Ages Of A And B: Solving An Age-Old Riddle

Let's dive into a classic age problem! We're trying to figure out the current ages of two individuals, A and B, based on the information we have: A's age is currently three times B's age, and in 20 years, A's age will be twice B's age. It's like a detective story, but with numbers instead of clues. Ready to put on your thinking caps and solve this age-old riddle?

Setting Up the Equations

Alright, guys, let's translate this word problem into math! This is where we transform everyday language into the precise language of equations. It might sound intimidating, but trust me, it's like learning a new superpower. Once you can turn words into equations, you can solve all sorts of problems!

First, we need to define our variables. Let's say:

• A's current age is represented by the variable 'a'.
• B's current age is represented by the variable 'b'.

Now, let's use the information we have to create our equations:

1. "A's age is the triple of B's": This means that a = 3b. This is our first equation, and it tells us the fundamental relationship between A and B's ages right now. It's like saying A is always three times older than B at this moment in time.
2. "In 20 years, A's age will be the double of B's": In 20 years, A's age will be a + 20, and B's age will be b + 20. So, the equation becomes a + 20 = 2(b + 20). This equation describes their ages in the future, 20 years from now. It shows how their ages will relate to each other after time has passed.

So, we have two equations:

• a = 3b
• a + 20 = 2(b + 20)

Now, the fun begins! We'll use these equations to find the values of 'a' and 'b', which will tell us the current ages of A and B.

Solving the System of Equations

Okay, now that we have our equations set up, it's time to solve them. There are a couple of ways we can tackle this, but substitution is often the easiest method for this type of problem. Here's how it works:

We know that a = 3b, so we can substitute '3b' for 'a' in the second equation. This means we're replacing 'a' in the second equation with its equivalent value in terms of 'b'.

So, the second equation, a + 20 = 2(b + 20), becomes:

• 3b + 20 = 2(b + 20)

Now, let's simplify and solve for 'b'. First, distribute the 2 on the right side:

• 3b + 20 = 2b + 40

Next, subtract 2b from both sides:

• b + 20 = 40

Then, subtract 20 from both sides:

• b = 20

So, B's current age is 20 years old! We've found our first piece of the puzzle.

Now that we know B's age, we can find A's age using the equation a = 3b. Substitute 20 for 'b':

• a = 3 * 20
• a = 60

So, A's current age is 60 years old! We've found the other piece of the puzzle.

Checking Our Work

Before we declare victory, let's make sure our answers make sense. It's always a good idea to check our work, especially in math problems, to avoid silly mistakes!

We found that A is currently 60 years old and B is currently 20 years old. Let's see if this fits the information we were given:

• "A's age is the triple of B's": Is 60 three times 20? Yes, it is! (60 = 3 * 20)
• "In 20 years, A's age will be the double of B's": In 20 years, A will be 80 (60 + 20) and B will be 40 (20 + 20). Is 80 twice 40? Yes, it is! (80 = 2 * 40)

Since our answers fit both conditions, we can be confident that we've solved the problem correctly!

The Solution

Alright, guys, after all that equation-solving and checking, we've finally arrived at the solution!

• A's current age is 60 years old.
• B's current age is 20 years old.

So, there you have it! A is 60 and B is 20. We successfully navigated the twists and turns of this age problem and emerged victorious. Give yourselves a pat on the back!

Why are these problems important?

You might be wondering, "Okay, we solved for A and B's ages... but why does this even matter?" That's a fair question! While these age problems might seem like abstract puzzles, they actually help us develop some really important skills.

• Problem-solving: These problems require us to break down a complex situation into smaller, manageable parts. We have to identify the key information, translate it into equations, and then use logical steps to find the solution. This is a skill that's valuable in all aspects of life, from figuring out how to fix a leaky faucet to developing a new business strategy.
• Critical thinking: We can't just blindly accept the information we're given. We need to analyze it, question it, and make sure it makes sense. For example, we checked our work at the end to make sure our answers were consistent with the original problem. This critical thinking skill helps us make better decisions and avoid being misled.
• Mathematical reasoning: These problems help us strengthen our understanding of mathematical concepts like variables, equations, and systems of equations. This is essential for anyone pursuing a career in science, technology, engineering, or mathematics (STEM), but it's also helpful for understanding financial statements, analyzing data, and making informed choices in our daily lives.
• Attention to detail: Even a small mistake can throw off the entire solution. We need to be careful when setting up the equations, performing the calculations, and checking our work. This attention to detail is important in any field where accuracy is critical.

So, the next time you encounter an age problem, don't dismiss it as just a silly math exercise. Think of it as an opportunity to sharpen your mind and develop skills that will benefit you in all areas of your life. It's like a workout for your brain! And who knows, maybe one day you'll be the one writing these problems for others to solve!