Solving for Three Consecutive Integers

Solving for three consecutive integers is a classic problem in mathematics that often tests one's understanding of basic algebra and equations. In this article, we will walk through a detailed exploration of how to find three consecutive integers when given a specific condition related to their sum.

Understanding the Problem

The problem states: 'There are three consecutive integers, and half the sum of the first and third number is 28.' Let's denote the first integer as x, the second integer as x 1, and the third integer as x 2.

Setting Up the Equation

Given the condition half the sum of the first and third integers is 28, we can set up the following equation:

Equation: (frac{x x 2}{2} 28)

Let's simplify this equation step by step:

Step 1: Combine the x terms and the constant.

(frac{2x 2}{2} 28)

Step 2: Simplify the fraction by dividing both the numerator and the denominator by 2.

(x 1 28)

Step 3: Subtract 1 from both sides to solve for x.

(x 27)

Identifying the Integers

Now that we have determined that x 27, let's find the three consecutive integers:

The first integer: x 27 The second integer: x 1 28 The third integer: x 2 29 Thus, the three consecutive integers are 27, 28, and 29.

To confirm the solution, we can check if half the sum of the first and third integers equals 28:

Verification: First integer: 27 Third integer: 29 Sum: 27 29 56 Half the sum: 56 / 2 28 This confirms our solution is correct.

Conclusion

Through the process of setting up and solving the equation, we have identified the three consecutive integers as 27, 28, and 29. This problem exemplifies the application of algebraic equations in solving real-world problems, demonstrating the importance of basic mathematical skills in understanding and resolving various scenarios in daily life.