Unraveling The Number Puzzle: Finding The Initial Value

by Admin 56 views
Unraveling the Number Puzzle: Finding the Initial Value

Hey there, math enthusiasts! Today, we're diving into a numerical riddle. We're going to use our problem-solving skills to figure out an initial number. This type of problem is a classic example of how algebra can be used in the real world, even if it doesn't always seem like it. I am going to explain step by step how to solve this math problem so that you can understand and remember it. So, let's get started.

Understanding the Problem: The Foundation of Solving

Before we start working on the problem, it is very important to first understand what the problem is asking. The first step in cracking any math problem is to thoroughly understand what is being asked. Let's break down the information presented: "The triple of a number decreased by ten times 820 results in 34,900." Okay, here are the key parts:

  • We're looking for an initial "number." This is our unknown, the treasure we're trying to find. We can represent this with a variable, like 'x'.
  • "The triple of a number" means three times the number. This translates to 3x.
  • "Decreased by ten times 820" means we subtract the result of 10 multiplied by 820. This is 10 * 820, which equals 8200.
  • "Results in 34,900" tells us that after performing the calculations, the answer is 34,900. This implies an equation where everything before that equals 34,900.

So, putting it all together, the core of the problem is represented by the equation: 3x - 8200 = 34,900. It is extremely important that you break down the problem in order to understand what the question is asking and what steps you need to take to solve the equation. The foundation of any math problem relies on the understanding of each key component that is presented in the problem.

Setting Up the Equation: Turning Words into Math

Now that we've broken down the problem, it's time to translate the word problem into a mathematical equation. It is also important to know that the equation we are going to write is based on the keywords and the breakdown of the problem that we went over. Remember, we are going to use the variable 'x' to represent our unknown number. Following our breakdown from before, the equation will be: 3x - 8200 = 34,900. This equation accurately captures the relationships described in the word problem: three times the number (3x), decreased by 8200, equals 34,900. Once you write the equation, all you need to do is solve it and find what x is equal to. So, you must make sure that the equation is written properly. A wrongly written equation will make the entire problem wrong. The equation will act as a roadmap so that you can find the final answer of what the question is asking for.

Solving for 'x': Finding the Initial Number

With our equation in hand, 3x - 8200 = 34,900, let's solve for 'x'. Solving for 'x' means isolating 'x' on one side of the equation to find its value. Remember, we need to apply opposite operations to keep the equation balanced. Here's how we'll do it step-by-step:

  1. Isolate the term with 'x': Our goal is to get the term with 'x' (3x) by itself. To do this, we need to get rid of the -8200. Since we're subtracting 8200, we do the opposite operation: add 8200 to both sides of the equation. This keeps the equation balanced.
    • 3x - 8200 + 8200 = 34,900 + 8200
    • This simplifies to 3x = 43,100
  2. Solve for 'x': Now we have 3x = 43,100. We want to find the value of x, not 3x. Since 'x' is multiplied by 3, we do the opposite: divide both sides of the equation by 3.
    • 3x / 3 = 43,100 / 3
    • This simplifies to x = 14,366.6667.

Therefore, the initial number is approximately 14,366.67. This process of using inverse operations, like adding and dividing, is fundamental in solving any algebraic equation. Make sure to keep the equation balanced by applying operations to both sides.

Verification: Double-Checking the Answer

We've found our answer, but it's always a good idea to check our work. Verification helps ensure we haven't made any mistakes along the way. To do this, we'll substitute our found value back into the original word problem to see if it makes sense. If the result equals 34,900, we know we're spot on.

  • The original problem says, "The triple of a number decreased by ten times 820 results in 34,900."
  • Our found number is 14,366.6667. Let's do the math.
    • Triple of the number: 3 * 14,366.6667 = 43,100.0001 (rounding difference)
    • Ten times 820: 10 * 820 = 8200
    • Subtraction: 43,100.0001 - 8200 = 34,900.0001

The result is approximately 34,900. The slight difference is due to rounding during the calculations, but the result is close enough to know we have solved the problem correctly. This step is a critical part of problem-solving. It checks whether the answer we came up with is actually correct. If the number does not match, then you can easily tell that the problem was done incorrectly.

Conclusion: Mastering the Numerical Riddle

So there you have it, guys! We've successfully solved the numerical riddle. We started with a word problem, broke it down into an equation, solved for our unknown variable, and then verified our answer. The initial number is approximately 14,366.67. This process demonstrates the power of algebra in turning a seemingly complex problem into a manageable one. Remember, the key is to understand the question, translate it into math, solve the equation carefully, and always verify your answer. Now you know how to solve problems like this one, it is time to practice and make sure you understand each step. With practice, you'll get more confident and be able to solve these problems faster. Keep practicing and keep up the great work!