Solving for Three Consecutive Integers That Sum to 12: A Comprehensive Mathematical Investigation

The problem of finding three consecutive integers that sum to 12 is a classic example of a mathematical puzzle that can be solved using algebraic techniques. This article provides a detailed exploration of the process, including multiple approaches to the solution.

The Problem and Setup

We are given the task of finding three consecutive integers such that their sum equals 12. Let's denote these integers as (x), (x 1), and (x 2). The equation representing the sum of these integers is:

(x (x 1) (x 2) 12)

Solution Approach 1: Simplifying the Equation

First, we simplify the equation by combining like terms:

(x x 1 x 2 12)

This simplifies to:

(3x 3 12)

We then proceed to isolate (x) by subtracting 3 from both sides of the equation:

(3x 9)

Finally, we divide both sides by 3 to solve for (x):

(x 3)

Therefore, the three consecutive integers are (3), (4), and (5).

Solution Approach 2: Direct Calculation

A more intuitive approach involves directly calculating the sum of three consecutive integers. One can quickly observe that (3 4 5 12). This approach, while faster, relies on understanding the properties of consecutive integers.

Solution Approach 3: Algebraic Equation Simplification

Let's use (x-1), (x), and (x 1) to represent the three consecutive integers. The equation representing their sum is:

((x-1) x (x 1) 12)

Again, simplifying this equation results in:

(3x 12)

Solving for (x) gives:

(x 4)

Thus, the integers are (3), (4), and (5).

Conclusion

The problem of finding three consecutive integers that sum to 12 can be solved using various algebraic techniques. Each approach provides a unique perspective on the problem, highlighting the elegance and simplicity of algebraic methods. Understanding these techniques not only helps in solving specific problems but also enhances general problem-solving skills.

Related Keywords

Consecutive Integers,
Sum of Integers,
Mathematical Problem Solving

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