Solving Integer Problems Finding The Right Equation

In the fascinating world of mathematics, problem-solving often involves translating word problems into algebraic equations. This article delves into a specific problem involving two positive integers, their distance on a number line, and their product. We will dissect the problem, explore different approaches, and identify the correct equation to solve for the greater integer. Let’s embark on this mathematical journey together.

Problem Statement: Cracking the Code

The problem presents a scenario with two positive integers that are 3 units apart on a number line. This immediately tells us that the difference between the two integers is 3. Additionally, we know that their product, the result of multiplying them together, is 108. The core question is: Which equation can be used to solve for m, the greater integer? We are given four options, each representing a different algebraic equation, and our task is to determine which one accurately reflects the problem's conditions.

Option A: $(m-12)(m-9)=108$

This equation suggests that the product of two expressions, $(m-12)$ and $(m-9)$, equals 108. However, directly applying the problem's conditions doesn't immediately lead to this form. The problem states that the two integers are 3 units apart, but this equation doesn't explicitly represent that relationship. While it's possible that a solution to this equation might coincidentally involve integers with a difference of 3, it's not a direct translation of the problem's information. To truly evaluate this option, we'd need to expand the equation and see if it aligns with the original problem's constraints.

Option B: $m(m-3)=108$

This equation appears to be a strong contender. It directly incorporates the information about the 3-unit difference. Here, m represents the greater integer, and $(m-3)$ would then represent the smaller integer. The equation states that the product of these two integers is 108, which perfectly aligns with the problem's statement. This equation seems to be a logical representation of the problem. Let's further analyze why this is a strong possibility. If we let the greater integer be m, the smaller integer, being 3 units apart, is indeed m-3. The problem explicitly states that their product is 108, thus the equation m( m-3) = 108 accurately captures this relationship. To strengthen our understanding, we could expand this equation to a quadratic form and then attempt to solve for m. This will either validate or invalidate it as the correct equation.

Option C: $(m+3)(m-3)=108$

This equation presents a different scenario. It involves the product of $(m+3)$ and $(m-3)$, which is a classic difference of squares pattern. Expanding this expression would result in $m^2 - 9$, and the equation would become $m^2 - 9 = 108$. While this equation might have a solution, it doesn't directly reflect the problem's initial condition that the two integers are 3 units apart and their product is 108. The structure of this equation leans more towards a scenario where we are dealing with the square of an integer and a constant difference. It is important to recognize that while mathematical relationships can often be expressed in multiple ways, the most direct translation of the word problem should be our focus. This equation, while mathematically valid in its own right, is a less direct representation of the problem's core conditions.

Option D: $m(m+3)=108$

This equation presents a contrasting relationship compared to option B. Here, m is multiplied by $(m+3)$. If m is the greater integer, then $(m+3)$ would be an even larger integer, which contradicts the problem's condition that the integers are only 3 units apart. This equation suggests that we are multiplying the greater integer by an integer that is 3 greater than itself, which doesn’t align with the problem’s description. The key is to understand that m represents the greater integer, and we need to find the smaller integer. If the integers are 3 units apart, the smaller integer must be m-3, not m+3. This subtle difference is crucial in correctly translating the word problem into an algebraic equation. Therefore, we can confidently rule out this option as it misrepresents the relationship between the two integers.

The Verdict: Decoding the Correct Equation

After careful analysis of each option, we can confidently identify option B, $m(m-3)=108$, as the correct equation. This equation accurately represents the problem's conditions: m is the greater integer, $(m-3)$ is the smaller integer (3 units less than m), and their product is 108. The other options either misrepresent the relationship between the integers or don't directly translate the given information.

To solidify our understanding, let's briefly solve the equation $m(m-3)=108$. Expanding this, we get $m^2 - 3m = 108$. Rearranging into a quadratic equation, we have $m^2 - 3m - 108 = 0$. This quadratic equation can be factored as $(m-12)(m+9) = 0$. This gives us two possible solutions for m: 12 and -9. Since the problem specifies positive integers, we discard -9. Therefore, m = 12. The smaller integer would then be 12 - 3 = 9. Indeed, 12 multiplied by 9 equals 108, confirming our solution.

Key Takeaways: Mastering the Art of Equation Formation

This problem highlights the importance of carefully translating word problems into algebraic equations. Here are some key takeaways:

• Identify the unknowns: Clearly define the variables you'll be using (in this case, m for the greater integer).
• Translate relationships: Pay close attention to the relationships described in the problem (the integers are 3 units apart, their product is 108) and represent them algebraically.
• Eliminate incorrect options: By understanding the relationships, you can often quickly eliminate options that misrepresent the problem's conditions.
• Verify the solution: Once you've identified a potential equation, solving it and checking the solution against the original problem is a crucial step.

By mastering these skills, you'll be well-equipped to tackle a wide range of mathematical word problems. Remember, the key is to break down the problem, translate the information into algebraic expressions, and carefully analyze the resulting equations.

Conclusion: The Power of Algebraic Representation

In this article, we successfully decoded an integer problem by translating its conditions into algebraic equations. We identified the correct equation, $m(m-3)=108$, and verified its solution. This exercise demonstrates the power of algebraic representation in solving mathematical puzzles. By understanding how to translate word problems into equations, we can unlock a world of mathematical problem-solving possibilities.

Which equation can be used to find the value of the greater integer, represented by m, given that two positive integers are 3 units apart on a number line and their product is 108?