Solving Age Puzzle: How Old Is Hannah?
Hey guys! Let's dive into a fun math problem today. We've got a classic age puzzle on our hands, and it’s a great way to flex those problem-solving muscles. This one involves figuring out how old Hannah is, given some clues about her and her siblings. So, grab your thinking caps, and let's get started!
Understanding the Problem
So, the core of the problem is this: Hannah is the oldest of four siblings. What's important here is that their ages are consecutive integers. This means their ages follow each other in order, like 1, 2, 3, 4 or 10, 11, 12, 13. There are no gaps in their ages. And we also know that if you add up all their ages, the total comes to 86. The big question here is, how old is Hannah? This is a classic algebra problem dressed up in a relatable scenario.
To really nail this, we need to break down what the question is giving us. Consecutive integers are the key. Think of it like a staircase where each step is one year older. The sum is our final total, the number we need to match when we’ve figured out everyone’s age. And, of course, Hannah's age is what we’re trying to find – the top step of that age staircase, if you will. We have to use all the given information, especially the consecutive integers, to get to the result. Without consecutive integers the possibilities would be endless, but this constraint lets us set up a clear mathematical approach and solve it step by step.
Setting Up the Equation
Now, let's translate this word problem into some math we can work with. This is where algebra comes to our rescue! Because we don’t know Hannah’s age, or the ages of her siblings, we can use variables to represent them. The trick here is to use the fact that their ages are consecutive to make things easier.
Let's say the youngest sibling is x years old. Since the next sibling is one year older, their age would be x + 1. The next one would be x + 2, and Hannah, being the oldest, would be x + 3 years old. See how we’re using x to represent all their ages in relation to each other? This is a crucial step in solving these kinds of problems, guys. If you name them all different variables, like a, b, c, and d, you're not going to be able to put them in one equation and solve for the ages. So, now that we have the ages represented algebraically, the next step is to add them up, because we know the sum of their ages.
We know the sum of their ages is 86, so we can write the equation like this: x + (x + 1) + (x + 2) + (x + 3) = 86. This equation is the heart of our problem. It takes all the information we have and puts it into a format we can solve. It’s like a recipe – we’ve got all the ingredients (the ages), and now we need to mix them together in the right way (the equation) to get the final result (Hannah’s age). This step of turning the words into an equation is often the hardest part, but once you’ve got it, the rest is just algebra!
Solving for x
Alright, we've got our equation: x + (x + 1) + (x + 2) + (x + 3) = 86. Now comes the fun part – simplifying and solving for x! This is where we put our algebra skills to the test.
First, let’s combine like terms. We've got four x’s, so that’s 4x. Then we have the numbers 1, 2, and 3, which add up to 6. So, our equation simplifies to 4x + 6 = 86. See how much cleaner that looks? Combining like terms makes the equation much easier to work with and reduces the chance of making mistakes. It’s like tidying up before you start cooking – a clean workspace makes everything flow better.
Now, we want to isolate x on one side of the equation. To do that, we need to get rid of the +6. We do this by subtracting 6 from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep things balanced. So, 4x + 6 - 6 = 86 - 6, which simplifies to 4x = 80. We are one step closer to finding the age of the youngest sibling, and hence the age of Hannah!
Finally, to solve for x, we need to get rid of the 4 that’s multiplying it. We do this by dividing both sides of the equation by 4. So, 4x / 4 = 80 / 4, which gives us x = 20. Woohoo! We’ve found x! But remember, x is the age of the youngest sibling. We’re not quite done yet; we still need to find Hannah’s age.
Finding Hannah's Age
Okay, so we’ve figured out that the youngest sibling is 20 years old. That’s a big step, but remember our original goal: to find Hannah’s age. This is where we go back to how we defined Hannah’s age in terms of x.
We said that Hannah is x + 3 years old, right? Well, now we know that x is 20, so we just need to substitute that value into the expression. So, Hannah’s age is 20 + 3 = 23 years old. And there we have it! We’ve cracked the code and found Hannah’s age.
It’s always a good idea to double-check our answer to make sure it makes sense in the context of the problem. If the youngest sibling is 20, then the other siblings are 21, 22, and 23. If we add those ages up (20 + 21 + 22 + 23), we get 86, which is the sum we were given in the problem. So, our answer checks out! This is the most important part of any math problem, or any task for that matter. Checking your result is the best way to ensure accuracy. You can even make up your own problem with different numbers, and practice solving them to get better at these math problems.
Conclusion
So, to wrap things up, Hannah is 23 years old. Awesome job, guys! We tackled this age puzzle by breaking it down into smaller steps: understanding the problem, setting up an equation, solving for x, and then using x to find Hannah’s age. This is a great example of how algebra can be used to solve real-world problems.
The key takeaway here is that word problems might seem tricky at first, but by translating them into mathematical expressions, they become much more manageable. This skill isn't just useful for math class; it's a life skill. Learning how to break down complex problems into smaller, solvable steps is essential in many areas of life.
And that’s it for this problem! I hope you guys had fun working through it with me. Remember, practice makes perfect, so keep those problem-solving skills sharp. Until next time, keep puzzling!