Solve For Initial Number: Step-by-Step Calculation

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Solve for Initial Number: Step-by-Step Calculation

Let's break down this math problem step by step, guys, and figure out the initial number. We're given a series of operations performed on an unknown number, and we need to reverse those operations to find our starting point. This involves a bit of algebraic thinking and working backward, but don't worry, we'll make it super clear. We will use a step-by-step approach to solve for the initial number, ensuring that each operation is carefully reversed to arrive at the correct answer. Understanding how to solve such problems enhances problem-solving skills and reinforces basic arithmetic principles. This method can be applied to various mathematical puzzles and real-world scenarios where reverse engineering is necessary to find the initial state or value.

Understanding the Problem

The problem describes a sequence of actions performed on a number, let’s call it x. These actions are:

  1. Subtract 5
  2. Divide by 7
  3. Multiply by 9
  4. Add 5
  5. Subtract 10

The final result after all these operations is 40. Our mission is to find the value of x. To do this, we will reverse the order of operations, applying the inverse operation at each step. This approach is fundamental in solving algebraic problems and understanding how mathematical operations affect the outcome. By systematically undoing each step, we can isolate the initial number and determine its value. This process not only helps in solving the specific problem but also enhances analytical and logical reasoning skills, which are crucial in mathematics and other fields.

Step 1: Reverse the Last Operation

The last operation was subtracting 10, which gave us 40. To reverse this, we need to add 10 back to 40:

40 + 10 = 50

So, before subtracting 10, the number was 50. This first step demonstrates the core strategy of reversing operations. By understanding that subtraction is undone by addition, we can systematically work backward through the problem. This principle applies universally in mathematics and is essential for solving equations and understanding inverse relationships. Recognizing and applying inverse operations is a key skill that simplifies complex problems and allows for efficient problem-solving. This foundational understanding builds confidence and accuracy in mathematical manipulations.

Step 2: Reverse the Second to Last Operation

Before subtracting 10, we added 5. Now, we subtract 5 from the result we got in step 1 (which was 50):

50 - 5 = 45

This tells us that before adding 5, the number was 45. Continuing to reverse the operations, we maintain the same logic: addition is undone by subtraction. This consistent approach makes the problem more manageable and reduces the chance of errors. Each step reinforces the concept of inverse operations and their application in solving mathematical problems. By carefully tracking each operation and its inverse, we ensure accuracy and clarity in our solution. This systematic process not only helps in finding the correct answer but also enhances our understanding of the underlying mathematical principles.

Step 3: Reverse the Third Operation

Before adding 5, we multiplied by 9. To reverse multiplication, we divide by 9:

45 / 9 = 5

So, before multiplying by 9, the number was 5. Here, we encounter the inverse operation of multiplication: division. This step further solidifies our understanding of how to reverse mathematical operations. Division effectively undoes the multiplication, allowing us to work backward and isolate the initial number. Applying this principle accurately is crucial for solving complex equations and understanding the relationships between different mathematical operations. This step reinforces the importance of knowing the inverse operations and applying them correctly to unravel the problem.

Step 4: Reverse the Fourth Operation

Before multiplying by 9, we divided by 7. To reverse division, we multiply by 7:

5 * 7 = 35

This means that before dividing by 7, the number was 35. We continue to reverse the operations, now dealing with the inverse of division, which is multiplication. By multiplying the result by 7, we effectively undo the division that occurred earlier in the problem. This step further illustrates the importance of understanding inverse operations and their role in simplifying complex calculations. It also highlights the systematic approach required to accurately reverse a series of mathematical operations, ensuring we move closer to finding the initial number.

Step 5: Reverse the First Operation

Finally, before dividing by 7, we subtracted 5. To reverse this, we add 5 back to the result from step 4 (which was 35):

35 + 5 = 40

Therefore, the initial number was 40. We've now arrived at the beginning of the problem by reversing all the operations. The final step involves undoing the initial subtraction by adding 5 to the intermediate result. This completes the process of systematically reversing each operation to find the initial number. It underscores the importance of understanding inverse operations and applying them accurately in a step-by-step manner. This method not only solves the problem but also reinforces fundamental mathematical principles and logical reasoning skills.

Verification

Let's check our answer by performing the original operations on 40:

  1. Subtract 5: 40 - 5 = 35
  2. Divide by 7: 35 / 7 = 5
  3. Multiply by 9: 5 * 9 = 45
  4. Add 5: 45 + 5 = 50
  5. Subtract 10: 50 - 10 = 40

The final result is indeed 40, so our initial number of 40 is correct! By verifying our solution, we confirm the accuracy of our step-by-step approach. This process reinforces the importance of checking our work and ensuring that the answer aligns with the original problem statement. Verification provides confidence in our solution and reinforces our understanding of the mathematical principles involved. It also highlights the value of systematic problem-solving and attention to detail.

Conclusion

The initial number is 40. By reversing each operation in the correct order, we successfully found the starting value. Remember, guys, always take it one step at a time and use inverse operations to solve these kinds of problems. Understanding and applying inverse operations is crucial for solving complex equations and understanding mathematical relationships. The step-by-step approach ensures accuracy and clarity in problem-solving. By verifying the solution, we confirm its correctness and reinforce our understanding of the mathematical principles involved. This method not only solves the specific problem but also enhances analytical and logical reasoning skills, which are valuable in mathematics and other fields.