Solving Math Problems Finding Numbers And Triangle Angles

Hey guys! Math problems can sometimes feel like decoding a secret message, right? But don't worry, we're going to break down a couple of common types of problems and solve them step by step. We'll look at a number puzzle and a triangle angle problem. Let's dive in!

Finding the Elusive Number

In this first problem, we're on a quest to find a specific number based on a set of clues. These clues are presented as a word problem, which we'll translate into a mathematical equation. The key here is to carefully identify the different parts of the problem and how they relate to each other. Think of it like piecing together a puzzle, where each piece of information leads us closer to the solution.

Our problem states: "If 15 is added to 5 times a number, the result is 18 more than 4 times a number." Sounds a bit like a riddle, doesn't it? Let's break it down: the main goal here is to find the value of "a number." Since we don't know what that number is yet, we'll represent it with a variable. A variable is just a symbol, usually a letter like x, y, or n, that stands for an unknown quantity. In this case, let's use x to represent our mystery number. So, x is our unknown, and our mission is to figure out its value.

Now, let's translate the words into mathematical expressions. When we see "5 times a number," that means we're multiplying 5 by our variable x. So, we can write that as 5x, or simply 5x. Next, we have "15 is added to 5 times a number." This means we're adding 15 to our 5x. So, we can write that part as 5x + 15. The problem then tells us that "the result is" something else. In math, "is" often means equals, so this phrase indicates an equality. We're setting the first part of the problem equal to something else.

On the other side of the equation, we have "18 more than 4 times a number." Similar to before, "4 times a number" translates to 4x. Then, "18 more than" means we're adding 18 to that. So, this part becomes 4x + 18. Now we have all the pieces to build our equation. We know that "15 added to 5 times a number" (5x + 15) is equal to "18 more than 4 times a number" (4x + 18). Putting it all together, our equation is: 5x + 15 = 4x + 18. This is the mathematical representation of our word problem, and it's the key to finding our mystery number.

Now comes the fun part – solving the equation! Our goal is to isolate the variable x on one side of the equation. This means we want to get x by itself, so we can see what it equals. We can do this by using inverse operations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. To isolate x, we need to get all the terms with x on one side of the equation and all the constant terms (the numbers without variables) on the other side. Let's start by subtracting 4x from both sides of the equation. This will eliminate the 4x term on the right side. Remember, whatever we do to one side of the equation, we must do to the other side to keep it balanced.

Subtracting 4x from both sides gives us: 5x + 15 - 4x = 4x + 18 - 4x. Simplifying this, we get x + 15 = 18. Now we have x on the left side, but it's not quite alone yet. We still have the +15 term. To get rid of that, we need to perform the inverse operation of addition, which is subtraction. So, we'll subtract 15 from both sides of the equation. This gives us: x + 15 - 15 = 18 - 15. Simplifying again, we get x = 3. And there we have it! We've solved for x, and we've found our mystery number. The number is 3. To make sure our answer is correct, we can plug it back into the original equation and see if it holds true. This is called checking our solution. Let's substitute 3 for x in our equation: 5x + 15 = 4x + 18. Substituting, we get: 5(3) + 15 = 4(3) + 18. Simplifying, we have: 15 + 15 = 12 + 18. And finally: 30 = 30. Since both sides of the equation are equal, our solution is correct. We've successfully found the number!

Cracking the Triangle Angle Code

Let's tackle another exciting problem! This time, we're venturing into the world of geometry, specifically triangles. Triangles are fascinating shapes with lots of interesting properties, and one of the most fundamental is that the sum of the three angles inside any triangle always adds up to 180 degrees. This is a crucial piece of information that we'll use to solve our problem. In our problem, we're given some clues about the angles of a particular triangle, and our mission is to figure out the measure of each angle. It's like being a detective and using clues to uncover hidden information.

The problem states: "If two angles of a triangle are in the ratio 3:2 and the sum of these two angles is equal to the third angle, find the angles of the triangle." This might sound a bit complicated at first, but let's break it down step by step. The first piece of information is that two angles are in the ratio 3:2. What does this mean? A ratio is simply a way of comparing two quantities. In this case, it's comparing the sizes of two angles. The ratio 3:2 tells us that one angle is 3 parts and the other angle is 2 parts, where each "part" represents a certain number of degrees. We don't know the exact size of each part yet, but we can represent it with a variable. Let's use x again. So, if one angle is 3 parts, we can represent it as 3x degrees. And if the other angle is 2 parts, we can represent it as 2x degrees. This is a crucial step in translating the ratio into algebraic terms.

Next, the problem tells us that "the sum of these two angles is equal to the third angle." This gives us another important relationship between the angles. We know that the two angles we've represented as 3x and 2x add up to something, and that "something" is the measure of the third angle. So, we can write this relationship as: Third angle = 3x + 2x. This means the third angle is the sum of the first two angles. Now, we come back to that fundamental property of triangles: the sum of the three angles in any triangle is always 180 degrees. We can use this fact to build an equation that will allow us to solve for x. We know the three angles are 3x, 2x, and 3x + 2x. So, if we add them together, they should equal 180 degrees. This gives us the equation: 3x + 2x + (3x + 2x) = 180. This equation represents the relationship between all three angles in the triangle and the total degrees in a triangle. Now that we have our equation, let's solve for x. First, we need to simplify the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, all the terms on the left side of the equation have x to the power of 1, so they are like terms. We can add their coefficients (the numbers in front of the variables) together.

We have 3x + 2x + 3x + 2x. Adding the coefficients, we get: (3 + 2 + 3 + 2)x = 10x. So, our simplified equation becomes: 10x = 180. Now, to isolate x, we need to get rid of the 10 that's multiplying it. The inverse operation of multiplication is division, so we'll divide both sides of the equation by 10. This gives us: 10x / 10 = 180 / 10. Simplifying, we get: x = 18. Great! We've found the value of x. But remember, x is just a part of our solution. It represents one "part" of the angles. We still need to find the actual measures of the three angles. We know the first angle is 3x, so we'll substitute 18 for x to find its measure: First angle = 3x = 3(18) = 54 degrees. The second angle is 2x, so we'll substitute again: Second angle = 2x = 2(18) = 36 degrees. The third angle is the sum of the first two, which we can also calculate: Third angle = 3x + 2x = 54 + 36 = 90 degrees. So, we've found all three angles of the triangle! They are 54 degrees, 36 degrees, and 90 degrees. To make sure our answer is correct, we can check if the angles add up to 180 degrees: 54 + 36 + 90 = 180. Yes, they do! We've successfully cracked the triangle angle code.

Conclusion

So, there you have it! We've tackled two different types of math problems, from finding a mystery number to discovering the angles of a triangle. The key to solving these problems is to break them down into smaller, manageable steps. Translate the words into mathematical expressions, build equations, and then use inverse operations to isolate the variables. And remember to always check your solutions! Math can be challenging, but it's also a rewarding puzzle to solve. Keep practicing, and you'll become a math master in no time!