Finding Natural Number 'm': A Step-by-Step Guide

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Hey guys! Let's dive into this math problem together and figure out how to determine the natural number 'm' when we know that n + p = 4 and n + m * p = 392. It might sound a bit tricky at first, but don't worry, we'll break it down step by step so it's super easy to follow. We'll explore the problem's context, lay out the solution in a clear and understandable way, and highlight the key concepts involved. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we fully understand what the problem is asking. We're given two equations:

  1. n + p = 4
  2. n + m * p = 392

Our goal is to find the value of 'm', where 'm' is a natural number. Remember, natural numbers are positive whole numbers (1, 2, 3, and so on). This means we're looking for a positive, non-fractional value for 'm' that fits both equations. To solve this, we'll need to use a bit of algebraic manipulation and logical deduction. The key here is to use the information we have in the first equation to help us simplify the second equation and isolate 'm'. We'll look at how 'n' and 'p' relate to each other, and then see how 'm' fits into the bigger picture. Understanding the constraints (like 'm' being a natural number) will also help us narrow down our possibilities and find the correct solution.

Laying Out the Solution

Now that we've wrapped our heads around the problem, let's map out a plan to solve it. Here's the strategy we'll use:

  1. Isolate a Variable: We'll start by isolating one variable (either 'n' or 'p') in the first equation (n + p = 4). This will allow us to express one variable in terms of the other.
  2. Substitute: Next, we'll substitute the expression we found in step 1 into the second equation (n + m * p = 392). This will give us an equation with only two variables: 'm' and the remaining variable from step 1.
  3. Simplify: We'll then simplify the equation from step 2 to make it easier to work with. This might involve combining like terms or rearranging the equation.
  4. Solve for 'm': Our main goal! We'll manipulate the simplified equation to isolate 'm' on one side of the equation. This will give us an expression for 'm' in terms of the other variable.
  5. Use the Natural Number Condition: Since 'm' must be a natural number, we'll use this information to find possible values for the other variable. This will likely involve testing different values or using some logical deduction.
  6. Find 'm': Finally, we'll substitute the values we found in step 5 back into the expression for 'm' to find the solution. We'll also double-check that our solution works in both original equations.

This step-by-step approach will help us tackle the problem systematically and avoid getting lost in the algebra. Remember, the key is to break down the problem into smaller, manageable steps.

Step-by-Step Solution

Okay, let's put our plan into action and solve for 'm'!

  1. Isolate a Variable: Let's isolate 'n' in the first equation: n + p = 4. We can rewrite this as: n = 4 - p.

  2. Substitute: Now, we'll substitute this expression for 'n' into the second equation: n + m * p = 392. Replacing 'n' with (4 - p), we get: (4 - p) + m * p = 392.

  3. Simplify: Let's simplify the equation: 4 - p + m * p = 392. Subtract 4 from both sides: -p + m * p = 388. We can factor out 'p': p * (m - 1) = 388.

  4. Solve for 'm': We now have an equation relating 'p' and 'm': p * (m - 1) = 388. To find 'm', we need to consider the factors of 388. The factors of 388 are: 1, 2, 4, 97, 194, and 388.

  5. Use the Natural Number Condition: Since 'm' is a natural number, 'm - 1' must be an integer. Also, 'p' must be a natural number (because n + p = 4, and if n is a natural number, p must also be a natural number to make the sum 4). We can now consider the possible pairs of factors of 388 and see which ones work for our equation. Remember that p < 4 because n + p = 4 and n needs to be a natural number.

  6. Possible factor pairs of 388 are:

    • 1 x 388
    • 2 x 194
    • 4 x 97

    Let's analyze each pair:

    • If p = 1, then m - 1 = 388, so m = 389. This is a possible solution.
    • If p = 2, then m - 1 = 194, so m = 195. This is another possible solution.
    • If p = 4, then m - 1 = 97, so m = 98. This is also a possible solution.

  7. Find 'm': Let's go back to our equation n = 4 - p and check if our solutions are valid:

    • If p = 1, then n = 4 - 1 = 3. We have m = 389, n = 3, p = 1. Let's check the second equation: 3 + 389 * 1 = 392. This solution works!

    • If p = 2, then n = 4 - 2 = 2. We have m = 195, n = 2, p = 2. Let's check the second equation: 2 + 195 * 2 = 392. This solution also works!

    • If p = 4, then n = 4 - 4 = 0. 0 is not a natural number, so this is not a solution

Therefore, there are two valid solutions that satisfy this equation. The natural number m can be 389 if p is 1, or m can be 195 if p is 2.

Key Concepts and Considerations

Let's recap the key concepts we used to solve this problem. Understanding these concepts will help you tackle similar problems in the future.

  • Algebraic Manipulation: We used algebraic manipulation to isolate variables and simplify equations. This is a fundamental skill in solving mathematical problems.
  • Substitution: Substitution is a powerful technique where we replace one variable with an equivalent expression. This helps us reduce the number of variables in an equation.
  • Factoring: Factoring expressions allows us to rewrite them in a more convenient form, often revealing relationships between variables.
  • Natural Numbers: Understanding the properties of natural numbers (positive whole numbers) helped us narrow down the possible solutions.
  • Logical Deduction: We used logical deduction to analyze the factors of 388 and determine which pairs satisfied the given conditions.
  • Checking Solutions: Always remember to check your solutions in the original equations to ensure they are valid. This helps prevent errors.

Another important consideration is the context of the problem. The fact that 'm', 'n', and 'p' were natural numbers was crucial in finding the solution. If they were allowed to be other types of numbers (like fractions or negative numbers), the solution process would be different, and there might be more possible solutions.

Conclusion

So there you have it! We successfully determined the natural number 'm' by breaking down the problem into smaller steps, using algebraic manipulation, and applying logical deduction. Remember, the key to solving these types of problems is to understand the concepts involved and approach the problem systematically. By isolating variables, substituting expressions, and using the given conditions, we were able to find the solutions for 'm'. And hey, if you ever get stuck on a similar problem, just remember this guide and take it one step at a time! You got this!