Finding The Number: A Step-by-Step Math Guide

by Tim Redaksi 46 views
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Hey everyone! Today, we're diving into a fun math problem: determining a number when you know what happens to it after some calculations. This is a classic type of algebra problem, and it's super useful for understanding how equations work. Let's break it down step-by-step so you can totally nail it! We'll start with the question: "If a number is multiplied by 4 and then reduced by 6, the result is 18. Determine the number and explain the solution steps."

Understanding the Problem

Okay, so the problem gives us a few clues. We know there's a secret number, and we know that if we perform certain operations on that number, we get 18. The operations are: First, multiply the number by 4, and then subtract 6 from the result. Our mission, should we choose to accept it (and we do!), is to figure out what that original number is. This type of problem is all about working backward, undoing the operations to get to the starting point. It's like unwinding a tangled ball of yarn – you have to go in reverse to get back to the beginning! The core concept here revolves around the idea of inverse operations. Every mathematical operation has an inverse, which is an operation that reverses the effect of the original operation. For example, the inverse of addition is subtraction, and the inverse of multiplication is division. We'll use these inverse operations to isolate the unknown number and find its value. The goal is to isolate the variable (the unknown number) on one side of the equation and the constant terms on the other side. This is achieved by performing the inverse operations in the reverse order of the original operations.

Let's break down the problem statement into a mathematical expression. We can represent the unknown number with a variable, let's say 'x'. The problem states that the number is first multiplied by 4. Mathematically, this can be written as 4 * x or simply 4x. Then, 6 is subtracted from the result. This operation is represented as 4x - 6. Finally, the problem states that this entire expression equals 18. So, we can write the equation as: 4x - 6 = 18. This equation encapsulates the entire problem statement in a concise mathematical form. Now our goal is to solve this equation and find the value of 'x'. This involves a series of steps that use the properties of equality and inverse operations. Each step in solving the equation must maintain the balance of the equation, ensuring that whatever operations we perform on one side, we perform on the other side as well. This guarantees that we are always working with equivalent equations, eventually leading us to the correct value of 'x'.

Now, let's think about the general approach. What have we been told? We're told that "if a number is multiplied by 4 and then reduced by 6, the result is 18." So, this can also be expressed as an equation. We're going to use algebraic principles, especially working backwards to solve this problem. The primary aim is to reveal the unknown number, so we will simplify the equation until the unknown number can be easily identified. The best way to approach this problem is to systematically undo each operation that has been performed on the unknown number. We'll start with the last operation that was performed (subtraction) and work our way backward. This approach ensures we are reversing the equation steps correctly.

Step-by-Step Solution

Alright, let's get into the step-by-step solution. We're essentially working backward to isolate the unknown number, as mentioned before. Here's how it goes:

  1. Write the Equation: First, let's translate the problem into a mathematical equation. Let's use 'x' to represent the unknown number. The problem tells us that "if a number is multiplied by 4 and then reduced by 6, the result is 18." So, our equation is: 4x - 6 = 18.

  2. Undo the Subtraction: The last thing that happened to our number was subtracting 6. To undo this, we need to do the opposite: add 6 to both sides of the equation. This is a crucial step in maintaining the balance of the equation. By adding 6 to both sides, we ensure that the equality remains true. This operation isolates the term with 'x' on one side and simplifies the other side. So, we have: 4x - 6 + 6 = 18 + 6. This simplifies to 4x = 24.

  3. Undo the Multiplication: Now, the number is being multiplied by 4. To undo this, we do the opposite: divide both sides of the equation by 4. This is the inverse operation, and it helps isolate 'x'. Dividing both sides ensures that the equality remains intact. Now, we have: 4x / 4 = 24 / 4. This simplifies to x = 6.

  4. The Answer: Therefore, the number is 6! We've successfully solved for 'x'. Congratulations, you've cracked the code!

Verification

Now, always a good practice in math, let's check our answer to make sure we're right. To verify our answer, we will substitute the value of x (which is 6) back into the original equation. The original equation was 4x - 6 = 18. Substituting x = 6, we get 4 * 6 - 6 = 18. Then, we simplify the left side of the equation. 4 multiplied by 6 is 24, so we now have 24 - 6 = 18. Next, we calculate 24 minus 6, which equals 18. So the equation becomes 18 = 18. This verifies that our solution, x = 6, is correct because the left side equals the right side. Our answer checks out. This confirmation is crucial to validate our mathematical solution and ensure the problem is solved correctly.

Let's substitute our answer (6) back into the original word problem. If we take 6, multiply it by 4 (6 * 4 = 24), and then subtract 6 (24 - 6 = 18), we get 18, which is what the problem stated. Therefore, our answer is correct. This method is essential for verifying any algebraic solution, ensuring that we didn't make any errors in our calculations. It gives us confidence in our understanding and ability to solve such problems. You can also see how this is done with mental math. For example, 4 * 6 = 24 and 24 - 6 = 18, so we know the answer must be 6.

Tips and Tricks for Similar Problems

Here are some tips and tricks that will help you solve similar problems in the future:

  • Read Carefully: Make sure you understand what the question is asking. Sometimes, the wording can be a little tricky. Take your time to break down each part of the problem.
  • Translate into an Equation: Practice turning word problems into mathematical equations. This is a fundamental skill in algebra.
  • Work Backwards: Always remember to work backward, undoing the operations one by one.
  • Check Your Answer: Always verify your answer by plugging it back into the original problem.

Conclusion

Great job, everyone! You've successfully found the number. This is a crucial first step in understanding and solving more complex algebraic problems. Keep practicing, and you'll become a pro at these problems in no time. Remember to always apply the steps we used today. Keep your cool and break down the problem to get to the end result.

Key Takeaways:

  • The most important thing is to understand what the question is asking and break it into parts.
  • Using inverse operations is critical when solving for the unknown.
  • Always verify your answer!

I hope this step-by-step guide has been helpful. If you have any more questions, feel free to ask. See you next time, and keep up the great work, everyone! Now you can easily deal with these types of questions. Keep practicing, and good luck!