Age Proportion Problem: Solving For Current Sum
Hey guys, let's tackle this interesting age proportion problem together! We'll break it down step by step to make sure everything's crystal clear. This type of problem often pops up in math quizzes and exams, so understanding the logic behind it is super important. We're going to use proportions and a little bit of algebra to figure out the solution. So, grab your thinking caps, and let's dive in!
Understanding the Problem
Okay, so the problem states that we have two friends, and their current ages are proportional to 3 and 4. What does that even mean? Well, it means that if one friend's age is 3 times a certain number, the other friend's age is 4 times the same number. Think of it like a ratio. If we let that number be 'x,' we can say the friends' current ages are 3x and 4x. This foundational understanding is crucial to solving the problem effectively. Make sure you fully grasp this concept before moving on. It's like laying the groundwork before building a house – you need a solid foundation! Now, why is this initial representation so important? Because it allows us to translate the word problem into algebraic expressions, which are much easier to manipulate and solve.
Now, the problem throws another curveball: After 2 years, their ages will be proportional to 4 and 5. So, we need to account for that time jump. Each friend will be 2 years older, right? So, their ages after 2 years will be 3x + 2 and 4x + 2. And these new ages are proportional to 4 and 5. That means we can set up another ratio equation. This is where the problem starts to get interesting because we're dealing with two different sets of proportions, and we need to find a way to connect them.
We're asked to find the sum of their current ages, which would be 3x + 4x, or 7x. So, our main goal is to find the value of 'x'. Once we find 'x,' we can easily calculate 7x and get our answer. Remember, stay focused on the ultimate goal, which is finding the sum. It's easy to get lost in the calculations, but always keep the final question in mind. This will help you stay on track and avoid making unnecessary steps. So, now that we have a clear understanding of what the problem is asking, let's move on to setting up the equations and solving for 'x'.
Setting Up the Equations
Alright, now let's translate this word problem into some sweet algebraic equations! This is where the magic happens, guys. We already know that the friends' current ages are 3x and 4x. And after 2 years, their ages are 3x + 2 and 4x + 2. The problem also tells us that after 2 years, their ages are proportional to 4 and 5. This is the key to setting up our equation.
Remember, when we say two things are proportional, it means their ratio is constant. So, we can write the following proportion: (3x + 2) / (4x + 2) = 4 / 5. This equation is the heart of the problem. It connects the two different time points (current ages and ages after 2 years) through the proportionality relationship. Make sure you understand how we derived this equation from the problem statement. It's crucial for solving similar problems in the future.
Think of this equation as a bridge connecting the past and the future ages of the friends. We're using the information about their ages after 2 years to figure out their current ages. It's like a detective piecing together clues to solve a mystery! Now, we have one equation with one unknown variable ('x'), which means we can solve for 'x'. This is a standard algebraic equation, and we'll use cross-multiplication to solve it. This technique is a fundamental tool in algebra, so make sure you're comfortable with it.
Before we jump into solving, let's take a quick breather and recap what we've done so far. We've successfully translated the word problem into a clear algebraic equation. This is a major step because it allows us to use the power of algebra to find the solution. Pat yourselves on the back for getting this far! Now, let's roll up our sleeves and get our hands dirty with the actual solving process. We're almost there, guys!
Solving for 'x'
Okay, guys, let's get down to business and solve for 'x'! We have the equation (3x + 2) / (4x + 2) = 4 / 5. To get rid of the fractions, we're going to use cross-multiplication. This means we multiply the numerator of the left side by the denominator of the right side, and vice versa. So, we get 5 * (3x + 2) = 4 * (4x + 2). This step is critical, so double-check your work to make sure you've cross-multiplied correctly. A small mistake here can throw off the entire solution.
Now, let's distribute the numbers on both sides of the equation. We get 15x + 10 = 16x + 8. Remember the distributive property? It's your best friend in these situations! We're just multiplying the numbers outside the parentheses by each term inside the parentheses. Now we have a linear equation, which is much easier to solve. Our goal is to isolate 'x' on one side of the equation. To do that, we'll subtract 15x from both sides and subtract 8 from both sides.
This gives us 10 - 8 = 16x - 15x, which simplifies to 2 = x. Woohoo! We found 'x'! Give yourselves a cheer because this is a huge accomplishment. We've solved the main unknown variable, which is the key to unlocking the answer to the entire problem. But hold on a second, we're not quite done yet. Remember, the problem asked for the sum of their current ages, not just the value of 'x'. So, we need to take this value of 'x' and plug it back into our original expressions for their ages.
Before we move on, let's pause and appreciate the journey we've taken so far. We started with a word problem, translated it into an algebraic equation, and then solved for 'x'. This is the essence of problem-solving in mathematics. It's about breaking down complex problems into smaller, manageable steps. Now, let's use this value of 'x' to find the sum of their ages and finally answer the question!
Finding the Sum of Current Ages
Alright, guys, we've conquered the most challenging part – finding 'x'! Now, let's bring it home and calculate the sum of the friends' current ages. We know that their current ages are 3x and 4x, and we found that x = 2. So, the first friend's age is 3 * 2 = 6 years old, and the second friend's age is 4 * 2 = 8 years old. This is where all our hard work pays off! We're finally seeing the light at the end of the tunnel.
Now, to find the sum of their ages, we simply add them together: 6 + 8 = 14 years. And there you have it! The sum of their current ages is 14 years. High fives all around! We've successfully solved the problem. We've taken a word problem, translated it into equations, solved for the unknown, and answered the question. This is a testament to your problem-solving skills!
So, the answer is 14. Looking at the answer choices, we see that option A) 14 is the correct answer. Always make sure to double-check your answer against the options provided. It's a simple step, but it can save you from making careless mistakes. And remember, showing your work is super important, especially in math problems. It not only helps you keep track of your steps but also allows you to identify any errors you might have made along the way.
Final Thoughts
Awesome job, everyone! We successfully tackled this age proportion problem. Remember, the key to solving these types of problems is to break them down into smaller, manageable steps. First, understand the problem and identify what you're trying to find. Then, translate the word problem into algebraic equations. Next, solve for the unknown variables. And finally, use those values to answer the original question.
Age proportion problems might seem tricky at first, but with practice, you'll become a pro at solving them. The most important thing is to stay calm, be patient, and trust your skills. And don't be afraid to ask for help if you get stuck. Math is a team sport, and we're all in this together!
Keep practicing, keep learning, and keep challenging yourselves. You've got this! And remember, every problem you solve makes you a little bit smarter and a little bit more confident. So, go out there and conquer the world of math!