealing with a multiplication and a subtraction. We need to double Kara's age and then subtract 3 to find Cassie's age. This kind of phrasing is common in algebra problems, so keep an eye out for similar wording. Understanding how to translate these words into mathematical expressions is a fundamental skill in algebra.
Second, we're told that "the sum of their ages is 42." This part is a bit more straightforward. It means if we add Cassie's age and Kara's age together, we'll get 42. This gives us a direct relationship that we can use in our equation. Remember, the goal is to use these relationships to create an equation that we can solve. So, let's keep these two pieces of information in mind as we move on to the next step. We're essentially translating a real-world scenario into the language of algebra, which is a super useful skill to have!
Defining the Variable
Alright, now that we've got a good handle on the problem, the next crucial step is to define our variable. In algebra, a variable is just a letter (usually x, y, or in this case, n) that represents an unknown quantity. It's like a placeholder for a number we need to figure out. In our problem about Cassie and Kara's ages, we're going to let the variable n represent Kara's age. This is a smart move because the problem asks us to find Kara's age, and it also makes it easier to express Cassie's age in terms of Kara's age.
Why is defining the variable so important? Well, it's the foundation upon which we build our entire equation. By clearly stating what n stands for, we're setting ourselves up for success in the next steps. It's like laying the groundwork before you start building a house – you need a solid foundation! Plus, it helps us stay organized and avoid confusion as we work through the problem. Trust me, in algebra, clarity is your best friend. So, with n representing Kara's age, we're ready to move on and express the other pieces of information in terms of n. This is where the fun really begins – we're about to translate those words into math!
Expressing Cassie's Age
Now comes the fun part – translating the word problem into mathematical expressions! We've already defined n as Kara's age, which is a great start. But remember, the problem gives us a relationship between Cassie's age and Kara's age. It says, "Cassie's age is 3 less than twice Kara's age." We need to take those words and turn them into an algebraic expression.
Let's break it down piece by piece. "Twice Kara's age" means we need to multiply Kara's age (n) by 2. So that's 2n, or simply 2n. Easy enough, right? Now, the next part says "3 less than" this. That means we need to subtract 3 from what we just got. So, Cassie's age can be expressed as 2n - 3. See how we took the words and turned them into a mathematical expression? This is a super important skill in algebra.
We now have expressions for both Kara's age (n) and Cassie's age (2n - 3). This is a huge step forward! We're almost ready to put it all together into a complete equation. By expressing Cassie's age in terms of n, we've created a direct link between their ages, which is exactly what we need to solve the problem. So, give yourself a pat on the back – you're doing great! Next up, we'll use this information to build the equation itself.
Building the Equation
Okay, we're on the home stretch now! We've defined our variable (n for Kara's age), and we've expressed Cassie's age in terms of n (2n - 3). Now it's time to use the final piece of information from the problem: "The sum of their ages is 42." This is the key that will unlock our equation.
What does "sum" mean? It means we need to add the two ages together. So, we're going to add Kara's age (n) to Cassie's age (2n - 3). And what should that equal? According to the problem, it equals 42. So, let's put it all together. Our equation is: n + (2n - 3) = 42. There it is! We've successfully translated the entire word problem into a concise algebraic equation.
This is a really important moment. We've taken a real-world scenario and turned it into something we can solve using algebra. The equation n + (2n - 3) = 42 represents the relationship between Cassie and Kara's ages, and it's the tool we'll use to find the answer. Take a moment to appreciate what we've done! Building the equation is often the hardest part of these problems, and we've nailed it. Now, all that's left is to solve the equation and find Kara's age. Let's move on to the next step and do just that!
Solving the Equation
Alright, we've got our equation: n + (2n - 3) = 42. Now comes the part where we actually find the value of n, which, if you remember, represents Kara's age. Solving equations is a fundamental skill in algebra, and it's like detective work – we're uncovering the hidden value of our variable.
The first thing we need to do is simplify the equation. Notice those parentheses? Let's get rid of them. Since we're adding the expression inside the parentheses, we can just drop them: n + 2n - 3 = 42. Now, we have some like terms on the left side – n and 2n. We can combine these to get 3n: 3n - 3 = 42. See how much simpler the equation looks already?
Next, we want to isolate the term with n on one side of the equation. To do that, we need to get rid of the -3. The opposite of subtracting 3 is adding 3, so we'll add 3 to both sides of the equation. This is a crucial rule in algebra: whatever you do to one side, you have to do to the other to keep the equation balanced. So, we get 3n - 3 + 3 = 42 + 3, which simplifies to 3n = 45.
We're almost there! Now we just have 3n = 45. This means 3 times n equals 45. To get n by itself, we need to do the opposite of multiplication, which is division. So, we'll divide both sides of the equation by 3: (3n)/3 = 45/3. This simplifies to n = 15. Woohoo! We've solved for n!
Finding Cassie's Age
So, we've figured out that n = 15. But hold on, what did n represent again? Ah, yes, n is Kara's age! So, we now know that Kara is 15 years old. But the problem gave us information about Cassie's age too, and sometimes it's good practice to find all the unknowns. Plus, it's a great way to double-check our work!
Remember that we expressed Cassie's age as 2n - 3. Now that we know n is 15, we can plug that value into the expression. So, Cassie's age is 2 * 15 - 3. Let's do the math: 2 * 15 is 30, and 30 - 3 is 27. So, Cassie is 27 years old. Awesome!
