Card Puzzle: Finding Blue, Purple, And Sky-Blue Card Counts

Hey guys! Ever stumbled upon a math problem that felt like unlocking a secret treasure chest? Well, today we're diving into one of those puzzles! We've got a box filled with 96 cards – some are blue, some are purple, and others are sky-blue. The challenge? Figuring out exactly how many of each color are hiding inside. It sounds like a fun little detective game, right? So, let's put on our thinking caps and get ready to crack this code!

Deciphering the Card Mystery: A Step-by-Step Solution

In this section, we'll break down the problem step by step, making it super easy to follow. We'll translate the word puzzle into math equations and then solve those equations to reveal the number of each colored card. Think of it like translating an ancient scroll – once we understand the language, the message becomes clear. So, grab a pen and paper, and let's get started on our mathematical adventure!

1. Setting Up the Equation: Decoding the Clues

The first step in solving this card conundrum is to translate the information we have into mathematical expressions. Let's start by assigning a variable to the unknown number of blue cards. We'll call this 'x'. Now, let's carefully examine the clues given in the problem. The problem states that the number of sky-blue cards is the same as the number of blue cards. This means if we have 'x' blue cards, we also have 'x' sky-blue cards. Easy peasy, right?

Now, let's move on to the purple cards. The problem tells us that the number of purple cards is triple the number of blue cards, plus six. This can be written as 3x + 6. So, for every blue card, there are three purple cards, plus an extra six. This is where the puzzle starts to get interesting!

We also know the total number of cards in the box is 96. This is a crucial piece of information because it allows us to create an equation that ties everything together. We can add the number of blue cards (x), the number of sky-blue cards (x), and the number of purple cards (3x + 6) and set that equal to 96. This gives us the equation: x + x + (3x + 6) = 96. This equation is the key to unlocking the mystery of the cards!

This equation is like a map leading us to the treasure. It represents the relationship between the number of cards of each color and the total number of cards in the box. By solving this equation, we'll be able to find the value of 'x', which will then allow us to determine the number of each colored card. So, let's move on to the next step and see how we can simplify and solve this equation!

2. Solving for 'x': Unveiling the Value of Blue Cards

Now that we have our equation, x + x + (3x + 6) = 96, it's time to roll up our sleeves and solve for 'x'. Think of 'x' as the secret code we need to crack to unlock the number of blue cards. To do this, we need to simplify the equation by combining like terms. This is like organizing our puzzle pieces so we can fit them together more easily.

First, let's combine the 'x' terms on the left side of the equation. We have x + x + 3x, which adds up to 5x. So, our equation now looks like this: 5x + 6 = 96. We're getting closer to isolating 'x' and revealing its value!

Next, we need to isolate the term with 'x' in it. To do this, we'll subtract 6 from both sides of the equation. Remember, whatever we do to one side of the equation, we have to do to the other to keep things balanced. Subtracting 6 from both sides gives us: 5x = 90. We're almost there!

Finally, to solve for 'x', we need to divide both sides of the equation by 5. This will give us the value of 'x', which represents the number of blue cards. Dividing both sides by 5, we get: x = 18. Eureka! We've cracked the code! This means there are 18 blue cards in the box. But our adventure isn't over yet – we still need to figure out how many sky-blue and purple cards there are.

Knowing the value of 'x' is like finding the first piece of a jigsaw puzzle. It gives us a starting point and allows us to piece together the rest of the picture. Now that we know there are 18 blue cards, we can use this information and the clues from the problem to determine the number of sky-blue and purple cards. So, let's move on to the final step and reveal the complete solution!

3. Finding the Other Colors: Completing the Card Count

With the value of 'x' (the number of blue cards) now revealed as 18, we're in the home stretch! It's like we've found the key to the treasure chest, and now we just need to open it and see what's inside. To find the number of sky-blue and purple cards, we'll simply plug the value of 'x' back into the expressions we set up earlier. This is where all our hard work pays off!

First, let's find the number of sky-blue cards. The problem told us that the number of sky-blue cards is the same as the number of blue cards. Since we know there are 18 blue cards, this means there are also 18 sky-blue cards. That was easy, right? It's like finding two identical puzzle pieces that fit perfectly together.

Now, let's tackle the purple cards. The problem stated that the number of purple cards is 3x + 6. To find this number, we'll substitute 18 for 'x' in this expression. This gives us 3 * 18 + 6. Let's do the math: 3 * 18 equals 54, and 54 + 6 equals 60. So, there are 60 purple cards in the box. We've found the final piece of the puzzle!

To make sure we've got everything right, let's add up the number of each color of card: 18 blue cards + 18 sky-blue cards + 60 purple cards = 96 cards. And that's the total number of cards in the box! We've successfully solved the mystery and determined the number of each colored card. High five!

This final step is like putting the last piece of a jigsaw puzzle in place. It completes the picture and gives us a sense of accomplishment. We've not only solved the problem but also confirmed that our solution is correct. So, let's celebrate our mathematical victory and move on to the next challenge!

Wrapping Up the Card Puzzle: A Sense of Accomplishment

Alright guys, we did it! We successfully navigated the world of variables and equations to uncover the hidden numbers of blue, purple, and sky-blue cards in the box. It's like we went on a mathematical quest and emerged victorious. This problem wasn't just about finding the right answer; it was about learning how to break down a problem, translate words into math, and then solve it step by step. These are skills that will come in handy in all sorts of situations, not just in math class.

Remember, the key to solving problems like this is to take it one step at a time. Don't get overwhelmed by the whole problem; instead, focus on understanding each clue and translating it into a mathematical expression. Once you have your equation set up, the rest is just a matter of following the rules of algebra. And if you ever get stuck, don't be afraid to ask for help or look for resources online.

Solving math problems can be like solving a riddle or cracking a code. It's a fun way to challenge your mind and improve your problem-solving skills. So, the next time you encounter a math puzzle, don't shy away from it. Embrace the challenge and see what you can discover. You might just surprise yourself with what you're capable of!

So, to recap, in our box of 96 cards, we discovered that there are 18 blue cards, 18 sky-blue cards, and a whopping 60 purple cards. We solved this using a combination of careful reading, algebraic manipulation, and a little bit of mathematical detective work. Give yourselves a pat on the back – you've earned it!

Now that we've conquered this card puzzle, who knows what other mathematical mysteries await us? Keep your minds sharp, your pencils ready, and your problem-solving skills honed. There's a whole world of mathematical adventures out there, just waiting to be explored. Until next time, keep those numbers crunching and those equations solving!