Math Puzzle: Triple A Number, Add 10, Equals 40!

Hey guys! Let's dive into this awesome math puzzle together. It's got a bit of everything – basic algebra and some geometry thrown in for good measure. We're going to break it down step by step, so don't worry if it looks a little intimidating at first. We'll solve for the mystery number, figure out the distance between those points, and then put it all together for the final answer. Math can be super fun when you treat it like a puzzle, and this one is a real brain-teaser! So, grab your thinking caps, and let's get started!

Unraveling the First Part: The Mystery Number

Okay, so the first part of our puzzle asks us to find a number. This is where we dust off our algebra skills. Algebra is super useful for solving these kinds of problems because it lets us use symbols and equations to represent what we're trying to figure out. In this case, we're looking for a number, so let's call that number 'x'. That's a classic algebraic move, right? Now, let's translate the words of the problem into a mathematical equation. The problem tells us to triple the number. What does that mean mathematically? It simply means multiply the number ('x') by 3. So, we get 3x. The problem then says to add 10 to this tripled number. So, we add 10 to our 3x, giving us 3x + 10. And finally, the problem tells us that this whole thing, 3x + 10, is equal to 40. So, we can write our equation as: 3x + 10 = 40. This is our algebraic representation of the first part of the puzzle, now it's time to solve this equation to discover the value of 'x'.

Solving the Algebraic Equation

Now that we have our equation, 3x + 10 = 40, we need to solve for 'x'. This involves isolating 'x' on one side of the equation. To do that, we'll use some basic algebraic manipulations. The first thing we want to do is get rid of that +10 on the left side. We can do this by subtracting 10 from both sides of the equation. Remember, in algebra, whatever you do to one side, you have to do to the other to keep things balanced. So, subtracting 10 from both sides, we get: 3x + 10 - 10 = 40 - 10. This simplifies to 3x = 30. Great! We're one step closer. Now, we have 3x = 30. This means 3 times 'x' is equal to 30. To find 'x', we need to undo this multiplication. We do that by dividing. So, we divide both sides of the equation by 3: 3x / 3 = 30 / 3. This simplifies to x = 10. Boom! We've found our mystery number. The value of x is 10. This means that the number we were looking for, the one that when tripled and added to 10 equals 40, is indeed 10. But we're not done yet! We've only solved the first part of the puzzle. Now we need to tackle the geometry part and then bring it all together.

Tackling the Geometry: Distance Between Points

The second part of our puzzle involves a little bit of geometry. Specifically, we need to find the distance between two points, P(1, 1) and Q(5, 1). Now, when you see coordinates like this, you might think of the coordinate plane, that familiar grid with the x and y axes. Points are located on this plane using pairs of numbers, like our (1, 1) and (5, 1). The first number in the pair tells you how far to move along the x-axis (horizontally), and the second number tells you how far to move along the y-axis (vertically). So, point P(1, 1) is located 1 unit to the right and 1 unit up from the origin (the point where the axes cross). Point Q(5, 1) is located 5 units to the right and 1 unit up from the origin. Now, we want to find the distance between these two points. There's a handy formula we can use for this, called the distance formula, but in this case, we can actually take a shortcut! Notice that both points have the same y-coordinate (they're both at y = 1). This means that the points lie on a horizontal line. When points lie on a horizontal or vertical line, finding the distance is super easy. We just need to look at the difference in their x-coordinates.

Calculating the Distance

Since our points P(1, 1) and Q(5, 1) lie on a horizontal line, finding the distance between them is actually quite straightforward. We just need to look at the difference in their x-coordinates. Point P has an x-coordinate of 1, and point Q has an x-coordinate of 5. The distance between them is simply the absolute value of the difference between these numbers. So, we calculate |5 - 1| = |4| = 4. This means the distance between points P and Q is 4 units. Think of it like counting the spaces on the number line between 1 and 5 – there are 4 spaces. Now, the problem isn't just asking for this distance. It's asking for half of this distance. So, we need to divide our distance of 4 by 2. 4 / 2 = 2. So, half the distance between points P and Q is 2 units. We've conquered the geometry part of the puzzle! We know the distance and half the distance. Now, it's time for the grand finale – putting it all together.

The Grand Finale: Putting It All Together

Alright, guys, we've done the hard work! We've solved for the mystery number (x = 10), and we've calculated half the distance between points P and Q (which is 2). Now comes the final step: dividing the mystery number by half the distance. The problem asks us to take the number we found in the first part (which was 10) and divide it by the value we found in the second part (which was 2). This is a simple division problem: 10 / 2. And the answer is... 5! So, there you have it. We've solved the entire puzzle. By carefully working through each part, translating the words into math, and using our algebra and geometry skills, we arrived at the final answer. Remember, math problems are often like puzzles – they have different pieces that need to fit together. And sometimes, the best way to solve a complex problem is to break it down into smaller, more manageable steps. We did it! What an awesome journey of solving the problems.

Conclusion: We Cracked the Code!

So, to recap, we started with a seemingly complex math puzzle: find a number that, when tripled and added to 10, equals 40, and then divide that number by half the distance between two points. We broke the problem down into two main parts: an algebra part and a geometry part. For the algebra part, we translated the words into an equation (3x + 10 = 40) and then solved for 'x', finding that the mystery number was 10. For the geometry part, we found the distance between points P(1, 1) and Q(5, 1), which was 4, and then calculated half that distance, which was 2. Finally, we divided the mystery number (10) by half the distance (2), giving us our final answer of 5. The answer is 5. This puzzle is a great example of how different areas of math, like algebra and geometry, can come together to solve a single problem. It also highlights the importance of careful reading, translating words into math, and breaking down complex problems into smaller steps. You guys rocked it! Keep practicing and challenging yourselves with these kinds of puzzles, and you'll become math masters in no time!