We've now found both Kara's age (15) and Cassie's age (27). But before we celebrate, let's make sure our answers make sense in the context of the original problem. This is a really important step in problem-solving – always check your work!
Checking the Answer
Okay, we've found that Kara is 15 and Cassie is 27. But how do we know if these ages are correct? This is where checking our answer comes in handy. It's like the final step in a puzzle – making sure all the pieces fit together. We need to go back to the original problem and see if our ages satisfy the conditions we were given.
The problem told us two things: first, that Cassie's age is 3 less than twice Kara's age, and second, that the sum of their ages is 42. Let's check the first condition. Twice Kara's age is 2 * 15, which is 30. Is Cassie's age (27) 3 less than 30? Yes, it is! So, our ages satisfy the first condition.
Now, let's check the second condition. The sum of their ages should be 42. Is 15 + 27 = 42? Yes, it is! So, our ages also satisfy the second condition. This is fantastic news! We've not only found the ages, but we've also confirmed that they're correct. This gives us a lot of confidence in our answer.
Answering the Question
We've done it! We've successfully navigated the algebra problem and found the ages of Kara and Cassie. We defined our variable, built an equation, solved it, and even checked our answer. That's a lot of algebra in one go! But now, we need to make sure we actually answer the question that was asked. This might seem obvious, but it's an important step that's easy to overlook.
The original problem asked us to create an equation to find Kara's age, n, and we did that: n + (2n - 3) = 42. But the problem also had some boxes to fill in, and those boxes represent the different parts of the equation. So, let's make sure we've got the right numbers to put in those boxes. Looking back at our equation, we can see that the boxes should contain the following numbers:
- The first box (coefficient of n) should be 2.
- The second box (constant to be subtracted) should be 3.
- The third box (the final sum) should be 42.
So, the completed equation would look like this: n + (2n - 3) = 42. We've not only solved for Kara's age, but we've also made sure we've answered the question in the specific format it was asked. That's attention to detail that will serve you well in algebra and beyond!
Key Takeaways
Wow, we've covered a lot in this algebra journey! We started with a word problem about Cassie and Kara's ages, and we ended up solving a full-fledged equation and finding the answers. Before we wrap up, let's quickly review the key steps we took. This will help solidify your understanding and make you even more confident in tackling similar problems in the future.
First, we understood the problem. We carefully read the word problem and identified the key pieces of information, especially the relationships between Cassie and Kara's ages. This is a crucial step because it sets the stage for everything else.
Next, we defined the variable. We let n represent Kara's age, which made it easier to express Cassie's age in terms of n. Defining the variable clearly is like laying the foundation for a building – it's essential.
Then, we expressed Cassie's age using an algebraic expression. We translated the words "3 less than twice Kara's age" into 2n - 3. This is a great example of how to turn words into math.
After that, we built the equation. We used the information that the sum of their ages is 42 to create the equation n + (2n - 3) = 42. This is where all the pieces came together.
We then solved the equation using algebraic techniques, step by step, until we found that n = 15, meaning Kara is 15 years old. This is the heart of the algebra process.
We also found Cassie's age by plugging the value of n back into the expression we created earlier. This showed us the power of using variables to relate different quantities.
And finally, we checked our answer to make sure it satisfied the conditions of the original problem. This is a crucial step to ensure accuracy and build confidence in your solution.
By following these steps, you can conquer algebra age problems with ease! Remember to take your time, break the problem down into smaller parts, and always check your work. You've got this!
Practice Problems
Alright, you've made it through the explanation and seen how we tackled the algebra problem involving Cassie and Kara's ages. Now it's your turn to put your newfound skills to the test! Practice is absolutely key to mastering any math concept, and algebra is no exception. So, let's dive into some practice problems that are similar to the one we just worked through. These problems will give you a chance to flex your algebraic muscles and build your confidence.
Remember those key steps we talked about – understanding the problem, defining the variable, expressing unknowns, building the equation, solving the equation, and checking your answer? Keep those in mind as you work through these problems. Don't be afraid to go back and review the steps if you get stuck. The goal is to become comfortable with the process and develop your problem-solving skills.
So, grab a pencil and paper, and let's get started! The more you practice, the more natural these steps will become, and the more confident you'll feel when you encounter algebra problems in the future. And remember, it's okay to make mistakes – that's how we learn! Just keep at it, and you'll be amazed at how much progress you can make.
Conclusion
We've reached the end of our algebraic adventure with Cassie and Kara's ages! We've journeyed from a word problem to a solution, and along the way, we've reinforced some fundamental algebra skills. We've seen how to translate words into mathematical expressions, how to build and solve equations, and how to check our answers to ensure accuracy. These are all essential skills that will serve you well in your algebra studies and beyond.
Remember, algebra is like a language – the more you practice, the more fluent you become. So, keep working on these types of problems, and don't be afraid to challenge yourself. The more you engage with the material, the better you'll understand it. And the better you understand it, the more confident you'll become in your problem-solving abilities.
So, congratulations on making it this far! You've taken a big step towards mastering algebra. Now, go out there and tackle those equations with confidence. You've got the tools and the knowledge – all you need is a little practice. Happy solving!
